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## Functions and Analysis

### Advanced Functions - Trigonometric Functions and Their Applications

#### Q.02

'Let \ \\sin \\theta=x \, then \ -1 \\leqq x \\leqq 1 \, and the equation is \ 1-2 x^{2}+2 k x+k-5=0 \ or \ 2 x^{2}-2 k x-k+4=0 \ The required condition is that the quadratic equation \\( (*) \\) has at least one real number solution in the range \ -1 \\leqq x \\leqq 1 \. Let \\( f(x)=2 x^{2}-2 k x-k+4 \\), and let the discriminant of \\( f(x)=0 \\) be \ D \. 1] The condition for both solutions to be in the range \ -1<x<1 \ is that the graph of \\( y=f(x) \\) intersects (including the case of tangency) with the portion of \ x \ axis between \ -1<x<1 \, and the following (i)〜(iv) simultaneously hold. (i) \ D \\geqq 0 \ (ii) \\( f(-1)>0 \\) (iii)\\( f(1)>0 \\) (iv) \ -1< \ axis \ <1 \'

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'Conditions for the existence of solutions to trigonometric equations'

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'CHECK 39 ⇒ Page 187 in this book. (1) \\sin 105^\\circ=\\sin \\left(60^\\circ+45^\\circ\\right)=\\sin 60^\\circ \\cos 45^\\circ+\\cos 60^\\circ \\sin 45^\\circ=\\frac{\\sqrt{3}}{2} \\cdot \\frac{1}{\\sqrt{2}}+\\frac{1}{2} \\cdot \\frac{1}{\\sqrt{2}}=\\frac{\\sqrt{6}+\\sqrt{2}}{4}\\cos 105^\\circ=\\cos \\left(60^\\circ+45^\\circ\\right)=\\cos 60^\\circ \\cos 45^\\circ-\\sin 60^\\circ \\sin 45^\\circ=\\frac{1}{2} \\cdot \\frac{1}{\\sqrt{2}}-\\frac{\\sqrt{3}}{2} \\cdot \\frac{1}{\\sqrt{2}}=\\frac{\\sqrt{2}-\\sqrt{6}}{4}\\tan 105^\\circ=\\tan \\left(60^\\circ+45^\\circ\\right)=\\frac{\\tan 60^\\circ+\\tan 45^\\circ}{1-\\tan 60^\\circ \\tan 45^\\circ}=\\frac{\\sqrt{3}+1}{1-\\sqrt{3} \\cdot 1}=\\frac{(\\sqrt{3}+1)^{2}}{1-3}=-2-\\sqrt{3}'

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'Find the maximum and minimum values of the following functions. Note that the range of θ is 0≤θ≤π. (1) y=sin 2θ+√3 cos 2θ (2) y=-4 sinθ+3 cosθ'

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'Express y = 4sin²θ - 4cosθ + 1 in terms of cosθ.'

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'(2) $\\cos \\theta + \\cos ^{2} \\theta = 1$ Therefore, from $1 - \\cos ^{2} \\theta = \\cos \\theta$, we have $\\sin ^{2} \\theta = \\cos \\theta$. \\[ \egin{array}{l} \\frac{\\sin ^{4} \\theta + \\cos ^{3} \\theta}{2 \\cos \\theta} = \\frac{\\left(\\sin ^{2} \\theta \\right)^{2} + \\cos ^{3} \\theta}{2 \\cos \\theta} = \\frac{\\cos ^{2} \\theta + \\cos ^{3} \\theta}{2 \\cos \\theta} \\\\ = \\frac{\\cos \\theta + \\cos ^{2} \\theta}{2} = \\frac{1}{2} \\end{array} \\]'

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'Let f(x)=x^{3}-3 x^{2}+2 x and g(x)=a x(x-2) (where a>1).'

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'(1) Find all the values of $x$ that satisfy the equation $\\sin 3 x=-\\sin x$.'

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'Find the number of real solutions of f(x)=x^{3}-3 x+1.'

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'(2) For a positive integer $n$, if $\\sin \\theta=\\frac{1}{n}$, then $\\cos \\theta= \\pm \\sqrt{1-\\sin^2 \\theta} = \\pm \\sqrt{1-\\left(\\frac{1}{n}\\right)^{2}} = \\pm \\frac{\\sqrt{n^2-1}}{n}$ (1). Thus, since $\\cos \\theta$ is rational, $\\sqrt{n^{2}-1}$ is also rational. Therefore, there exist coprime positive integers $p, q(p \\geqq 0, q>0)$ such that\n\ \\sqrt{n^{2}-1}=\\frac{p}{q} \\]\n Expanding both sides by squaring results in $n^{2}-1=\\frac{p^{2}}{q^{2}}$. Since $n^{2}-1$ is an integer, $\\frac{p^{2}}{q^{2}}$ is also an integer. Considering $p, q$ \overlinee coprime and $q$ is a positive integer, we get\n\\[----y=q \\]\n Therefore, $n^{2}-1=p^{2}$, which implies\n\\[ n^{2}-p^{2}=1 \\n Hence, $(n+p)(n-p)=1$, with $n+p$ being a positive integer and $n-p$ being an integer, we have $n+p=1, n-p=1$. Solving this system we get $\\quad n=1, p=0$. Therefore, if for a positive integer $n$, $\\sin \\theta=\\frac{1}{n}$, then $n=1$.'

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'Exercise Example 10 Trigonometric Functions and Chebyshev Polynomials'

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'Assume that the function f satisfies f((x+y)/2) ≤ (1/2){ f(x)+f(y)} for real numbers x, y. Prove that the function f satisfies f((x1+x2+...+xn)/n) ≤ (1/n){ f(x1)+f(x2)+...+f(xn)} for n real numbers x1, x2, ..., xn.'

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'Using radian measure, convert the following angles to radians.'

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'(2) 1 + tan^2 θ = 1/cos^2 θ leads to cos^2 θ = 1/(1+2^2) = 1/5 therefore cos θ = ±1/√5'

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'Calculate trigonometric functions based on the following conditions. (1) π<θ<2π, hence sin θ<0, thus sin^2 θ+cos^2 θ=1, so sin θ=-√(1-cos^2 θ)=-√(1-(12/13)^2)=-5/13 also, tan θ=sin θ/cos θ=(-5/13)÷(12/13)=-5/12'

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'(1) From sin3x = -sinx, we have 3sinx - 4sin^3x = -sinx, which simplifies to 4sinx(1+sinx)(1-sinx) = 0. Therefore, sinx = 0, ±1. Since 0 ≤ x ≤ 2π, we get x = 0, π/2, π, 3π/2, 2π.'

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'Question 2: \\\sin x+ \\sin 2 x+\\sin 3 x+\\sin 4 x = \\text{What}\'

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'Using radians, convert the following radians to degrees.'

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'Radians and Trigonometric Functions\nFind the arc length and area of a sector with radius r, and central angle θ radians.\nArc length: rθ\nArea: 12r^{2}θ'

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'Prove the definite integral properties of odd and even functions:'

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'Example 47 | Trigonometric Function Graphs (1)\\nDraw the graphs of the following functions.\\n(1) y=sin(θ-π/2)\\n(2) y=sinθ+1\\n(3) y=tan(θ+π/2)'

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'Example 98 | Trigonometric Equations and Inequalities (4)'

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"Let y = ax² + bx + c (a ≠ 0), then y' = 2ax + b, which leads to the equation of line as y - (aα² + bα + c) = (2aα + b)(x - α), i.e., y = (2aα + b)x - aα² + c. Similarly, the equation of another line is y = (2aβ + b)x - aβ² + c. The x-coordinate of the intersection point P is the solution to the following equation: (2aα + b)x - aα² + c = (2aβ + b)x - aβ² + c. Since a ≠ 0 and α ≠ β, x = a(β² - α²) / 2a(β - α) = (α + β) / 2."

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'Using the addition formula, find the following values.'

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'Define the trigonometric functions sin θ, cos θ, tan θ of a general angle θ on the coordinate plane.'

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'Exercise example 3 10 Trigonometric Functions and Chebyshev Polynomials (continued)'

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'Example 55 | Addition Formula of Tangents of Three Angles'

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'Solve problems involving trigonometric equations, trigonometric inequalities, and finding maximum and minimum values of trigonometric functions.'

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'Example 54 | Values of Trigonometric Functions (Addition Theorem)'

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'I came up with the idea of using coordinates to represent shapes in a plane.'

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'Find the maximum and minimum values of y=2sin ^{2}θ+3sinθcosθ+6cos ^{2}θ when 0≤θ<2π.'

