Chapter 10

Verify that equation is an identity. \(\frac{\cot \alpha+1}{\cot \alpha-1}=\frac{1+\tan \alpha}{1-\tan \alpha}\)

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(2 \sin ^{-1} \frac{1}{4}\right)$$

Write each expression as a product of trigonometric functions or values. $$\sin 9 B-\sin 3 B$$

Express solutions to the nearest hundredth. (Hint: In Exercise 83 , the equation has three solutions.) $$\sin ^{3} x+\sin x=1$$

Verify that equation is an identity. \(\frac{\sin ^{4} \alpha-\cos ^{4} \alpha}{\sin ^{2} \alpha-\cos ^{2} \alpha}=1\)

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(2 \arctan \frac{4}{3}\right)$$

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(2 \cos ^{-1} \frac{1}{4}\right)$$

Express solutions to the nearest hundredth. (Hint: In Exercise 83 , the equation has three solutions.) $$e^{x}=\sin x+3$$

Verify that equation is an identity. \(\cot s+\tan s=\sec s \csc s\)

Determining Wattage Amperage is a measure of the amount of electricity that is moving through a circuit, while voltage is a measure of the force pushing the electricity. The wattage \(W\) consumed by an electrical device can be determined by calculating the product of amperage \(I\) and voltage \(V .\) (Source: Wilcox, G. and C. Hesselberth, Electricity for Engineering Technology, Allyn \& Bacon.) (a) A household circuit has voltage $$V=163 \sin 120 \pi t$$ when an incandescent light bulb is turned on with amperage $$I=1.23 \sin 120 \pi t$$ Graph the wattage $$W=V I$$ that is consumed by the light bulb over the interval \(0 \leq t \leq 0.05\) (b) Determine the maximum and minimum wattages used by the light bulb. (c) Use identities to find values for \(a, c,\) and \(\omega\) so that $$ W=a \cos \omega t+c $$ (d) Check your answer in part (c) by graphing both expressions for \(W\) on the same coordinate axes. (e) Use the graph from part (a) to estimate the average watt. age used by the light. How many watts do you think this incandescent light bulb is rated for?

Give the exact real number value of each expression. Do not use a calculator. $$\sin \left(2 \cos ^{-1} \frac{1}{5}\right)$$

Daylight Hours in New Orleans The seasonal variation in length of daylight can be modeled by a sine function. For example, the daily number of hours \(h\) of daylight in New Orleans is approximated by $$h=\frac{35}{3}+\frac{7}{3} \sin \frac{2 x \pi}{365}$$ where \(x\) is the number of days after March 21 (disregarding leap year). (Source: Bushaw, Donald et al., A Sourcebook of Applications of School Mathematics, The Mathematical Association of America.) (a) On what date will there be about 14 hours of day. light? (b) What date has the least number of hours of daylight? (c) When will there be about 10 hours of daylight?

Verify that equation is an identity. \(\frac{\sin ^{2} \gamma}{\cos \gamma}=\sec \gamma-\cos \gamma\)

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(2 \tan ^{-1}(-2)\right)$$

Mach Number for a Plane An airplane flying faster than sound sends out sound waves that form a cone as illustrated in the figure. The cone intersects the ground to form a hyperbola. As this hyperbola passes over a particular point on the ground, a sonic boom is heard at that point. If \(\alpha\) is the angle at the vertex of the cone, then $$\sin \frac{\alpha}{2}=\frac{1}{m}$$ where \(m\) is the Mach number for the speed of the plane. (See Exercise 114 in the previous section.) We assume that \(m>1 .\) Find the measure of \(\alpha,\) in degrees, if \(m=1.5 .\) Round to the nearest tenth. GRAPH CANT COPY

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

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\tan \theta+\cot \theta=\sec \theta \csc \theta$$

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}\right)$$

The equation $$0.342 D \cos \theta+h \cos ^{2} \theta=\frac{16 D^{2}}{V^{2}}$$ is used to reconstruct accidents in which a vehicle vaults into the air after hitting an obstruction. \(V\) is the velocity (in feet per second) of the vehicle when it hits the obstruction, \(D\) is the distance (in feet) from the obstruction to the vehicle's landing point, and \(h\) is the difference in height (in feet) between the landing point and the takeoff point. Angle \(\theta\) is the takeoff angle-the angle between the horizontal and the path of the vehicle. Find \(\theta\) to the nearest degree if \(V=60, D=80,\) and \(h=2\)

Verify that equation is an identity. \(\frac{\cos \theta}{\sin \theta \cot \theta}=1\)

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

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\cos ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}\right)$$

Maximum Viewing Angle The bottom of a 10 -foot-high movie screen is located 2 feet above the eyes of the view ers, all of whom are sitting at the same level. A viewer seated 5 feet from the screen has the maximum viewing angle \(x\) determined by the equation $$\frac{\tan x+0.4}{1-0.4 \tan x}=2.4$$ Find the maximum viewing angle (to the nearest degree).

Verify that equation is an identity. \(\sin ^{4} \theta-\cos ^{4} \theta=2 \sin ^{2} \theta-1\)

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\tan ^{-1} \frac{3}{4}+\tan ^{-1} \frac{12}{5}\right)$$

moves along a straight line. The distance \(s\) that the particle is from a starting point at time \(t\) is $$s(t)=\sin t+2 \cos t$$ Find all values of \(t\) in \(\left[0, \frac{\pi}{2}\right)\) that satisfy each equation. (a) \(s(t)=\frac{2+\sqrt{3}}{2}\) (b) \(s(t)=\frac{3 \sqrt{2}}{2}\)

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

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\sin ^{-1} \frac{8}{17}+\tan ^{-1} \frac{4}{3}\right)$$

Verify that equation is an identity. \(\left(1-\cos ^{2} \alpha\right)\left(1+\cos ^{2} \alpha\right)=2 \sin ^{2} \alpha-\sin ^{4} \alpha\)

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\tan ^{-1} \frac{5}{12}-\tan ^{-1} \frac{3}{4}\right)$$

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

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

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\sin ^{-1} \frac{3}{5}+\cos ^{-1} \frac{5}{13}\right)$$

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

Give the exact real number value of each expression. Do not use a calculator. $$\sin \left(\sin ^{-1} \frac{1}{2}+\tan ^{-1}(-3)\right)$$

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

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

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\cos ^{-1} \frac{\sqrt{3}}{2}-\sin ^{-1}\left(-\frac{3}{5}\right)\right)$$

Verify that equation is an identity. \(\frac{\csc \theta+\cot \theta}{\tan \theta+\sin \theta}=\cot \theta \csc \theta\)

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

Use a calculator to find each value. $$\cos \left(\tan ^{-1} 0.5\right)$$

Verify that equation is an identity. \(\sec ^{4} x-\sec ^{2} x=\tan ^{4} x+\tan ^{2} x\)

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

Use a calculator to find each value. $$\sin \left(\cos ^{-1} 0.25\right)$$

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

Use a calculator to find each value. $$\tan (\arcsin 0.12251014)$$

Verify that equation is an identity. \(\frac{\sec ^{4} s-\tan ^{4} s}{\sec ^{2} s+\tan ^{2} s}=\sec ^{2} s-\tan ^{2} s\)

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

Use a calculator to find each value. $$\cot (\arccos 0.58236841)$$

Verify that equation is an identity. \(\frac{\tan s}{1+\cos s}+\frac{\sin s}{1-\cos s}=\cot s+\sec s \csc s\)