Chapter 8

\(57-62\) . Use a graphing device to find the solutions of the equation, correct to two decimal places. $$ \sin 2 x=x $$

\(55-60\) Write the product as a sum. $$ \cos x \sin 4 x $$

Write the expression in terms of sine only. $$ 5(\sin 2 x-\cos 2 x) $$

Verify the identity. $$ \frac{\sin x-1}{\sin x+1}=\frac{-\cos ^{2} x}{(\sin x+1)^{2}} $$

\(55-60\) Write the product as a sum. $$ \cos 5 x \cos 3 x $$

Write the expression in terms of sine only. $$ 3 \sin \pi x+3 \sqrt{3} \cos \pi x $$

Verify the identity. $$ \frac{\sin w}{\sin w+\cos w}=\frac{\tan w}{1+\tan w} $$

\(57-62\) . Use a graphing device to find the solutions of the equation, correct to two decimal places. $$ 2^{\sin x}=x $$

\(55-60\) Write the product as a sum. $$ 3 \cos 4 x \cos 7 x $$

(a) Express the function in terms of sine only. (b) Graph the function. $$ g(x)=\cos 2 x+\sqrt{3} \sin 2 x $$

Phases of the Moon As the moon revolves around the earth, the side that faces the earth is usually just partially illuminated by the sun. The phases of the moon describe how much of the surface appears to be in sunlight. An astronomical measure of phase is given by the fraction \(F\) of the lunar disc that is lit. When the angle between the sun, earth, and moon is \(\theta\left(0 \leq \theta \leq 360^{\circ}\right),\) then $$ F=\frac{1}{2}(1-\cos \theta) $$ Determine the angles \(\theta\) that correspond to the following phases: (a) \(F=0\) (new moon) (b) \(F=0.25\) (a crescent moon) (c) \(F=0.5 \quad\) (first or last quarter) (d) \(F=1\) (full moon)

Verify the identity. $$ \frac{(\sin t+\cos t)^{2}}{\sin t \cos t}=2+\sec t \csc t $$

\(57-62\) . Use a graphing device to find the solutions of the equation, correct to two decimal places. $$ \sin x=x^{3} $$

\(55-60\) Write the product as a sum. $$ 11 \sin \frac{x}{2} \cos \frac{x}{4} $$

(a) Express the function in terms of sine only. (b) Graph the function. $$ f(x)=\sin x+\cos x $$

Equations and Identities Which of the following statements is true? A. Every identity is an equation. B. Every equation is an identity. Give examples to illustrate your answer. Write a short paragraph to explain the difference between an equation and an identity.

Verify the identity. $$ \sec t \csc t(\tan t+\cot t)=\sec ^{2} t+\csc ^{2} t $$

\(57-62\) . Use a graphing device to find the solutions of the equation, correct to two decimal places. $$ \frac{\cos x}{1+x^{2}}=x^{2} $$

\(61-66\) Write the sum as a product. $$ \sin 5 x+\sin 3 x $$

Let \(g(x)=\cos x .\) Show that \(\frac{g(x+h)-g(x)}{h}=-\cos x\left(\frac{1-\cos h}{h}\right)-\sin x\left(\frac{\sin h}{h}\right)\)

\(57-62\) . Use a graphing device to find the solutions of the equation, correct to two decimal places. $$ \cos x=\frac{1}{2}\left(e^{x}+e^{-x}\right) $$

\(61-66\) Write the sum as a product. $$ \sin x-\sin 4 x $$

Show that if \(\beta-\alpha=\pi / 2\) , then $$ \sin (x+\alpha)+\cos (x+\beta)=0 $$

Verify the identity. $$ \frac{1+\sec ^{2} x}{1+\tan ^{2} x}=1+\cos ^{2} x $$

Range of a Projectile If a projectile is fired with velocity \(v_{0}\) at an angle \(\theta,\) then its range, the horizontal distance it travels (in feet), is modeled by the function $$ R(\theta)=\frac{v_{0}^{2} \sin 2 \theta}{32} $$ (See page \(614 .\) ) If \(v_{0}=2200 \mathrm{ft} / \mathrm{s}\) , what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 \(\mathrm{ft}\) away?