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'Exercise Example 10 Trigonometric Functions and Chebyshev Polynomials (continued) To find the 5th degree polynomial of cos 5θ'

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'Example 97 | Trigonometric equation (using sum and product formulas)'

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'Prove the following trigonometric identity:\n\n(4) \\\cos 20^\\circ \\cos 40^\\circ \\cos 80^\\circ\'

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'I considered using an infinite number of trigonometric functions to represent a periodic function.'

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'(1) For any angle θ, plot the region in the xy-plane consisting of points (x, y) that satisfy -2≤xcosθ+ysinθ≤y+1, and determine its area. (2) For any angles α, β, plot the region in the xy-plane consisting of points (x, y) that satisfy -1≤x²cosα+ysinβ≤1, and determine its area. [Hitotsubashi University]'

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'Investigate the maximum and minimum of trigonometric functions in the given equation, and solve the problems including applications to geometry.'

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'Trigonometric functions and Chebyshev polynomials'

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'(3) From $\\sin \\theta +\\sin ^{2} \\theta=1$, we have $1-\\sin ^{2} \\theta=\\sin \\theta$, therefore $\\cos ^{2} \\theta=\\sin \\theta$. Substituting into the equation $\\cos ^{2} \\theta+2 \\cos ^{4} \\theta=\\cos ^{2} \\theta+2(\\cos ^{2} \\theta)^{2}=\\sin \\theta+2 \\sin ^{2} \\theta=\\sin \\theta+2(1-\\sin \\theta)=2-\\sin \\theta \\ldots 1$. From $\\sin \\theta+\\sin ^{2} \\theta=1$, we have $\\sin ^{2} \\theta+\\sin \\theta-1=0$, solving gives $\\sin \\theta=\\frac{-1 \\pm \\sqrt{5}}{2}$. Since $-1 \\leq \\sin \\theta \\leq 1$, we get $\\sin \\theta=\\frac{-1+\\sqrt{5}}{2}$ (1), substituting back gives $\\cos ^{2} \\theta+2 \\cos ^{4} \\theta=2-\\frac{-1+\\sqrt{5}}{2}=\\frac{5-\\sqrt{5}}{2}$.'

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'What does it mean to solve a math problem, similar to navigating the ocean?'

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'(2) \\sin 15 ^ {\\circ} = \\sin \\left(60 ^ {\\circ} -45 ^ {\\circ} \\right) = \\sin 60 ^ {\\circ} \\cos 45 ^ {\\circ} - \\cos 60 ^ {\\circ} \\sin 45 ^ {\\circ} = \\frac{\\sqrt{3}}{2} \\cdot \\frac{1}{\\sqrt{2}} - \\frac{1}{2} \\cdot \\frac{1}{\\sqrt{2}} = \\frac{\\sqrt{6}-\\sqrt{2}}{4} \\cos 15 ^ {\\circ} = \\cos \\left(60 ^ {\\circ} -45 ^ {\\circ} \\right) = \\cos 60 ^ {\\circ} \\cos 45 ^ {\\circ} + \\sin 60 ^ {\\circ} \\sin 45 ^ {\\circ} = \\frac{1}{2} \\cdot \\frac{1}{\\sqrt{2}} + \\frac{\\sqrt{3}}{2} \\cdot \\frac{1}{\\sqrt{2}} = \\frac{\\sqrt{6}+\\sqrt{2}}{4} \\tan 15 ^ {\\circ} = \\tan \\left(60 ^ {\\circ} -45 ^ {\\circ} \\right) = \\frac{\\tan 60 ^ {\\circ} - \\tan 45 ^ {\\circ}}{1+\\tan 60 ^ {\\circ} \\tan 45 ^ {\\circ}} = \\frac{\\sqrt{3}-1}{1+\\sqrt{3} \\cdot 1} = \\frac{(\\sqrt{3}-1)^{2}}{\\sqrt{3}+1)(\\sqrt{3}-1)} = \\frac{3-2\\sqrt{3}+1}{3-1} = 2-\\sqrt{3}'

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'Exercise example 10 Trigonometric functions and Chebyshev polynomials (continued)'

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'Example 50 => Page 180\n(1) is the graph of y=cosθ translated symmetrically about the θ axis. The graph is shown on the right. Also, the period is 2π.'

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'Let the angle formed between the two straight lines and the positive direction of the x-axis be α and β respectively. The acute angle θ we seek is tanα=√3/2, tanβ=-3√3. Therefore, tanθ=tan(β-α)=(-3√3-√3/2)÷{1+(-3√3)∙√3/2}=√3. Since 0<θ<π/2, then θ=π/3'

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'124\n—Mathematics II\n(2) Left side = \\ frac { \\ cos \\ theta(1- \\ sin \\ theta) + \\ cos \\ theta(1+ \\ sin \\ theta)}{(1+ \\ sin \\ theta)(1- \\ sin \\ theta)}= \\ frac {2 \\ cos \\ theta}{1- \\ sin ^{2} \\ theta} \\ frac {2 \\ cos \\ theta}{ \\ cos ^{2} \\ theta}= \\ frac {2}{ \\ cos \\ theta} Therefore, \\ frac { \\ cos \\ theta}{1+ \\ sin \\ theta}+ \\ frac { \\ cos \\ theta}{1- \\ sin \\ theta}= \\ frac {2}{ \\ cos \\ theta}'

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'(1) f(θ)=\\frac{1}{2} \\sin θ=\\frac{1}{2} \\sin (θ+2 \\pi)=f(θ+2 \\pi)\nTherefore, the fundamental period is 2 \\pi\n(2) f(θ)=\\cos (-2 θ)=\\cos (-2 θ-2 \\pi)=\\cos \\{-2(θ+ \\pi)\\}=f(θ+\\pi)\nTherefore, the fundamental period is \\pi'

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'(4) \\[ \egin{aligned} \\sin x+\\sin 2 x+\\sin 3 x & =(\\sin 3 x+\\sin x)+\\sin 2 x \\\\ & =2 \\sin 2 x \\cos x+\\sin 2 x \\\\ & =\\sin 2 x(2 \\cos x+1) \\\\ \\cos x+\\cos 2 x+\\cos 3 x & =(\\cos 3 x+\\cos x)+\\cos 2 x \\\\ & =2 \\cos 2 x \\cos x+\\cos 2 x \\\\ & =\\cos 2 x(2 \\cos x+1) \\end{aligned} \\]'

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'Translate the given text into multiple languages.'

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'Question 145 Conditions for a function to have extremum in a range'

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'Given the equation \\[ \egin{array}{l} 2 \\cdot 2 \\sin \\theta \\cos \\theta-2 \\sin \\theta+2 \\sqrt{3} \\cos \\theta-\\sqrt{3}=0 \\\\ 2 \\sin \\theta(2 \\cos \\theta-1)+\\sqrt{3}(2 \\cos \\theta-1)=0 \\end{array} \\] Therefore, \\( (2 \\sin \\theta+\\sqrt{3})(2 \\cos \\theta-1)=0 \\) which implies \ \\sin \\theta=-\\frac{\\sqrt{3}}{2}, \\cos \\theta=\\frac{1}{2} \ Considering \ 0 \\leqq \\theta<2 \\pi \, from \ \\sin \\theta=-\\frac{\\sqrt{3}}{2} \ we get \ \\theta=\\frac{4}{3} \\pi, \\frac{5}{3} \\pi \ and from \ \\cos \\theta=\\frac{1}{2} \ we get \ \\theta=\\frac{\\pi}{3}, \\frac{5}{3} \\pi \\] Therefore, the solutions \overlinee \\[ \\theta=\\frac{\\pi}{3}, \\frac{4}{3} \\pi, \\frac{5}{3} \\pi \'

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'An inequality that involves trigonometric functions is called a trigonometric inequality, and solving a trigonometric inequality involves finding the range of angles (solution) that satisfy the inequality.'

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'The graph is a vertical shrink by half of the function y=tanθ. The graph on the right is the shrunken version. The period is π and the asymptote is the line θ=π/2+nπ (n is an integer).'

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'Using sum and double angle formulas, prove the following equations (3 times angle formula).'

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'Find the values of θ that satisfy the following equations for 0≤θ<2π:'

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'Solve the following equations and inequalities for \0 \\leqq \\theta<2 \\pi\.'

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'Explain the definitions of the trigonometric functions sin, cos, and tan.'

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'Prove the following trigonometric relationships based on the definition of -sin(θ): (i) tan(θ) = sin(θ) / cos(θ) (ii) sin^2(θ) + cos^2(θ) = 1 (iii) 1 + tan^2(θ) = 1 / cos^2(θ)'

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'Consider the function y=sin x-cos 2 x(0 ≤ x <2π).'

