Chapter 6

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

Use a graphing utility to complete the table and graph the functions in the same viewing window. Make a conjecture about \(y_{1}\) and \(y_{2}\) $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\\\\hline y_{1} & & & & & & & \\\\\hline y_{2} & & & & & & & \\\\\hline\end{array}$$ $$y_{1}=\frac{\cos x}{1-\sin x}, \quad y_{2}=\frac{1+\sin x}{\cos x}$$

Powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. $$\cos ^{3} x \sin ^{2} x=\left(\sin ^{2} x-\sin ^{4} x\right) \cos x$$

Use the product-to-sum formulas to write the product as a sum or difference. $$6 \sin 45^{\circ} \cos 15^{\circ}$$

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

Use a graphing utility to complete the table and graph the functions in the same viewing window. Make a conjecture about \(y_{1}\) and \(y_{2}\) $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\\\\hline y_{1} & & & & & & & \\\\\hline y_{2} & & & & & & & \\\\\hline\end{array}$$ $$y_{1}=\sec ^{4} x-\sec ^{2} x, \quad y_{2}=\tan ^{2} x+\tan ^{4} x$$

Powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. $$\sin ^{4} x+\cos ^{4} x=1-2 \cos ^{2} x+2 \cos ^{4} x$$

Use the product-to-sum formulas to write the product as a sum or difference. $$6 \sin \frac{\pi}{3} \cos \frac{\pi}{3}$$

Use a graphing utility to approximate the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\). $$x \tan x-1=0$$

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. $$\cos x \cot x+\sin x$$

Verify the identity. $$\tan \leftAnswers will vary.(\sin ^{-1} x\right)=\frac{x}{\sqrt{1-x^{2}}}$$

Use the product-to-sum formulas to write the product as a sum or difference. $$4 \cos \frac{\pi}{3} \sin \frac{5 \pi}{6}$$

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

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. $$\sin x(\cot x+\tan x)$$

Verify the identity. $$\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}$$

Use the product-to-sum formulas to write the product as a sum or difference. $$5 \sin \theta \sin 3 \theta$$

The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each wave has amplitude \(A,\) period \(T,\) and wavelength \(\lambda .\) The models for two such waves are $$\begin{aligned} &y_{1}=A \cos 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right) \text { and } y_{2}=A \cos 2 \pi\left(\frac{t}{T}+\frac{x}{\lambda}\right)\\\ &\text { Show that } y_{1}+y_{2}=2 A \cos \frac{2 \pi t}{T} \cos \frac{2 \pi x}{\lambda} \end{aligned}$$

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. $$\frac{1}{\sin x}\left(\frac{1}{\cos x}-\cos x\right)$$

Verify the identity. $$\tan \left(\sin ^{-1} \frac{x-1}{4}\right)=\frac{x-1}{\sqrt{16-(x-1)^{2}}}$$

Use a graphing utility to approximate the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\). $$\sec ^{2} x+0.5 \tan x=1$$

Use the product-to-sum formulas to write the product as a sum or difference. $$3 \sin 2 \alpha \sin 3 \alpha$$

A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by \(y=\frac{1}{3} \sin 2 t+\frac{1}{4} \cos 2 t\) where \(y\) is the distance from equilibrium (in feet) and \(t\) is the time (in seconds). (a) Use a graphing utility to graph the model. (b) Use the identity \(a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C)\) where \(C=\arctan (b / a), a>0,\) to write the model in the form \(y=\sqrt{a^{2}+b^{2}} \sin (B t+C) .\) Use the graphing utility to verify your result. (c) Find the amplitude of the oscillations of the weight. (d) Find the frequency of the oscillations of the weight.

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. $$\frac{1}{2}\left(\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}\right)$$

Verify the identity. $$\tan \left(\cos ^{-1} \frac{x+1}{2}\right)=\frac{\sqrt{4-(x+1)^{2}}}{x+1}$$

Use a graphing utility to approximate the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\). $$\csc ^{2} x+0.5 \cot x=5$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{25-x^{2}}, \quad x=5 \sin \theta$$

The length \(s\) of a shadow cast by a vertical gnomon (a device used to tell time) of height \(h\) when the angle of the sun above the horizon is \(\theta\) (see figure) can be modeled by $$s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}.$$ (a) Verify that the expression for \(s\) is equivalent to \(h \cot \theta\) (b) Use a graphing utility to complete the table. Let \(h=5\) feet. $$\begin{array}{|l|l|l|l|l|l|l|} \hline \theta & 15^{\circ} & 30^{\circ} & 45^{\circ} & 60^{\circ} & 75^{\circ} & 90^{\circ} \\ \hline s & & & & & & \\ \hline \end{array}$$ (c) Use your table from part (b) to determine the angles of the sun that result in the maximum and minimum lengths of the shadow. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is \(90^{\circ} ?\)

Use a graphing utility to approximate the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\). $$6 \sin ^{2} x-7 \sin x+2=0$$

Use the product-to-sum formulas to write the product as a sum or difference. $$\sin (x+y) \sin (x-y)$$

Determine whether the statement is true or false. Justify your answer. $$\sin \left(x-\frac{11 \pi}{2}\right)=\cos x$$

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

Use a graphing utility to approximate the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\). $$2 \tan ^{2} x+7 \tan x-15=0$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin 5 \theta-\sin \theta$$

Let \(x=\pi / 3\) in the identity in Example 7 and define the functions \(f\) and \(g\) as follows. \(f(h)=\frac{\sin [(\pi / 3)+h]-\sin (\pi / 3)}{h}\) \(g(h)=\cos \frac{\pi}{3}\left(\frac{\sin h}{h}\right)-\sin \frac{\pi}{3}\left(\frac{1-\cos h}{h}\right)\) (a) What are the domains of the functions \(f\) and \(g ?\) (b) Use a graphing utility to complete the table. (c) Use the graphing utility to graph the functions \(f\) and \(g\). (d) Use the table and graph to make a conjecture about the values of the functions \(f\) and \(g\) as \(h \rightarrow 0^{+}\).

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

Use inverse functions where necessary to solve the equation. $$\tan ^{2} x+\tan x-12=0$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin 3 \theta-\sin \theta$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{x^{2}+100}, \quad x=10 \tan \theta$$

Use inverse functions where necessary to solve the equation. $$\tan ^{2} x-\tan x-2=0$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\cos 6 x+\cos 2 x$$

Verify the identity. $$\cos (n \pi+\theta)=(-1)^{n} \cos \theta, \quad n \text { is an integer. }$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{49-x^{2}}, \quad x=7 \sin \theta$$

(a) verify the identity and (b) determine whether the identity is true for the given value of \(x\). Explain. $$\frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x}, \quad x=0$$

Use inverse functions where necessary to solve the equation. $$\sec ^{2} x-6 \tan x+4=0$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin x+\sin 5 x$$

Verify the identity. $$\sin (n \pi+\theta)=(-1)^{n} \sin \theta, \quad n \text { is an integer. }$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{64-x^{2}}, \quad x=8 \cos \theta$$

Use inverse functions where necessary to solve the equation. $$\sec ^{2} x+\tan x-3=0$$

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

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