\(61-66\) Write the sum as a product. $$ \cos 4 x-\cos 6 x $$

Verify the identity. $$ \frac{\sec x}{\sec x-\tan x}=\sec x(\sec x+\tan x) $$

Damped Vibrations The displacement of a spring vibrating in damped harmonic motion is given by $$ y=4 e^{-3 t} \sin 2 \pi t $$ Find the times when the spring is at its equilibrium position \((y=0) .\)

\(61-66\) Write the sum as a product. $$ \cos 9 x+\cos 2 x $$

(a) If \(L\) is a line in the plane and \(\theta\) is the angle formed by the line and the \(x\) -axis as shown in the figure, show that the slope \(m\) of the line is given by $$ m=\tan \theta $$ (b) Let \(L_{1}\) and \(L_{2}\) be two nonparallel lines in the plane with slopes \(m_{1}\) and \(m_{2},\) respectively. Let \(\psi\) be the acute angle formed by the two lines (see the following figure). Show that $$ \tan \psi=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$ (c) Find the acute angle formed by the two lines $$ \begin{aligned} y=\frac{1}{3} X+1 & \text { and } \quad y=-\frac{1}{2} x-3 \end{aligned} $$ (d) Show that if two lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other. [Hint: First find an expression for \(\cot \psi . ]\)

Verify the identity. $$ \frac{\sec x+\csc x}{\tan x+\cot x}=\sin x+\cos x $$

Hours of Daylight In Philadelphia the number of hours of daylight on day \(t\) (where \(t\) is the number of days after January 1 ) is modeled by the function $$ L(t)=12+2.83 \sin \left(\frac{2 \pi}{365}(t-80)\right) $$ (a) Which days of the year have about 10 hours of daylight? (b) How many days of the year have more than 10 hours of daylight?

\(61-66\) Write the sum as a product. $$ \sin 2 x-\sin 7 x $$

Verify the identity. $$ \sec v-\tan v=\frac{1}{\sec v+\tan v} $$

\(61-66\) Write the sum as a product. $$ \sin 3 x+\sin 4 x $$

Verify the identity. $$ \frac{\sin A}{1-\cos A}-\cot A=\csc A $$

\(67-72\). Find the value of the product or sum. $$ 2 \sin 52.5^{\circ} \sin 97.5^{\circ} $$

Verify the identity. $$ \frac{\sin x+\cos x}{\sec x+\csc x}=\sin x \cos x $$

\(67-72\). Find the value of the product or sum. $$ 3 \cos 37.5^{\circ} \cos 7.5^{\circ} $$

Verify the identity. $$ \frac{1-\cos x}{\sin x}+\frac{\sin x}{1-\cos x}=2 \csc x $$

Interference Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled by \(f_{1}(t)=C \sin \omega t\) and \(f_{2}(t)=C \sin (\omega t+\alpha) .\) The two sound waves interfere to produce a single sound modeled by the sum of these functions $$ f(t)=C \sin \omega t+C \sin (\omega t+\alpha) $$ (a) Use the Addition Formula for Sine to show that \(f\) can be written in the form \(f(t)=A \sin \omega t+B \cos \omega t,\) where \(A\) and \(B\) are constants that depend on \(\alpha\) . (b) Suppose that \(C=10\) and \(\alpha=\pi / 3 .\) Find constants \(k\) and \(\phi\) so that \(f(t)=k \sin (\omega t+\phi)\)

\(67-72\). Find the value of the product or sum. $$ \cos 37.5^{\circ} \sin 7.5^{\circ} $$

Verify the identity. $$ \frac{\csc x-\cot x}{\sec x-1}=\cot x $$

Addition Formula for sine In the text we proved only the Addition and Subtraction Formulas for Cosine. Use these formulas and the cofunction identities $$ \sin x=\cos \left(\frac{\pi}{2}-x\right) $$ $$\cos x=\sin \left(\frac{\pi}{2}-x\right)$$ to prove the Addition Formula for sine. \([\text { Hint: To get started, }\) use the first cofunction identity to write $$ \begin{aligned} \sin (s+t) &=\cos \left(\frac{\pi}{2}-(s+t)\right) \\ &=\cos \left(\left(\frac{\pi}{2}-s\right)-t\right) \end{aligned} $$ and use the Subtraction Formula for Cosine. \(]\)

\(67-72\). Find the value of the product or sum. $$ \sin 75^{\circ}+\sin 15^{\circ} $$

Verify the identity. $$ \frac{\csc ^{2} x-\cot ^{2} x}{\sec ^{2} x}=\cos ^{2} x $$

Addition Formula for Tangent Use the Addition Formulas for Cosine and Sine to prove the Addition Formula for Tangent. [Hint: Use $$ \tan (s+t)=\frac{\sin (s+t)}{\cos (s+t)} $$ and divide the numerator and denominator by \(\cos s \cos t ]\)

\(67-72\). Find the value of the product or sum. $$ \cos 25^{\circ}-\cos 195^{\circ} $$

Verify the identity. $$ \tan ^{2} u-\sin ^{2} u=\tan ^{2} u \sin ^{2} u $$

\(67-72\). Find the value of the product or sum. $$ \cos \frac{\pi}{12}+\cos \frac{5 \pi}{12} $$