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'How to memorize the addition formula, double angle, and half angle formulas'

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'Master the trigonometric equations and conquer example 123!'

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'(1) \ \\cos \\theta=\\frac{12}{13} \\quad \ [Quadrant 4 \ ] \\n(2) \ \\tan \\theta=2 \\sqrt{2} \\quad \ [Quadrant 3]'

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'Example 5: Practical maximum and minimum of trigonometric functions'

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'Given α is an angle in the second quadrant with sinα=3/5, and β is an angle in the third quadrant with cosβ=-4/5, find the values of sin(α-β) and cos(α-β).'

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'Prove the equation \\\frac{\\sin \\alpha+\\sin 2 \\alpha}{1+\\cos \\alpha+\\cos 2 \\alpha}=\\tan \\alpha\.'

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'Master the addition principle and conquer example 130!'

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"\2\\sin x=t\, let's substitute this in. Hence, \0 \\leq x < 2 \\pi\, so \-1 \\leq t \\leq 1\. Furthermore, from equation (1), we have\\n\\ny = 2 t^2 + t - 1 = 2 (t^2 + \\frac{1}{2}t) - 1 = 2 (t + \\frac{1}{4})^2 - 2 (\\frac{1}{4})^2 - 1 = 2 (t + \\frac{1}{4})^2 - \\frac{9}{8}\\n\ =t\. Consider the range of \t\. Convert the quadratic equation to standard form. Therefore, \y\ takes the maximum value of 2 when \t=1\ and the minimum value of \-\\frac{9}{8}\ when \t=-\\frac{1}{4}\."

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'Solve the following equations and inequalities for 0≤θ<2π. (1) sin(2θ-π/3) = √3/2 (2) sin(2θ-π/3) < √3/2'

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'Equations that hold true for trigonometric functions, where n is an integer.'

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'Find the maximum and minimum values of the following functions.'

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'Maximum and minimum of trigonometric functions (using t=sinθ+cosθ)'

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'Solve the following equations and inequalities for 0 ≤ θ < 2π.'

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'Find the maximum and minimum values of the functions and their corresponding θ values. (1) y=sin ^{2}θ+cosθ+1 (0≤θ<2π) (2) y=3sin^{2}θ-4sinθcosθ-1/cos^{2}θ (0≤θ≤π/3)'

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'Trigonometric Functions Graph (3) ... Scaling and Translation'

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'Find the maximum value, minimum value, and the corresponding values of θ of the function y=7sin^2θ-4sinθcosθ+3cos^2θ(0 ≤ θ ≤ π/2).'

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'Equations and inequalities involving trigonometric functions (substitution)'

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'Explain the extension from trigonometric ratios to trigonometric functions, and provide the definitions of sine θ, cosine θ, tangent θ for a general angle θ.'

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'Find the angle formed by two lines using the addition formula of tangent (tan)'

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'The figure above shows the graphs of (1) y=a sin bθ and (2) y=a cos bθ. Find the values of constants a and b. Note that a>0, b>0.'

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'Double angle and half angle formulas along with trigonometric values'

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'Find the maximum and minimum values of the function y = 3sinθ-2sin³θ (0 ≤ θ ≤ 7/6π), and the corresponding values of θ.'

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'Find the values of theta that satisfy the following equations.'

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'Find the values of θ that satisfy the following equations for 0 ≤ θ < 2π.'

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'Equations and inequalities involving trigonometric functions (using composition)'

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'By scaling the graph of y = cos^2 θ by a factor of 2 in the y-axis direction based on the line y = 1, we obtain a graph that is obtained by translating the graph of y = cos^2 θ downwards by 1 unit in the y-axis, then scaling vertically by a factor of 2 relative to the θ-axis, and further translating downwards by 1 unit in the y-axis, hence the equation is y = a(cos^2 θ - b) + 1. Find the option that matches the graph.'

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'Using the three addition formulas with β=α: (1) Calculate the following using the formulas: (a) sin 2α (b) Provide another expression for cos 2α: cos^2α - sin^2α, 2 cos^2α - 1, 1 - 2 sin^2α (c) tan 2α (2) Replace all values with θ/2 and calculate: (a) sin^2(θ/2) (b) cos^2(θ/2) (c) tan^2(θ/2)'

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'Find the maximum and minimum values of the function y=√3sinθ-cosθ (0≤θ<2π) and their corresponding values of θ. Also, plot the graph of the function.'

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'Maximum and minimum of trigonometric functions (utilizing composition)'

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'Equation involving trigonometric functions (using sin^2θ + cos^2θ = 1)'

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'The angle sizes of trigonometric functions learned so far, such as \ \\sin \\theta, \\cos \\theta \, were represented using units of degrees like \ 30^{\\circ}, 360^{\\circ} \. This is known as the degree system where 1 degree is equal to \ \\frac{1}{90} \ of a right angle.'

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'In trigonometry, there are formulas to transform the product of sine and cosine into sum and difference, and vice versa.'

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'System of inequalities involving trigonometric functions'

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'Derive the expression after dividing 3 sin² θ - 4 sin θ cos θ - 1 by cos² θ, and find the maximum and minimum values in the range 0 ≤ θ ≤ π/3.'

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'For the function f(x) = sin(2x) − 2 sin(x) − 2 cos(x) + 1 (0 ≤ x ≤ π)'

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'When x > 1, since 4(x²-1) > 0 and 1/(x²-1) > 0, we can conclude the following inequality based on the arithmetic mean being greater than or equal to the geometric mean. 4(x²-1)+1/(x²-1)+4 ≥ 2√(4(x²-1)・1/(x²-1))+4 = 8. Therefore, 4x² +1/((x+1)(x-1)) ≥ 8, with equality holding when 4(x²-1)=1/(x²-1). In this case, (x²-1)²=1/4. Since x > 1, x²-1=1/2, which means x²=3/2, so x=√(3/2)=√6/2. Hence, the minimum value of 4x² + 1/((x+1)(x-1)) is 8, with x equal to 2√(3/2) = √(6)/2.'

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'Solve the following inequalities for 0 ≤ θ < 2π.'

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'Basic Example 124 Solve the following equation for 0 ≤ θ < 2π: 2sin²θ + cosθ - 1 = 0'

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'Using the addition formula, find the following values.'

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'If the function $f(x)=ax^{3}+bx^{2}+cx+d$ has a maximum value of 0 at $x=1$ and the graph of the curve $y=f(x)$ looks like the figure on the right,'

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'(1) \\( \\cos \\left(\\theta+\\frac{\\pi}{4}\\right)=-\\frac{\\sqrt{3}}{2} \\)\\n(2) \2 \\sin 2 \\theta>\\sqrt{3} \'

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'Graph of trigonometric functions and translation/scaling'

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'Find the maximum and minimum values of the following functions and the corresponding values of θ.'

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'System of inequalities involving trigonometric functions'

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'Which of the following graphs does not match the graph of !ν within the range of 0 to π? The answer choices are: (0) y = sin(2θ + π/2) (1) y = sin(2θ - π/2) (2) y = cos{2(θ + π)} (3) y = cos{2(θ - π)}'

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'Maximum and minimum of trigonometric functions (reducing to quadratic functions)'

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'Basics of radians, arc length and area of a sector'

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'Find the maximum and minimum values of the function y=3sinθ+4cosθ.'

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'When 0 ≤ θ ≤ π and sinθ+cosθ=√3/2, find the value of the following expression.'

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'Inequality involving trigonometric functions (using sin^2θ + cos^2θ = 1)'

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'Calculate the area enclosed by the curve y=|x^2-1| and the line y=3.'

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'Prove that \ \\sin 3 \\alpha = 3 \\sin \\alpha - 4 \\sin ^{3} \\alpha \.'

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'It is said that feeling that studying is enjoyable is important, but why does this mindset affect memory?'

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'Explain the difference between physical change and chemical change.'

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'(1) In the example above, calculate the magnitude of acceleration of point P.'

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'Find the polar equation of the curve \\( \\left(x^{2}+y^{2}\\right)^{3}=4 x^{2} y^{2} \\). Also, sketch the general shape of this curve, considering the origin \ \\mathrm{O} \ as the pole and the positive part of the \ x \-axis as the initial line.'

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'Please describe the characteristics of the graph of y=√(ax) (where a ≠ 0).'

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'Points to consider when sketching the outline of a function graph'

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'\\[\\left(\\sin ^{-1} x\\right)^{\\prime}=\\frac{1}{\\sqrt{1-x^{2}}}(-1<x<1)\\]'

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'Given a>0, let f(x)=\\sqrt{a x-2}-1 (x \\geqq \\frac{2}{a}) be the function. Find the range of values of a when the graph of the function y=f(x) and its inverse function y=f^{-1}(x) share two distinct points.'

