Chapter 6

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathrm{0}<\boldsymbol{\theta}<\pi / 2 .\) Assume \(a>\mathrm{0}.\) $$\sqrt{a^{2}-u^{2}}, \quad u=a \sin \theta$$

Use inverse functions where necessary to solve the equation. $$2 \sin ^{2} x+5 \cos x=4$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\cos (\phi+\alpha)-\cos (\phi-\alpha)$$

Verify the identity. $$\begin{aligned} &a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \cos (B \theta-C), \text { where }\\\ &C=\arctan (a / b) \text { and } b>0. \end{aligned}$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathrm{0}<\boldsymbol{\theta}<\pi / 2 .\) Assume \(a>\mathrm{0}.\) $$\sqrt{a^{2}-u^{2}}, \quad u=a \cos \theta$$

Use inverse functions where necessary to solve the equation. $$2 \cos ^{2} x+7 \sin x=5$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\cos \left(\theta+\frac{\pi}{2}\right)-\cos \left(\theta-\frac{\pi}{2}\right)$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathrm{0}<\boldsymbol{\theta}<\pi / 2 .\) Assume \(a>\mathrm{0}.\) $$\sqrt{a^{2}+u^{2}}, \quad u=a \tan \theta$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. $$3 \tan ^{2} x+5 \tan x-4=0, \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin \left(x+\frac{\pi}{2}\right)+\sin \left(x-\frac{\pi}{2}\right)$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathrm{0}<\boldsymbol{\theta}<\pi / 2 .\) Assume \(a>\mathrm{0}.\) $$\sqrt{u^{2}-a^{2}}, \quad u=a \sec \theta$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. $$\cos ^{2} x-2 \cos x-1=0, \quad[0, \pi]$$

Use the sum-to-product formulas to find the exact value of the expression. $$\sin 75^{\circ}+\sin 15^{\circ}$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{\left(9+x^{2}\right)^{3}}, \quad x=3 \tan \theta$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. $$4 \cos ^{2} x-2 \sin x+1=0, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

Explain why the equation is not an identity and find one value of the variable for which the equation is not true. $$\sqrt{\sec ^{2} x-1}=\tan x$$

Use the sum-to-product formulas to find the exact value of the expression. $$\cos 120^{\circ}+\cos 60^{\circ}$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{\left(x^{2}-16\right)^{3}}, \quad x=4 \sec \theta$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. $$2 \sec ^{2} x+\tan x-6=0, \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$

Use the sum-to-product formulas to find the exact value of the expression. $$\cos \frac{3 \pi}{4}-\cos \frac{\pi}{4}$$

Use a graphing utility to solve the equation for \(\theta,\) where \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$\sin \theta=\sqrt{1-\cos ^{2} \theta}$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval \([0,2 \pi],\) and (b) solve the trigonometric equation and verify that the \(x\) -coordinates of the maximum and minimum points of \(f\) are among its solutions. (The trigonometric equation is found using calculus.) Trigonometric Equation \(2 \cos 2 x=0\) \(-2 \sin 2 x=0\) \(2 \sin x \cos x-\sin x=0\) \(-2 \sin x \cos x-\cos x=0\) \(\cos x-\sin x=0\) \(2 \cos x-4 \sin x \cos x=0\) Function \(f(x)=\sin 2 x\)

Explain why the equation is not an identity and find one value of the variable for which the equation is not true. $$1-\cos \theta=\sin \theta$$

Use the sum-to-product formulas to find the exact value of the expression. $$\sin \frac{5 \pi}{4}-\sin \frac{3 \pi}{4}$$

Use a graphing utility to solve the equation for \(\theta,\) where \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$\cos \theta=-\sqrt{1-\sin ^{2} \theta}$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval \([0,2 \pi],\) and (b) solve the trigonometric equation and verify that the \(x\) -coordinates of the maximum and minimum points of \(f\) are among its solutions. (The trigonometric equation is found using calculus.) Trigonometric Equation \(2 \cos 2 x=0\) \(-2 \sin 2 x=0\) \(2 \sin x \cos x-\sin x=0\) \(-2 \sin x \cos x-\cos x=0\) \(\cos x-\sin x=0\) \(2 \cos x-4 \sin x \cos x=0\) Function \(f(x)=\cos 2 x\)

Explain why the equation is not an identity and find one value of the variable for which the equation is not true. $$1+\tan x=\sec x$$

Find the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your answers. $$\sin 6 x+\sin 2 x=0$$

