Chapter 10

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{\tan ^{2} t+1}{\tan t \csc ^{2} t}=\tan t$$

Write each expression as an algebraic expression in \(u, u>0\). $$\sec \left(\cos ^{-1} u\right)$$

Verify that equation is an identity. \(\frac{\tan ^{2} t-1}{\sec ^{2} t}=\frac{\tan t-\cot t}{\tan t+\cot t}\)

Write each expression as an algebraic expression in \(u, u>0\). $$\cot \left(\tan ^{-1} u\right)$$

Verify that equation is an identity. \(\frac{1-\cos x}{1+\cos x}=\csc ^{2} x-2 \csc x \cot x+\cot ^{2} x\)

Write each expression as an algebraic expression in \(u, u>0\). $$\sin (\arccos u)$$

Verify that equation is an identity. \(\sin ^{2} \alpha \sec ^{2} \alpha+\sin ^{2} \alpha \csc ^{2} \alpha=\sec ^{2} \alpha\)

Write each expression as an algebraic expression in \(u, u>0\). $$\tan (\arccos u)$$

Verify that equation is an identity. \((\sec \alpha+\csc \alpha)(\cos \alpha-\sin \alpha)=\cot \alpha-\tan \alpha\)

Write each expression as an algebraic expression in \(u, u>0\). $$\cot (\arcsin u)$$

Verify that equation is an identity. \(\frac{1-\sin \theta}{1+\sin \theta}=\sec ^{2} \theta-2 \sec \theta \tan \theta+\tan ^{2} \theta\)

Write each expression as an algebraic expression in \(u, u>0\). $$\cos (\arcsin u)$$

Verify that equation is an identity. \(\frac{\sin \theta}{1-\cos \theta}-\frac{\sin \theta \cos \theta}{1+\cos \theta}=\csc \theta\left(1+\cos ^{2} \theta\right)\)

Write each expression as an algebraic expression in \(u, u>0\). $$\sin \left(\sec ^{-1} \frac{u}{2}\right)$$

Verify that equation is an identity. \(\frac{1+\sin \theta}{1-\sin \theta}-\frac{1-\sin \theta}{1+\sin \theta}=4 \tan \theta \sec \theta\)

Write each expression as an algebraic expression in \(u, u>0\). $$\cos \left(\tan ^{-1} \frac{3}{u}\right)$$

Verify that equation is an identity. \(\frac{1-\cos \theta}{1+\cos \theta}=2 \csc ^{2} \theta-2 \csc \theta \cot \theta-1\)

Write each expression as an algebraic expression in \(u, u>0\). $$\tan \left(\sin ^{-1} \frac{u}{\sqrt{u^{2}+2}}\right)$$

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{\cos ^{4} x-\sin ^{4} x}{\cos ^{2} x}=1-\tan ^{2} x$$

Verify that equation is an identity. \((2 \sin x+\cos x)^{2}+(2 \cos x-\sin x)^{2}=5\)

Write each expression as an algebraic expression in \(u, u>0\). $$\sec \left(\cos ^{-1} \frac{u}{\sqrt{u^{2}+5}}\right)$$

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{\csc t+1}{\csc t-1}=(\sec t+\tan t)^{2}$$

Verify that equation is an identity. \(\sin ^{2} x(1+\cot x)+\cos ^{2} x(1-\tan x)+\cot ^{2} x=\csc ^{2} x\)

Write each expression as an algebraic expression in \(u, u>0\). $$\sec \left(\operatorname{arccot} \frac{\sqrt{4-u^{2}}}{u}\right)$$

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{2\left(\sin x-\sin ^{3} x\right)}{\cos x}=\sin 2 x$$

Verify that equation is an identity. \(\sec x-\cos x+\csc x-\sin x-\sin x \tan x=\cos x \cot x\)

Write each expression as an algebraic expression in \(u, u>0\). $$\csc \left(\arctan \frac{\sqrt{9-u^{2}}}{u}\right)$$