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'Key points in substitution method of definite integration'

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'Find the equation of the curve C2 obtained by rotating the curve C1: 3x^2+2\\sqrt{3}xy+5y^2=24 counterclockwise by π/6 radians around the origin.'

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'Find the change in the value of the function, maximum and minimum values, and graph of the function\n(3) Let f(x)=sin(π cos x).\n(1) Find the value of f(π + x) - f(π - x).\n(2) Find the value of f(π / 2 + x) + f(π / 2 - x).\n(3) Draw the graph of y=f(x) in the range 0 ≤ x ≤ 2π (no need to check concavity).\n[Similar to Tokyo University of Science]'

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'Consider the change in the values of the function, maximum and minimum, curve C: {x=sin(θ) cos(θ), y=sin^3(θ) + cos^3(θ)} (-π / 4 ≤ θ ≤ π / 4).'

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'Why is it possible to calculate the definite integrals $\\int_{0}^{\\sqrt{2}} \\frac{d x}{x^{2}+2}$ and $\\int_{0}^{\\frac{a}{2}} \\sqrt{a^{2}-x^{2}} d x$ successfully by substituting $x=\\sqrt{2} \\tan \\theta$ and $x=a \\sin \\theta$?'

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'Find the asymptotes of the function y = x + 1 + 1 / (x - 1).'

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'Find the volume V of the solid obtained by rotating the area enclosed by the curve x=tanθ, y=cos2θ (-π/2<θ<π/2) and the x-axis around the x-axis once.'

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"Using Euler's formula, express trigonometric functions as exponential functions and derive the following equations."

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'Express the curves represented by the following polar equations in rectangular coordinates.'

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'\\(\\left(\\cos ^{-1} x\\right)^{\\prime}=-\\frac{1}{\\sqrt{1-x^{2}}}(-1<x<1)\\)'

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'Basic 2: Translation and Determination of Fraction Functions'

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'When the coordinates of point P moving on the coordinate plane at time t are given as x=4cos(t), y=sin(2t), find the magnitudes of the velocity and acceleration of point P at t=π/3.'

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'When the graph of the function $y=\x0crac{2 x+c}{a x+b}$ passes through the point $(-2, \x0crac{9}{5})$ and has the two lines $x=-\x0crac{1}{3}$, $y=\x0crac{2}{3}$ as asymptotes, find the values of the constants $a, b, c$.'

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'Prove that for a point P(x, y) moving on the circumference of an ellipse A x^{2}+B y^{2}=1 (A>0, B>0) with speed 1, the following statements hold true.'

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'When the coordinates of a point P moving on a coordinate plane at time t are given by the following expressions, find the magnitude of the velocity and acceleration of point P.'

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"When the coordinates (x, y) of the moving point P on the coordinate plane at time t are represented as {x=sin t y=12 cos 2 t}, find the maximum value of the magnitude of P's velocity."

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'Prove that the inequality \ b \\sin \\frac{a}{2}>a \\sin \\frac{b}{2} \ holds when \ 0<a<b<2\\pi \.'

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'When the point P moves along the number line, its coordinate x as a function of time t is given by x=2cos(πt+π/6), find the velocity v and acceleration α at t=2/3.'

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'On the coordinate plane with the origin O, consider the curve $C: \\frac{x^{2}}{4}+y^{2}=1$ where point P(1, $\\frac{\\sqrt{3}}{2}$) is taken.'

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'Prove that the equation f(x)=x^{2} has at least 2 real solutions in the range 0<x<2 when the function f(x) is continuous and f(0)=-1, f(1)=2, f(2)=3.'

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'(1) \ \\sin 175^{\\circ} < \\sin 35^{\\circ} < \\sin 140^{\\circ} \'

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'Let 0° ≤ θ ≤ 180°. Solve the following equation.'

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'A certain parabola was moved parallel to the x-axis by 1 unit and parallel to the y-axis by -2 units, then symmetrically moved with respect to the x-axis, resulting in the equation of the parabola as y=-x^2-3x+3. Find the original equation of the parabola.'

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'Find the sine, cosine, and tangent of the following angles.'

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'Using the table of trigonometric ratios at the end, find the following values of θ.'

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'Chapter 4 Geometry Measurement 163 EX \ \\quad 0^{\\circ} \\leqq \\theta \\leqq 180^{\\circ} \, when \ y=\\sin ^{4} \\theta+\\cos ^{4} \\theta \, let \ \\sin ^{2} \\theta=t \'

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'In triangle ABC, if sin A: sin B: sin C = 5: 16: 19, find the measure of the largest angle in this triangle.'

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'Basics of Trigonometry: Find the trigonometric ratios for a specific angle θ.'

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'Find the quadratic functions represented by the following graphs.'

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'Express the following trigonometric functions in terms of angles between 0 degrees and 90 degrees. Also, find their values using the trigonometric table at the end.'

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'By using the law of cosines, we find the value of a.'

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'Find the equation of a parabola that satisfies the following conditions: Condition: The equation of the parabola is y = 2x^{2} + ax + b. The parabola obtained by shifting this parabola 2 units in the x-axis direction and -3 units in the y-axis direction coincides with the equation y = 2x^{2}.'

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'In triangle ABC, if sinA: sinB: sinC = 3: 5: 7, find the ratio of cosA: cosB: cosC. (Tohoku Gakuin University)'

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'Calculate the trigonometric functions and show the results.'

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'(1) \\sin 111^{\\circ}\\n(2) \\cos 155^{\\circ}\\n(3) \\tan 173^{\\circ}'

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'For 0° ≤ θ ≤ 180°, find the range of θ that satisfies the following inequalities.'

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'In triangle ABC, if sin A: sin B: sin C = 5: 7: 8, then cos C = __.'

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'From 2sinθ = sqrt(2) to sinθ = 1 / sqrt(2). The points P and Q on the semicircle with radius 1, where the y-coordinate is 1 / sqrt(2), are the points to consider. The required θ is ∠AOP and ∠AOQ.'

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'Extension of trigonometric ratios: Find the trigonometric ratios when the angle is in the range of 0° to 360°.'

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'(4) Solve the equation. Given 0 ≤ θ ≤ 180°. Solve the equation: √2 sinθ = tanθ'

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'In triangle ABC, if sin A:sin B:sin C = 3:5:7, find the ratio of cos A:cos B:cos C.'

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'Explain the definition and relationships of trigonometric ratios. (1) Definition of trigonometric ratios (2) Relationships of trigonometric ratios (3) Trigonometric ratios in special angles'

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'When would you use the Extended Examples and Exercises page?'

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'In examples analyzing the motion of function graphs and geometric shapes, what digital content can be used to connect visual images with mathematical equations for learning purposes?'

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'In order, \\\\( \\cos 20^{\\circ}, \\\\ \\sin 10^{\\circ}, \\\\ \\frac{1}{\\tan 35^{\\circ}} \\\\\\\n'

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'Explain the relationship between necessary and sufficient conditions.'

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'Supplement for sine, cosine, and tangent of 0°, 90°, and 180°\n\nWhen θ=0°, in the definition formula of trigonometric ratios with r=1 and point P₀ with coordinates (1,0),\nsin 0°=0, \ncos 0°=1, \ntan 0°=0\n\nWhen θ=90°, in the definition formula of trigonometric ratios with r=1 and point P₁ with coordinates (0,1),\nsin 90°=1, \ncos 90°=0\n\nWhen θ=180°, in the definition formula of trigonometric ratios with r=1 and point P₂ with coordinates (-1,0),\nsin 180°=0, \ncos 180°=-1, \ntan 180°=0'

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'Find the following.\n(1) Values of \ \\sin 15^{\\circ}, \\cos 73^{\\circ}, \\tan 25^{\\circ} \\n(2) Acute angles \ \\alpha, \eta, \\gamma \ that satisfy \ \\sin \\alpha=0.4226, \\cos \eta=0.7314 \, and \ \\tan \\gamma=8.1443 \\n(3) Approximate value of \ x \ and angle \ \\theta \ in the right figure. Round \ x \ to two decimal places.'

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'θ is the mutual relationship of trigonometric ratios from 0° to 180°'

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'Find the value of cosine from the sine ratio formula'

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'The desired solution is that since the graph of the function y=|x^2-6x-7| either intersects or entirely lies above the graph of y=2x+2,'

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"Using De Morgan's Law, please provide a specific example with sets A, B, and C."

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'Let θ be such that 0° ≤ θ ≤ 180°. If sin θ = 1/3, find the values of cos θ and tan θ.'

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'In triangle ABC, if sinA/sqrt(3)=sinB/sqrt(7)=sinC holds true, find the measure of the largest angle.'