Use the figure, which shows two lines whose equations are \(y_{1}=m_{1} x+b_{1}\) and \(y_{2}=m_{2} x+b_{2}\). Assume that both lines have positive slopes. Derive a formula for the angle between the two lines. Then use your formula to find the angle between the given pair of lines. $$\begin{aligned} &y=x\\\ &y=\sqrt{3} x \end{aligned}$$

Use a graphing utility to solve the equation for \(\theta,\) where \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$\sec \theta=\sqrt{1+\tan ^{2} \theta}$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval \([0,2 \pi],\) and (b) solve the trigonometric equation and verify that the \(x\) -coordinates of the maximum and minimum points of \(f\) are among its solutions. (The trigonometric equation is found using calculus.) Trigonometric Equation \(2 \cos 2 x=0\) \(-2 \sin 2 x=0\) \(2 \sin x \cos x-\sin x=0\) \(-2 \sin x \cos x-\cos x=0\) \(\cos x-\sin x=0\) \(2 \cos x-4 \sin x \cos x=0\) Function \(f(x)=\sin ^{2} x+\cos x\)

Verify that for all integers \(n, \sin \left[\frac{(12 n+1) \pi}{6}\right]=\frac{1}{2}.\)

Find the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your answers. $$\cos 2 x-\cos 6 x=0$$

Use the figure, which shows two lines whose equations are \(y_{1}=m_{1} x+b_{1}\) and \(y_{2}=m_{2} x+b_{2}\). Assume that both lines have positive slopes. Derive a formula for the angle between the two lines. Then use your formula to find the angle between the given pair of lines. $$\begin{aligned} &y=x\\\ &y=\frac{1}{\sqrt{3}} x \end{aligned}$$

Use a graphing utility to solve the equation for \(\theta,\) where \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$\tan \theta=\sqrt{\sec ^{2} \theta-1}$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval \([0,2 \pi],\) and (b) solve the trigonometric equation and verify that the \(x\) -coordinates of the maximum and minimum points of \(f\) are among its solutions. (The trigonometric equation is found using calculus.) Trigonometric Equation \(2 \cos 2 x=0\) \(-2 \sin 2 x=0\) \(2 \sin x \cos x-\sin x=0\) \(-2 \sin x \cos x-\cos x=0\) \(\cos x-\sin x=0\) \(2 \cos x-4 \sin x \cos x=0\) Function \(f(x)=\cos ^{2} x-\sin x\)

Find the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your answers. $$\frac{\cos 2 x}{\sin 3 x-\sin x}-1=0$$

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln |\cos \theta|-\ln |\sin \theta|$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval \([0,2 \pi],\) and (b) solve the trigonometric equation and verify that the \(x\) -coordinates of the maximum and minimum points of \(f\) are among its solutions. (The trigonometric equation is found using calculus.) Trigonometric Equation \(2 \cos 2 x=0\) \(-2 \sin 2 x=0\) \(2 \sin x \cos x-\sin x=0\) \(-2 \sin x \cos x-\cos x=0\) \(\cos x-\sin x=0\) \(2 \cos x-4 \sin x \cos x=0\) Function \(f(x)=\sin x+\cos x\)

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=2^{x}+3$$

Find the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your answers. $$\sin ^{2} 3 x-\sin ^{2} x=0$$

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln |\cot \theta|+\ln |\sin \theta|$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval \([0,2 \pi],\) and (b) solve the trigonometric equation and verify that the \(x\) -coordinates of the maximum and minimum points of \(f\) are among its solutions. (The trigonometric equation is found using calculus.) Trigonometric Equation \(2 \cos 2 x=0\) \(-2 \sin 2 x=0\) \(2 \sin x \cos x-\sin x=0\) \(-2 \sin x \cos x-\cos x=0\) \(\cos x-\sin x=0\) \(2 \cos x-4 \sin x \cos x=0\) Function \(f(x)=2 \sin x+\cos 2 x\)

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=-2^{x-3}$$

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. Use a graphing utility to verify your results. $$y=-\frac{1}{2}(x-10)+14$$

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln |\sec x|+\ln |\sin x|$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=2^{-x}-1$$

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. Use a graphing utility to verify your results. $$y=x^{2}-3 x-40$$

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln |\tan x|-\ln |\sin x|$$

Find the smallest positive fixed point of the function \(f .\) IA fixed point of a function \(f\) is a real number \(c\) such that \(f(c)=c .]\) $$f(x)=\cos x$$