Chapter 7

Solve the equation by first using a Sum-to-Product Formula. $$\sin 5 \theta-\sin 3 \theta=\cos 4 \theta$$

Verify the identity. $$\cot ^{2} \theta \cos ^{2} \theta=\cot ^{2} \theta-\cos ^{2} \theta$$

Write the expression in terms of sine only. $$\sin x+\cos x$$

Write the product as a sum. $$\cos x \sin 4 x$$

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

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

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

Write the product as a sum. $$\cos 5 x \cos 3 x$$

Total Internal Reflection When light passes from a more-dense to a less-dense medium- from glass to air, for example- the angle of refraction predicted by Snell's Law (sce Exercise 57 ) can be \(90^{\circ}\) or larger. In this case the light beam is actually reflected back into the denser medium. This phenomenon, called total internal reflection, is the principle behind fiber optics. Set \(\theta_{2}=90^{\circ}\) in Snell's Law, and solve for \(\theta_{1}\) to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. (Note that the index of refraction from glass to air is the reciprocal of the index from air to glass.)

Use a graphing device to find the solutions of the equation, correct to two decimal places. $$\cos x=\frac{x}{3}$$

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

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

Write the product as a sum. $$3 \cos 4 x \cos 7 x$$

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) \(F=0 \quad\) (new moon) (b) \(F=0.25\) (a crescent moon) (c) \(F=0.5\) (first or last quarter) (d) \(F=1 \quad\) (full moon)

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

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

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

Write the product as a sum. $$11 \sin \frac{x}{2} \cos \frac{x}{4}$$

Equations and ldentities 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.

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

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

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

Write the sum as a product. $$\sin 5 x+\sin 3 x$$

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

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

Verify the identity. $$\frac{1+\tan ^{2} u}{1-\tan ^{2} u}=\frac{1}{\cos ^{2} u-\sin ^{2} u}$$

Write the sum as a product. $$\sin x-\sin 4 x$$

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

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

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

Write the sum as a product. $$\cos 4 x-\cos 6 x$$

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 576.) 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 ft away?

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

Write the sum as a product. $$\cos 9 x+\cos 2 x$$

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 \((y=0)\).

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

Write the sum as a product. $$\sin 2 x-\sin 7 x$$

(a) Graph the function and make a conjecture, then (b) prove that your conjecture is true. $$y=\sin ^{2}\left(x+\frac{\pi}{4}\right)+\sin ^{2}\left(x-\frac{\pi}{4}\right)$$

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?

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

Write the sum as a product. $$\sin 3 x+\sin 4 x$$

(a) Graph the function and make a conjecture, then (b) prove that your conjecture is true. $$y=-\frac{1}{2}[\cos (x+\pi)+\cos (x-\pi)]$$

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

Find the value of the product or sum. $$2 \sin 52.5^{\circ} \sin 97.5^{\circ}$$

What makes the equation \(\sin (\cos x)=0\) different from all the other equations we've looked at in this section? Find all solutions of this equation.

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

Find the value of the product or sum. $$3 \cos 37.5^{\circ} \cos 7.5^{\circ}$$

Adding an Echo A digital delay device echoes an input signal by repeating it a fixed length of time after it is received. If such a device receives the pure note \(f_{1}(t)=5 \sin t\) and echoes the pure note \(f_{2}(t)=5 \cos t,\) then the combined sound is \(f(t)=f_{1}(t)+f_{2}(t)\) (a) Graph \(y=f(t)\) and observe that the graph has the form of a sine curve \(y=k \sin (t+\phi)\) (b) Find \(k\) and \(\phi\)

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

Find the value of the product or sum. $$\cos 37.5^{\circ} \sin 7.5^{\circ}$$