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'(1) Using the trigonometric table, find the values of sine, cosine, and tangent for 128°.\n(2) Let sin 27° = a. Express the cosine of 117° in terms of a.'

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'θ (trigonometric equation) that satisfies the trigonometric identity'

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'Prove that for a triangle ABC with angles A, B, and C, denoted as A, B, and C, the following equations hold true.'

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'Trigonometric relationships when θ is an acute angle'

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'The following two equations are also valid. \ \egin{\overlineray}{l} b^{2}=c^{2}+a^{2}-2 c a \\cos B \\\\ c^{2}=a^{2}+b^{2}-2 a b \\cos C \\ \\end{\overlineray} \\] Summarizing this as the cosine rule: \\[ \egin{\overlineray}{l} a^{2}=b^{2}+c^{2}-2 b c \\cos A \\\\ b^{2}=c^{2}+a^{2}-2 c a \\cos B \\\\ c^{2}=a^{2}+b^{2}-2 a b \\cos C \\ \\end{\overlineray} \\] Prove the following equalities in triangle ABC from the cosine rule. \\[ \\cos A = \\frac{b^{2}+c^{2}-a^{2}}{2 b c} , \\quad \\cos B = \\frac{c^{2}+a^{2}-b^{2}}{2 c a}, \\quad \\cosC = \\frac{a^{2}+b^{2}-c^{2}}{2 a b} \'

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'Prove the following equations hold for the interior angles A, B, C of triangle ABC:'

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"Let's review the sine theorem and cosine theorem!"

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'Find the values of trigonometric functions for obtuse angles'

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'Solve the equations: sin aθ = sin bθ, sin aθ = cos bθ'

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'Since $\\ frac { \\ pi} {6} <1 < \\ frac { \\ pi} {3}, \\ frac { \\ pi} {2} <2 < \\ frac {2} {3} \\ pi, \\ frac {5} {6} \\ pi <3 < \\ pi$, so $\\ frac {1} {2} < \\ sin 1 < \\ frac {\\ sqrt {3}} {2}, \\ frac {\\ sqrt {3}} {2} < \\ sin 2 <1,0 < \\ sin 3 < \\ frac {1} {2}$. Therefore, the minimum positive value in $\\ sin 1, \\ sin 2, \\ sin 3, \\ sin 4$ is $\\ sin 3$, and the maximum value is $\\ sin 2$.'

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'(1) Find the value(s) of theta that satisfy the equation $\\cos ^{2} \\theta+\\sqrt{3} \\sin \\theta \\cos \\theta=1$ under the condition $0 \\leq \\theta <2 \\pi$.'

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"Let's consider the relationship between the movement (trajectory) of the coffee cups in an amusement park and trigonometric functions. When disk 1 completes one full clockwise rotation, while disk 2 with half the radius completes two anticlockwise rotations, what kind of trajectory does point C on disk 2 trace?"

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'Solve the following equations and inequalities for 0 ≤ θ < 2π. (1) cos 2θ = √3 cosθ + 2 (2) sin 2θ < sinθ'

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'How to draw a graph of a cubic function - creating a table of increasing and decreasing'

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'Maximum and minimum of trigonometric functions (1)'

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'For the function y=sin 2x(sin x+cos x-1), let t=sin x+cos x, express the range of y in terms of the range of t.'

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'Prove the equation 1 + sin θ - cos θ / 1 + sin θ + cos θ = tan(θ/2).'

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'Using the addition formula, find the following values.'

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'Find the maximum and minimum values of the 1372nd homogeneous equation 𝑓(𝜃)=sin^{2}𝜃+sin𝜃cos𝜃+2cos^{2}𝜃 (0≤𝜃≤𝜋/2).'

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'Translate the given text into multiple languages.'

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'Find the range of constant k for which the curve y=x^3-2x+1 and the line y=x+k share 3 distinct points.'

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'Basic Example 134 Solution of Triangular Equations and Inequalities (Composite)\nSolve the following equations and inequalities when 0 ≤ θ < 2π:\n(1) sin θ-√3 cos θ=-1\n(2) sin θ- cos θ<1\nBasics 123,133'

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'The sum and difference of two angles α and β, represented in terms of the trigonometric functions of α and β, are known as the trigonometric addition formula.'

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'Solve the following equation or inequality when 0 ≤ θ < 2π. 2) sin 2θ + sin θ - cos θ > 1/2'

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'Find the values of $a$ and $b$ such that the maximum value of the function $f(\\theta)=a \\cos ^{2} \\theta+(a-b) \\sin \\theta \\cos \\theta+b \\sin ^{2} \\theta$ is $3+\\sqrt{7}$ and the minimum value is $3-\\sqrt{7}$.'

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'Why is the graph of y=sinθ in Example 118(3) not scaled by a factor of 1/2 in the θ direction?'

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'Calculate the values of the following trigonometric functions.'

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'For the function \ y=\\sin 2 \\theta+\\sin \\theta+\\cos \\theta \:'

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'Solve the following equations and inequalities when $0 \\leq \\theta < 2 \\pi$:\n1) $2 \\sin 2\\theta = \\tan \\theta + \\frac{1}{\\cos \\theta}$\n2) $\\sin 2\\theta + \\sin \\theta - \\cos \\theta > \\frac{1}{2}$'

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'The following are the graphs of functions (1) and (2). Calculate the values from A to H. (1) y=sin θ (2) y=cos θ'

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'Plot the graphs of the following functions and determine their periods:'

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'Chapter 7 Integral Calculus\nLet the parabola y=\\frac{1}{2}x^{2} be denoted as C, and let point P(a,\\frac{1}{2}a^{2}) lie on C. Here, a>0. Consider point P\nand let l be the tangent to C, and Q be the intersection of l with the x-axis. Also, let m be the line passing through point Q and perpendicular to l. Answer the following questions:\n(1) Find the equations of lines l and m.\n(2) Let the intersection of line m with the y-axis be A. Define the area of triangle APQ as S. Furthermore, define the area enclosed by the y-axis, line segment AP, and curve C as T. Determine the minimum value of S-T and the corresponding value of a.'

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'Among sin 1, sin 2, sin 3, sin 4, the negative value is A. The minimum value of the positive values is B, and the maximum value is C.'

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'Find the maximum value, minimum value, and the corresponding values of θ of the function f(θ) = 8sin^3θ - 3cos2θ - 12sinθ + 7 defined for 0 ≤ θ ≤ 2π. [Tokyo University of Science]'

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'Let a>1 be 190° practice. For the function y=2x^{3}-9x^{2}+12x where 1≤x≤a, (1) find the minimum value. (2) find the maximum value.'

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'In the $x y$ plane, the curve $y=f(x)$ always passes through two fixed points regardless of the value of $a$. What are the coordinates of these two fixed points? Determine the range of $a$ values for which $f(x)$ does not have extremum.'

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'Please explain the periodicity of trigonometric functions.'

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'For the interior angles A, B, and C of a triangle ABC with angles of 120 degrees, answer the following questions:'

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'Find the sine, cosine, and tangent values of 195 degrees.'

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'Find the general term of the sequence {an} defined by the following conditions using the substitutions in the parentheses.'

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'Calculate the values of the following trigonometric functions.'

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'Prove the formulas of product to sum and sum to product'

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'Find the maximum and minimum values of the function y=2sinθ+2cos²θ-1 (-π/2 ≤ θ ≤ π/2), and the values of θ that give the maximum and minimum values.'

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'Find the maximum and minimum values of the following functions. Also, determine the values of θ at those points.\n(1) y = cos θ - sin θ (0 ≤ θ < 2π)\n(2) y = √3 sin θ - cos θ (π ≤ θ < 2π)'

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'Find the maximum and minimum values of the given functions. Also, determine the values of θ at that time.'

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'Using the half-angle formula, find the following values. (1) $\\sin \\frac{3}{8} \\pi$ (2) $\\cos \\frac{3}{8} \\pi$ (3) $\\tan \\frac{3}{8} \\pi$'

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"Let's think about the solution method for trigonometric equations and inequalities (quadratic equations). There is a way to solve trigonometric equations and inequalities that involve multiple trigonometric functions, like in basic example 124."

