Advanced Functions - Trigonometric Functions and Their Applications | AI tutor The No.1 Homework Finishing Free App

'Exercise 8 III (2)'

'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 \'

'Conditions for the existence of solutions to trigonometric equations'

'Find the period of y = \\sin k \\theta.'

'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}'

'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'

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

'(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} \\]'

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

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

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

'(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 $n=1$.'

'Exercise Example 10 Trigonometric Functions and Chebyshev Polynomials'

'106 4 sin^4 α'

'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.'

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

'Find the period of the function y = tan kθ.'

'(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'

'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'

'(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.'

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

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

'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'

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

'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)'

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

"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."

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

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

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

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

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

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

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

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

'What is uniform circular motion?'

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

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

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

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

'(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]'

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

'Trigonometric functions and Chebyshev polynomials'

'(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 $\\cos ^{2} \\theta+2 \\cos ^{4} \\theta=2-\\frac{-1+\\sqrt{5}}{2}=\\frac{5-\\sqrt{5}}{2}$.'

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

'(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}'

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

'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.'

'From the condition, tan𝛼+tan𝛽+tan𝛾=tan𝛼tan𝛽tan𝛾'

'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'

'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}'

'(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'

'(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} \\]'

'Translate the given text into multiple languages.'

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

'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 \'

'Solve the following equations and inequalities.'

'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.'

'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).'

'Relationships of trigonometric functions'

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

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

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

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

'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)'

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

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

'Master the trigonometric equations and conquer example 123!'

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

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

'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).'

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

'Master the addition principle and conquer example 130!'

"\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}\."

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

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

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

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

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

'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)'

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

'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).'

'Equations and inequalities involving trigonometric functions (substitution)'

'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.'

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

'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.'

'Trigonometric functions interrelationships'

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

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

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

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

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

'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.'

'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)'

'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.'

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

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

'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.'

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

'System of inequalities involving trigonometric functions'

'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.'

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

'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.'

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

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

'Prove the following equations.'

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

'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,'

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

'Graph of trigonometric functions and translation/scaling'

'Solve the following trigonometric equations.'

'Prove the following equation.'

'Trigonometric functions around you'

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

'Find the following values.'

'System of inequalities involving trigonometric functions'

'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)}'

'Transformation of trigonometric expressions'

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

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

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

'Trigonometric Function Graph (1) - Zoom In/Out'

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

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

'Prove the double angle formulas.'

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

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

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

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

'Find the value of sin(45 degrees).'

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

'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.'

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

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

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

'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.'

'Key points in substitution method of definite integration'

'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.'

'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]'

'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).'

'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 $x=a \\sin \\theta$?'

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

'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.'

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

'Find the following indefinite integrals.'

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

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

'Basic 2: Translation and Determination of Fraction Functions'

'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.'

'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 $a, b, c$.'

'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.'

'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.'

"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."

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

'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.'

'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.'

'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.'

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

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

''

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

'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.'

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

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

'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 \'

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

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

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

'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.'

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

'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}.'

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

'Calculate the trigonometric functions and show the results.'

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

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

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

'From figure (1) to P(-1, 1)'

'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.'

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

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

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

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

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

'Variable transformation'

'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?'

'Summary of trigonometric functions'

'Sine rule or cosine rule?'

"Let's review the Sine Rule and Cosine Rule!"

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

'Explain the relationship between necessary and sufficient conditions.'

'Applications of triangles'

'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'

'Using trigonometric tables to find angles'

'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.'

'Find the sin θ of triangle AED.'

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

'Solve the mathematics problem'

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

'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,'

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

'(6) Trigonometric ratios-117'

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

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

'Value of trigonometric symmetry identity'

'Trigonometric ratios of 15 degrees'

'(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.'

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

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

'Sine rule or cosine rule?'

'Trigonometric relationships when θ is an acute angle'

'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} \'

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

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

'Find the values of trigonometric functions for obtuse angles'

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

'Explain how to use the book to find solutions.'

'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 $\\ sin 2$.'

'Find the following values.'

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

"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?"

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

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

'Maximum and minimum of trigonometric functions (1)'

'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.'

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

'Trigonometric function value (1)'

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

'Chapter 4 Trigonometric Functions 195'

'Example 128 Using Addition Theorem'

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

'Translate the given text into multiple languages.'

'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.'

'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'

'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.'

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

'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 $3-\\sqrt{7}$.'

'Solving trigonometric equations (basic)'

'sin 15 degrees'

'Period of trigonometric functions'

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

'Calculate the values of the following trigonometric functions.'

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

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

'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'

'Prove the following equations.'

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

'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.'

'Graph of trigonometric functions (1)'

'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.'

'Trigonometric function relationships'

'(1) \\sin \\frac{13}{4} \\pi'

'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]'

'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.'

''

'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.'

'Please explain the periodicity of trigonometric functions.'

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

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

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

'Calculate the values of the following trigonometric functions.'

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

'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.'

'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)'

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

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

"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."

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

'(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)'

'Graphs of Trigonometric Functions (2)'

'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) $\\ tan \\ left(- \\ frac {10} {7} \\ pi \\ right)$'

'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.'

'Prove the following trigonometric identities.'

'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 )'

'Addition formulas'

'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) $3 \\sin \\theta+2 \\cos \\theta$'

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

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

'Proof of trigonometric identities'

'Convert the given angles to radians.'

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

"Let's draw your growth curve."

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

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

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

'(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} \'

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

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

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

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

'Relationships between trigonometric functions (1)'

'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.'

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

'Plot the graph of the following functions.'

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

'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.'

'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'