Verify that equation is an identity. \(\sin ^{3} \theta+\cos ^{3} \theta=(\cos \theta+\sin \theta)(1-\cos \theta \sin \theta)\)

Solve each problem. A painting 3 feet high and 6 feet from the floor will cut off an angle $$\theta=\tan ^{-1}\left(\frac{3 x}{x^{2}+4}\right)$$ to an observer. Assume that the observer is \(x\) feet from the wall where the painting is displayed and that the eyes of the observer are 5 feet above the ground. (IMAGE CAN'T COPY). Find the value of \(\theta\) for each value of \(x\) to the nearest degree. (a) \(x=3\) (b) \(x=6\) (c) \(x=9\) (d) Derive the given formula for \(\theta\). (Hint: Use right triangles and the identity for \(\tan (\theta-\alpha) .)\) (e) Graph the function for \(\theta\) with a calculator, and determine the distance that maximizes the angle.

Solve each problem. Suppose an airplane flying faster than sound goes directly over you. Assume that the plane is flying at a constant altitude. At the instant you feel the sonic boom from the plane, the angle of elevation to the plane is $$\alpha=2 \arcsin \frac{1}{m}$$ where \(m\) is the Mach number of the plane's speed. (The Mach number is the ratio of the speed of the plane to the speed of sound.) Find \(\alpha\) to the nearest degree for each value of \(m\) (a) \(m=1.2\) (b) \(m=1.5\) (c) \(m=2\) (d) \(m=2.5\)

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{1}{\sec t-1}+\frac{1}{\sec t+1}=2 \cot t \csc t$$

The distance or displacement \(y\) of a weight attached to an oscillating spring from its natural position is modeled by \(y=4 \cos 2 \pi t,\) where \(t\) is time in seconds. Potential energy is the energy of position and is given by \(P=k y^{2},\) where \(k\) is a constant. The weight has the greatest potential energy when the spring is stretched the most. (a) Write \(P\) in terms of the cosine function. (b) Use an identity to write \(P\) in terms of \(\sin 2 \pi t\).

Two types of mechanical energy are kinetic energy and potential energy. Kinetic energy is the energy of motion, and potential energy is the energy of position. A stretched spring has potential energy, which is converted to kinetic energy when the spring is released. If the potential energy of a weight attached to a spring is $$P(t)=k \cos ^{2} 4 \pi t$$ where \(k\) is a constant and \(t\) is time in seconds, then its kinetic energy is given by $$K(t)=k \sin ^{2} 4 \pi t$$ The total mechanical energy \(E\) is given by the equation \(E(t)=P(t)+K(t).\) (a) If \(k=2,\) graph \(P, K,\) and \(E\) in the window \([0,0.5]\) by \([-1,3],\) with \(\mathrm{Xscl}=0.25\) and \(\mathrm{Yscl}=1 .\) Interpret the graph. (b) Make a table of \(K, P,\) and \(E,\) starting at \(t=0\) and incrementing by \(0.05 .\) Interpret the results. (c) Use a fundamental identity to derive a simplified expression for \(E(t)\)

Let the energy stored in the inductor be $$L(t)=3 \cos ^{2} 6,000,000 t$$ and the energy in the capacitor be $$C(t)=3 \sin ^{2} 6,000,000 t$$ where \(t\) is time in seconds. The total energy \(E\) in the circuit is given by \(E(t)=L(t)+C(t)\) (a) Graph \(L, C,\) and \(E\) in the window \(\left[0,10^{-6}\right]\) by \([-1,4],\) with \(\mathrm{Xscl}=10^{-7}\) and \(\mathrm{Yscl}=1 .\) Interpret the graph. (b) Make a table of \(L, C,\) and \(E,\) starting at \(t=0\) and incrementing by \(10^{-7}\). Interpret your results. (c) Use a fundamental identity to derive a simplified expression for \(E(t)\)