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'Solve the following equations and inequalities for 0 ≤ θ < 2π. (1) cos 2θ - 3cosθ + 2 = 0 (2) sin 2θ > cosθ'

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'(2) \ \\sin \\theta=\\frac{\\sqrt{6} \\pm \\sqrt{2}}{4} \,\\n\ \\cos \\theta=\\frac{-\\sqrt{6} \\pm \\sqrt{2}}{4} \ (complex conjugate in the same order)'

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'Express the given values in terms of trigonometric functions of angles from 0 to $\\ frac {\\ pi} {4}$. (1) $\\ sin \\ frac {5} {9} \\ pi$ (2) $\\ cos \\frac {7} {5} \\ pi$ (3) $\\ tan \\ left(- \\ frac {10} {7} \\ pi \\ right)$'

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'Let f(x)=3x^3+ax^2+(3a+4)x. (1) In the xy-plane, the curve y=f(x) always passes through two fixed points. Find the coordinates of these two fixed points. (2) Determine the range of values for a so that f(x) does not have extremum.'

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'140 \\quad \\n¥( \\theta=\\frac{\\pi}{4}, \\frac{\\pi}{3}, \\frac{3}{4} \\pi, \\frac{5}{4} \\pi, \\frac{5}{3} \\pi, \\frac{7}{4} \\pi )'

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'Express the following expressions in the form of $r \\sin(\\theta+\\alpha)$. Given that $r>0,-\\pi<\\alpha \\leqq \\pi$.\n(1) $\\cos \\theta-\\sqrt{3} \\sin \\theta$\n(2) $3 \\sin \\theta+2 \\cos \\theta$'

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"Let OB'=r, and let α be the angle between OB' and the positive direction of the x-axis."

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'Prove that the following equations hold when t = tan(θ/2) (t ≠ ±1).'

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'Choose the appropriate one for each of the following answer groups: A and C. The order of the options is not relevant.'

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'Express the following trigonometric ratios in terms of angles less than 45°.'

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'In figure (a), find the values of \ \\sin \\theta, \\cos \\theta, \\tan \\theta \.'

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'Find the range of values for θ that satisfies the following inequalities when 0° ≤ θ ≤ 180°.'

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'(1) \ \\sin \\theta = \\frac{\\sqrt{3}}{2} \\\nOn the semicircle with radius 1, the points P and Q are the points where the y-coordinate is \ \\frac{\\sqrt{3}}{2} \ as shown in the right figure. The angles to be determined are \ \\angle AOP \\text { and } \\angle AOQ\\\nTherefore\\n\ \\theta = 60^{\\circ}, 120^{\\circ} \'

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'Let θ be an acute angle. When sin θ = 12/13, find the values of cos θ and tan θ.'

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'Using the diagram on the right, find the values of sin 15°, cos 15°, tan 15°.'

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'Let 0°<θ<180°. When 4cosθ+2sinθ=√2, find the value of tanθ.'

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'Let 0°≤θ≤180°. When one of sinθ, cosθ, tanθ takes a specific value, find the other 2 values.'

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'Relationships between trigonometric functions (1)'

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'Let θ be between 0° and 180°. Find the range of values of θ for which the quadratic equation x^2-(cosθ)x+cosθ=0 has two distinct real solutions, both of which are within the range -1<x<2.'

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'Let θ be an acute angle. Find the value of (sinθ+cosθ)² when tanθ=√7.'

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'Find the value of sin 140 degrees + cos 130 degrees + tan 120 degrees.'

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'Trigonometry is a method devised to measure things like distance to faraway objects and heights that cannot be directly measured, and its history dates back to ancient times. Here, we will discuss the method of calculating the height of a mountain using trigonometry.'

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'Find the range of values of θ that satisfy the following inequalities when 0° ≤ θ ≤ 180°: (1) sin θ > 1/2 (2) cos θ ≤ 1/√2 (3) tan θ < √3'

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'Find the value of cos²20°+cos²35°+cos²45°+cos²55°+cos²70°.'

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'\\Therefore\\cos ^{2} 20^{\\circ}+\\cos ^{2} 35^{\\circ}+\\cos ^{2} 45^{\\circ}+\\cos ^{2} 55^{\\circ}+\\cos ^{2} 70^{\\circ} \\ = \\cos ^{2} 20^{\\circ}+\\cos ^{2} 35^{\\circ}+\\cos ^{2} 45^{\\circ}+\\sin ^{2} 35^{\\circ}+\\sin ^{2} 20^{\\circ} \\ = \\left(\\sin ^{2} 20^{\\circ}+\\cos ^{2} 20^{\\circ}\\right)+\\left(\\sin ^{2} 35^{\\circ}+\\cos ^{2} 35^{\\circ}\\right)+\\cos ^{2} 45^{\\circ} \\ = 1+1+\\left(\\frac{1}{\\sqrt{2}}\\right)^{2}=\\frac{5}{2}'

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'Find the value of (sin⁴θ + 4 cos²θ - cos⁴θ + 1) / 3(1 + cos²θ).'

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'Find the maximum and minimum values of the following function, as well as the corresponding θ values.'

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'Let θ be an acute angle. Find the other two values when one of sin θ, cos θ, tan θ takes the following value.'

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'Refer to the trigonometric table and answer the following question. When θ = 37°, find the values of sin θ, cos θ, tan θ.'

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'Using the diagram on the right, find the values of sin 22.5 degrees, cos 22.5 degrees, and tan 22.5 degrees.'

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'Find the value of θ that satisfies the following equation when 0° ≤ θ ≤ 180°: (6) √3 tanθ + 1 = 0'

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'Find the maximum and minimum values of y = sin^2θ + cosθ - 1 for 0° ≤ θ ≤ 180°. Also, determine the values of θ at those points.'

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'In the proposition "if p then q", let the set of elements satisfying condition p be P, and the set of elements satisfying condition q be Q. When the proposition "if q then p" is true, regarding its contrapositive, ∎ holds true. Choose the correct option to fill in the blank.'

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'Simplify the following expression: (2) tan(45° + θ) tan(45° - θ) tan 30° (0° < θ < 45°)'

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'Values and transformations of trigonometric functions for obtuse angles'

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'For PR 0° ≤ θ ≤ 180°, find the values of θ that satisfy the following equation: (6)√3 tan θ + 1 = 0'

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'Basic Trigonometric Relations 108 Let θ be an acute angle. (1) When sin θ = 2/√13, find the values of cos θ and tan θ. (2) When tan θ = √5/2, find the values of sin θ and cos θ.'

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'Express the following trigonometric ratios as trigonometric ratios of angles between 0° and 90°, and find their values using the trigonometric table. (1) sin 111° (2) cos 155° (3) tan 173°'

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'Question 5 (2) Find the tangent of the second largest angle in triangle ABC.'

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'In triangle ABC, if sin A: sin B: sin C = 5:16:19, find the size of the largest angle in this triangle.'

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'Find the sine, cosine, and tangent of the following angles. (1) 135 degrees (2) 150 degrees (3) 1'

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'In triangle ABC, if ∠A=α, ∠B=β, ∠C=90 degrees, prove the following inequalities hold: (1) sinα+sinβ>1 (2) cosα+cosβ>1'

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'Prove the interrelations of the following trigonometric identities: $\\sin^2 \\theta + \\cos^2 \\theta = 1, \\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta}, 1 + \\tan^2 \\theta = \\frac{1}{\\cos^2 \\theta}$.'

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'Choose two options that are equal to sin 44° from the following choices. (1) sin 46° (2) cos 46° (3) sin 136° (4) cos 136°'

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'Find the values of θ that satisfy the following equation when 0° ≤ θ ≤ 180°: 2sinθ = √2'

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'Show the definitions of the following trigonometric ratios: $\\sin \\theta = \\frac{y}{r}, \\cos \\theta = \\frac{x}{r}, \\tan \\theta = \\frac{y}{x}$'

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'Prove the inequality sin 29 degrees < tan 29 degrees < cos 29 degrees.'

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'Let 0° ≤ θ ≤ 180°. If sinθ+cosθ = 1/√5, find the values of the following expressions: (1) tan^3θ+1/tan^3θ (2) sin^3θ-cos^3θ'

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'Please explain how to convert obtuse angle trigonometric ratios to acute angle trigonometric ratios using formulas.'

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'Find the sine, cosine, and tangent of the following angles.\n1. 25°\n2. 45°\n3. 75°\n4. 89°'

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'What curve does the given parametric representation represent?'

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'Find the parametric representation of a cycloid. A cycloid is the curve traced by a point \ P \ on a circle which rolls without slipping along a straight line. Determine the coordinates of \ P \ on the circle when it has rolled through an angle \ \\theta \.'

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"Reverse-engineer from the 'self you want to become'."

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'Find the curve of the equation x = t + \\frac{1}{t}, y = t^{2} + \\frac{1}{t^{2}}, t > 0.'

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'Chapter 4 Equations and Curves\n17 Parabolas\n18 Ellipses\n19 Hyperbolas\n20 Translation of Quadratic Curves\n21 Quadratic Curves and Lines\n22 Parametric Representations of Curves\n23 Polar Coordinates and Polar Equations'

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'What are good things to work on after solving basic examples and standard examples?'

