Chapter 16

Simplify. $$\sec x \sin x$$

Prove each identity. $$\cos (2 \pi-x)=\cos x$$

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$1+\cot ^{2} x=\sec ^{2} x$$

Simplify. $$\sec x \sin x \cot x$$

Trajectories: An object thrown at an angle \(\theta\) and with an initial velocity of \(\nu_{0}\) follows the path given by $$y=x \tan \theta-\frac{16.1 x^{2}}{\nu_{0}^{2}} \sec ^{2} \theta \quad \text { feet }$$ If \(\nu_{0}=376 \mathrm{ft} / \mathrm{s}\) and \(\theta=35.5^{\circ},\) find \(y\) when \(x=125 \mathrm{ft}\).

Simplify. $$\csc \theta \tan \theta-\tan \theta \sin \theta$$

Prove each identity. $$\tan \left(360^{\circ}-\beta\right)=-\tan \beta$$

Simplify. $$\frac{\cos x}{\cot x \sin x}$$

Pendulum: The period \(T\) for a pendulum of length \(L,\) Fig. \(16-20,\) is approximately given by $$T \cong 2 \pi \sqrt{\frac{L}{g}\left(1+\frac{1}{4} \sin ^{2} \frac{\theta}{2}-\frac{9}{64} \sin ^{4} \frac{\theta}{2}\right)}$$ where \(g=32.2 \mathrm{ft} / \mathrm{s}^{2}\) and \(\theta\) is the angle between the pendulum and the vertical. For a pendulum of length \(1.25 \mathrm{ft}\), find \(T\) when \(\theta=7.83^{\circ} .\)

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$3 \sin (x / 2)-1=2 \sin ^{2}(x / 2)$$

Simplify. $$\cot \theta \tan ^{2} \theta \cos \theta$$

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$\sin x=\cos x$$

Simplify. $$\frac{\tan x\left(\csc ^{2} x-1\right)}{\sin x+\cot x \cos x}$$

Prove each identity. $$\frac{1+\tan x}{1-\tan x}=\tan \left(\frac{\pi}{4}+x\right)$$

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$4 \cos ^{2} x+4 \cos x=-1$$

Simplify. $$\frac{\sin \theta}{\cos \theta \tan \theta}$$

Prove each identity. $$\frac{1-\cos x}{\sin x}=\tan \frac{x}{2}$$

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$1+\sin x=\sin x \cos x+\cos x$$

Simplify. $$\frac{\sin ^{2} x+\cos ^{2} x}{1-\cos ^{2} x}$$

Prove each identity. $$\frac{1-\tan ^{2} \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}=\cos \theta$$

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$3 \tan x=4 \sin ^{2} x \tan x$$

Simplify. $$\frac{1}{\sec ^{2} x}+\frac{1}{\csc ^{2} x}$$

Shear Stress: The shear stress \(s_{s}\) on a cross section of a bar in tension, Fig. \(16-12,\) is related to the axial stress \(s_{x}\) by the formula $$ s_{s}=s_{x} \sin \theta \cos \theta $$ where \(\theta\) is the angle between the axis of the bar and the normal to the cross section. Use the double-angle identities to write this expression with just a single trigonometric ratio.

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$3 \sin x \tan x+2 \tan x=0$$

Simplify. $$\sin \theta(\csc \theta+\cot \theta)$$

Simplify. $$\csc x-\cot x \cos x$$

Project: Use the formula for the addition of a sine wave and a cosine wave to express each following expression as a single sine function. (a) \(y=47.2 \sin \omega t+64.9 \cos \omega t\) (b) \(y=8470 \sin \omega t+7360 \cos \omega t\) (c) \(y=1.83 \sin \omega t+2.74 \cos \omega t\) (d) \(y=84.2 \sin \omega t+74.2 \cos \omega t\)

Simplify. $$\frac{\sec x-\csc x}{1-\cot x}$$

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$\cos x \sin 2 x=0$$

Simplify. $$\frac{1}{1+\sin x}+\frac{1}{1-\sin x}$$

Simplify. $$\sec ^{2} x\left(1-\cos ^{2} x\right)$$

Simplify. $$\cos \theta \sec \theta-\frac{\sec \theta}{\cos \theta}$$

Simplify. $$\cot ^{2} x \sin ^{2} x+\tan ^{2} x \cos ^{2} x$$

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. Applications Writing: Explain in your own words the difference between a trigonometric identity and a trigonometric equation.

Prove each identity. (All identities in this chapter can be proven. ) $$\tan x \cos x=\sin x$$

Prove each identity. (All identities in this chapter can be proven. ) $$\tan x=\frac{\sec x}{\csc x}$$

Prove each identity. (All identities in this chapter can be proven. ) $$\frac{\sin x}{\csc x}+\frac{\cos x}{\sec x}=1$$

Prove each identity. (All identities in this chapter can be proven. ) $$\sin \theta=\frac{1}{\cot \theta \sec \theta}$$

Prove each identity. (All identities in this chapter can be proven. ) $$\left(\cos ^{2} \theta+\sin ^{2} \theta\right)^{2}=1$$

Prove each identity. (All identities in this chapter can be proven. ) $$\frac{\csc \theta}{\sec \theta}=\cot \theta$$

Prove each identity. (All identities in this chapter can be proven. ) $$\csc x-\sin x=\cot x \cos x$$

Prove each identity. (All identities in this chapter can be proven. ) $$1=(\csc x-\cot x)(\csc x+\cot x)$$

Prove each identity. (All identities in this chapter can be proven. ) $$\tan x=\frac{\tan x+\sin x}{1+\cos x}$$

Prove each identity. (All identities in this chapter can be proven. ) $$\frac{\sin \theta+1}{1-\sin \theta}=(\tan \theta+\sec \theta)^{2}$$

Prove each identity. (All identities in this chapter can be proven. ) $$\frac{1+\sin \theta}{1-\sin \theta}=\frac{1+\csc \theta}{\csc \theta-1}$$

Prove each identity. (All identities in this chapter can be proven. ) $$(\sec \theta-\tan \theta)(\tan \theta+\sec \theta)=1$$

Applications.In calculus, we take the derivative of tan \(x\) by taking the derivative of \(\sin x / \cos x,\) and get the expression $$ \frac{(\cos x)(\cos x)-(\sin x)(-\sin x)}{\cos ^{2} x} $$ Show that this expression is equal to sec \(^{2} x\)

Applications.Alternating Current: Given \(R=Z \cos \theta\) and \(X_{L}-X_{C}=Z \sin \theta,\) where \(R\) is the resistance, \(X_{L}\) is the inductive reactance and \(X_{C}=\) capacitive reactance, evaluate and simplify (a) \(R^{2}+\left(X_{L}-X_{C}\right)^{2}\) (b) \(\frac{X_{L}-X_{C}}{R}\)