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'(2) Express the curves represented by the following polar equations in terms of Cartesian coordinates x, y:\n(A) r=√3 cos θ+sin θ\n(B) r^{2}(1+3 cos^{2} θ)=4'

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"How is mathematics useful in society? The ways in which mathematics is 'useful' have also changed over time. In the past, when discussing the application of mathematics, it was often associated with the keyword 'cutting-edge science and technology'. The importance of cutting-edge science and technology in society goes without saying, however, it was not something that was familiar in our daily lives. In recent years, the situation has changed. This is because mathematics has started to penetrate into various aspects of our daily lives."

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'(1) Since cos(-x) = cos x and (-x)^2 sin(-x) = -x^2 sin x, cos x is an even function and x^2 sin x is an odd function. Therefore, ∫_(-π/3)^(π/3) ( cos x + x^2 sin x ) dx = 2 ∫_0^(π/3) cos x dx = 2 [ sin x ]_0^(π/3) = √3'

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'Describe the curves represented by the following polar equations in rectangular coordinates.'

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'(50)(2) When n is odd, $\\int_{-1}^{1} T_{n}(x) d x=0$, and when n is even, $\\int_{-1}^{1} T_{n}(x) d x=\\frac{1}{n+1}-\\frac{1}{n-1}$'

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'In (1), if the range of the function is 1 ≤ y < 3/2, find the domain.'

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'For a positive number a, consider the point A(a, a^{2}) on the parabola y=x^{2}, and let l be the line rotated -30 degrees around point A. Let B be the intersection point of line l and the parabola y=x^{2} that is not A. Additionally, let (a, 0) be C and the origin be O. Find the equation of line l. Furthermore, let S(a) be the area enclosed by the line segments OC and CA and the parabola y=x^{2}, and let T(a) be the area enclosed by the line segments AB and CA and the parabola y=x^{2}. Find c=lim_{a→∞} T(a)/S(a).'

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'For real numbers θ satisfying (3) 0 ≤ θ < 2π, let z = cosθ + i sinθ. Prove the equation |1 - z| = 2 sin(θ/2) holds.'

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'Show the concavity and convexity of the function that meets the following conditions.'

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'4 (cos^2 x)’ = 2 cos x (cos x)’ = -2 sin x cos x'

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'Important Example\nMaximum and minimum of 13|ux + vy|\nWhen real numbers x, y, u, v satisfy the equations x^2 + y^2 = 1 and (u-2)^2 + (v-2√3)^2 = 1, find the maximum and minimum values of ux + vy.'

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'Let c be a constant satisfying -1<c<1. Find a continuous function f(x) that satisfies the relation f(x)+f(cx)=x^2 for all real numbers x.'

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'Find the inflection point of the function at the following x values.'

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'\\( 134\\left\\{\egin{array}{l}x=(a+b) \\cos \\theta-b \\cos \\frac{a+b}{b} \\theta \\\\ y=(a+b) \\sin \\theta-b \\sin \\frac{a+b}{b} \\theta\\end{array}\\right. \\)'

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'Let the curve represented by the parameter variables \\( x=\\sin t, y=\\cos \\left(t-\\frac{\\pi}{6}\\right) \\sin t(0 \\leqq t \\leqq \\pi) \\) be denoted by \ C \.'

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'For the equation r=\\frac{1}{1+a \\cos θ}, (1) prove that when a= ±1, it represents a parabola, and when |a|<1, it represents an ellipse. (2) Prove that the curve represented by the above equation intersects the y-axis at y= ±1 regardless of the value of a. (3) When |a|<1, let the part in the first quadrant of the ellipse and enclosed by the x-axis and y-axis be denoted as D. Find the volume of the solid obtained by rotating the figure D around the x-axis.'

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'(2) Continuation of (1): Therefore, let the angle between \\overrightarrow{AB} and \\overrightarrow{AC} be θ. Then, \\cos \\theta=\\frac{\\overrightarrow{AB}\\cdot\\overrightarrow{AC}}{\\left|\\overrightarrow{AB}\\right|\\left|\\overrightarrow{AC}\\right|}=\\frac{-2a+6}{3\\sqrt{a^{2}-2a+14}}. Since \\sin \\theta>0, we have \\sin \\theta=\\sqrt{1-\\cos ^{2} \\theta}=\\sqrt{1-\\frac{(-2a+6)^{2}}{9(a^{2}-2a+14)}}=\\frac{1}{3}\\sqrt{\\frac{5a^{2}+6a+90}{a^{2}-2a+14}}'

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'In the case of 0<θ<π/2, if dL/dθ=0, then cosθ=1/√3. Let α be the θ that satisfies this condition, then tanα=√(1/cos²α-1)=√2. From (3-√7)/√2<2/√2<(3+√7)/√2, we get tanθ₁<tanα<tanθ₂. Thus, θ₁<α<θ₂. Therefore, the table of increase and decrease of L is as shown on the right for θ₁<θ<θ₂. Hence, L attains its maximum value when θ=α. Since sinα=√(1-cos²α)=√6/3, the desired maximum value is 2sinα-√2/(3cosα)=√6/3. In this case, cosθ=1/√3.'

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"Problem 94 Responding to small changes\n(1) How much will the area S of ΔABC increase by?\n(2) How much will the length y of side CA increase by?\nUsing the following formula.\nFormula for small changes Δy≒y'Δx\nAnswer: When angle B increases by 1 degree\nStarting from S≒√3sin(x)."

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'Find the coordinates of a point obtained by reflecting about the real axis and rotating by -π/2 about the origin.'

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'Example 41 | Calculating Definite Integrals (2)\nFind the definite integral ∫_{0}^{π} sin(mx)cos(nx)dx. Here, m and n are natural numbers.'

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'Practice properties of the function f(x) = x sin(1/x) (x > 0) 134'

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'Prove that for the inverse function y=g(x) of y=tan x (-π/2<x<π/2), g(1/2)+g(1/3)=π/4.'

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'Verify the increasing and decreasing behavior of the function $y=f(x)$ from the table below and find the extreme values.'

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'Find the conditions under which the curve y=x^4+ax^3+bx^2+cx+d has a multiple tangent line.'

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'Calculate the value of sin(π/5) sin(2π/5) sin(3π/5) sin(4π/5).'

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'Finding the sum of an infinite series using a recurrence relation'

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'What you learn in this chapter builds upon what you have learned so far. Using this knowledge, it is essential to analyze the geometry of shapes further. In this chapter, we will apply the methods of analytical geometry to study the properties of shapes not previously covered, mainly focusing on the characteristics of conic sections such as ellipses, hyperbolas, and parabolas. Additionally, we will briefly touch upon methods to represent curves with equations, including parametric representations and polar coordinates as well as polar equations.'

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'(1) The coordinates of a point Q on curve C are given by the parametric equations, where the parameter t ranges from -π/2 to 0, as (√2/cos t, √2 tan t). The equation of the tangent line l at point Q is [√2/cos t x-√2 tan t y=2], which is equivalent to [x-sin t y=√2 cos t]'

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'Find the maximum value, minimum value, and the corresponding value of x for the given function.'

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'As the point on the curve moves infinitely far away, the curve approaches a certain straight line, which is called the asymptote of the curve.'

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'As $x$ gets infinitely large, what value does $y$ approach?'

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'When S=4, \2 \\sqrt{k^{2}+1}=4\ solves to \k=\\sqrt{3}\. Therefore, \\\cos \\alpha=\\frac{1}{2}, \\sin \\alpha=\\frac{\\sqrt{3}}{2}\. Since \0<\\alpha<\\frac{\\pi}{2}\, we have \\\alpha=\\frac{\\pi}{3}\. Hence, \\eta=\\frac{4}{3} \\pi\. In the range where \\\frac{\\pi}{3} \\leqq x \\leqq \\theta\, the area enclosed by the curves \y=\\sin x\, \y=\\sqrt{3} \\cos x\, and the line \x=\\theta\ is denoted as \T\. For \T<4\ to hold true, it must be that \\\frac{\\pi}{3}<\\theta<\\frac{4}{3} \\pi\.'

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'Transform the given radical function into the form of a square root function.'

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'Explain why the function f(x) is discontinuous: f(x)={ x^2 + 1 (x ≠ 0), 0 (x = 0) }'

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'The derivatives of trigonometric functions are as follows. Note that the angles are in radians.'

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'Find all the tangent lines of the curve y = x cos x that pass through the origin.'

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'The n-th term a_{n} is a_{n} = \\cos n \\pi . Let k be a natural number. When n=2k-1, \\cos n \\pi = \\cos (2k-1) \\pi = \\cos (-\\pi) = -1. When n=2k, \\cos n \\pi = \\cos 2k \\pi = 1. Therefore, the sequence \\{a_{n}\\} oscillates. Hence, the n-th term of the sequence \\{a_{n}\\} is a_{n}=(-1)^{n}, which does not converge to 0, so this infinite series diverges.'

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'On the xy plane, with the origin as the pole and the positive part of the x-axis as the starting line in polar coordinates, let the curve represented by the polar equation r=2+cosθ(0 ≤ θ ≤ π) be denoted as C. Find the volume of the solid obtained by rotating the region enclosed by C and the x-axis around the x-axis one full revolution.'

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'Practice What kind of curve do the following polar equations represent? Answer in rectangular coordinates.'

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'Solve the composite trigonometric function. When 42sin(x-π/6)-1=0 (0≤x≤π), the solutions are x=π/3, π'

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'The region represented by the given system of simultaneous inequalities is such that the x-coordinate of the intersection points of the curve y=sin x and the line y=t-x is denoted by α, where sin α=t-α and 0<α<t. In this case, V(t)=π ∫_{0}^{α} sin^2 x d x+1/3 π sin^2 α·(t-α). From (1), we have V(t)=π ∫_{0}^{α} sin^2 x d x + 1/3 π sin^3 α。'

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'Let 33θ be a real number, and n be an integer. If z=sinθ+i*cosθ, express the real part and imaginary part of the complex number zn in terms of cos(nθ) and sin(nθ).'

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'(f ∘ g)(x) = f(g(x)) = 12 - 3 ⋅ (-1)^2 = 9, so (f ∘ g)(x) = \egin{cases} -3x^2 + 12x & (x ≥ 0) \\\\ 9 & (x < 0) \\end{cases}'

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'Express the equation r^2 (cos^2 θ − sin^2 θ) = r sin θ(1 − r sin θ) + 1 using x = r cos θ, y = r sin θ, derived from cos 2θ = cos^2 θ − sin^2 θ'

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'(1) Let f′(t)=−e^(−t)sin(t)+e^(−t)cos(t)=−e^(−t)(sin(t)−cos(t)) = −√2 e^(−t)sin(t−π/4) If f′(t)=0, then sin(t−π/4)=0. Since t−π/4>−π/4, t=π/4+(n−1)π (n=1,2, ...)'

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'When a function y is represented using the parameter θ as x=1-cosθ, y=θ-sinθ'

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'For 0 ≤ θ ≤ π, cos(θ/2) ≥ 0. For 0 ≤ θ ≤ π/2, cos(θ) ≥ 0. For π/2 ≤ θ ≤ π, cos(θ) ≤ 0. Also, cos(θ) * cos(θ/2) = 1/2 * (cos(3/2 * θ) + cos(θ/2)).'

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'Solve the following equation for 0 ≤ θ < 2π. Also, find its general solution. (1) sin θ = √3/2'

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'Show that the following conditions are met and find the value of cos 36 degrees: (1) When θ = 36 degrees, sin 3θ = sin 2θ'

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'Comprehensive Exercise Part 2 Mathematics II Chapter 4 Trigonometric Functions'

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'Find the maximum and minimum values in the [ ] and the corresponding values of x.'

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'Find the maximum and minimum values of the given functions. Also, determine the values of θ at those points. Consider 1620 ≤ θ ≤ π. (1) y=sinθ−√3 cosθ (2) y=sin(θ−π/3)+sinθ'

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'In the given figure, as the point P(x, y) moves along the circumference of the unit circle, the point T(1, m) moves through all points on the line x = 1. From this definition, prove that x = r cos θ, y = r sin θ holds true.'

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'Prove the equation \ \\frac{\\cos \\theta}{1+\\sin \\theta}+\\tan \\theta=\\frac{1}{\\cos \\theta} \.'

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'Example 146 \\(\\ y=4 \\sin ^{2} \\theta-4 \\cos \\theta+1 \\rightarrow y = 4\\left(1-\\cos ^{2} \\theta\\right)-4 \\cos \\theta+1\\) is a quadratic function of \\\cos \\theta\.'

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'Determine the value of the constant a such that the absolute value of the minimum value of the function y=2sin3x+cos2x-2sinx+a is equal to the maximum value.'

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'Using the addition formula, find the following values.'

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'(1) \ \\sin 2 \\theta=\\cos 3 \\theta \ [Practice \\( 156(2) \\) ] The general solution is'

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'The period of the function f(θ) = 2sin3θ + 1 is A square, and the maximum value of f(θ) is B square. [Shonan Institute of Technology]'

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'(1) The function y=f(x) takes a maximum value at x=α and a minimum value at x=β. Show that the midpoint M of the line segment connecting the two points (α, f(α)) and (β, f(β)) lies on the curve y=f(x).'

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'How to memorize the addition theorem and double-angle/half-angle formulas?'

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'Practice plotting the graphs of the following functions and find their periods.'

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'Prove that the value of $\\tan \\alpha \\tan \eta+\\tan \eta \\tan \\gamma+\\tan \\gamma \\tan \\alpha$ is constant when positive real numbers $\\alpha, \eta, \\gamma$ satisfy $\\alpha+\eta+\\gamma=\\frac{\\pi}{2}$.'

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'Solution to the equations sin aθ=sin bθ, sin aθ=cos bθ'

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'1. Sine Addition Formula: \\( \\sin (\\alpha \\pm \eta)=\\sin \\alpha \\cos \eta \\pm \\cos \\alpha \\sin \eta \\)\n2. Cosine Addition Formula: \\( \\cos (\\alpha \\pm \eta)=\\cos \\alpha \\cos \eta \\mp \\sin \\alpha \\sin \eta \\)\n3. Tangent Addition Formula: \\( \\tan (\\alpha \\pm \eta)=\\frac{\\tan \\alpha \\pm \\tan \eta}{1 \\mp \\tan \\alpha \\tan \eta} \\)'

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'(2) If \ \\tan \\frac{\\theta}{2}=\\frac{1}{2} \, find the values of \ \\cos \\theta, \\tan \\theta, \\tan 2 \\theta \.'

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'Find the trigonometric functions of angle $\\theta$.'

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'Using the addition theorem, find the following values.'

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'Solving the equation sin aθ=sin bθ, sin aθ=cos bθ'

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'Find the values of sin θ, cos θ, tan θ when θ is the following values.'

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'(3) Prove that the value of $\\tan \\alpha \\tan \eta + \\tan \eta \\tan \\gamma + \\tan \\gamma \\tan \\alpha$ is constant when positive real numbers $\\alpha, \eta, \\gamma$ satisfy $\\alpha+\eta+\\gamma=\\frac{\\pi}{2}$.'

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'Radian measure and trigonometric functions Radian'

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'Solve the following equation for 0 ≤ θ < 2π. Also find its general solution.'

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'When n is a natural number and θ is a real number, answer the following question. (1) Prove cos(n+2)θ-2cosθcos(n+1)θ+cosnθ=0.'

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'Explain the properties of definite integrals of even and odd functions.'

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'Periodic Function with Period 4\nFor a function \\( f(x) \\), if there exists a non-zero constant \ p \ such that the equation \\( f(x+p)=f(x) \\) holds for all values of \ x \, then \\( f(x) \\) is called a periodic function with period \ p \. In this case, since \\( f(x+2p)=f(x+3p)=\\cdots =f(x) \\), the periods \ 2p, 3p, \\cdots \ are also valid periods, and there are infinitely many periods for a periodic function.\n\nProblem: Calculate the period of the function \\( y = \\cos(5\\theta) \\).'

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'Problem to find the solution of the triangle inequality'

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'Solve the following equation. Also, find its general solution. (4) sinθ=-1'

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'Properties and Graphs of Trigonometric Functions'

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'Practice drawing the graphs of the following functions. Also, determine their periods.'

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'(4) Given the equation $2 \\sin \\theta \\cdot \\frac{\\sin \\theta}{\\cos \\theta}=-3$, we have $2 \\sin ^{2} \\theta=-3 \\cos \\theta \\quad$, and $\\cos \\theta \\neq 0$. Therefore, $2\\left(1-\\cos ^{2} \\theta\\right)=-3 \\cos \\theta$, which simplifies to $2 \\cos ^{2} \\theta-3 \\cos \\theta-2=0$. This gives us $(\phantom{\rule{0ex}{0ex}}cos\phantom{\rule{0ex}{0ex}}theta-2)(2\phantom{\rule{0ex}{0ex}}cos\phantom{\rule{0ex}{0ex}}theta+1)$