Chapter 10

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$2 \cos ^{2} x+2 \cos x=1$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\sin \theta(\csc \theta-\sin \theta)$$

Verify that each equation is an identity. $$\frac{\tan (A+B)-\tan B}{1+\tan (A+B) \tan B}=\tan A$$

Use a calculator to give each real-number value of \(y .\) $$y=\arccos 0.44624593$$

Use an identity to write each expression as a single trigonometric function value. $$\frac{\sin 158.2^{\circ}}{1+\cos 158.2^{\circ}}$$

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$\cos ^{2} x-2 \cos x+3=0$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$(\sec \theta+\csc \theta)(\cos \theta-\sin \theta)$$

Verify that each equation is an identity. $$\frac{\sin (x-y)}{\sin (x+y)}=\frac{\tan x-\tan y}{\tan x+\tan y}$$

Verify that each equation is an identity. $$\frac{2 \cos 2 \alpha}{\sin 2 \alpha}=\cot \alpha-\tan \alpha$$

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\csc \theta \sec \theta \tan \theta$$

Verify that each equation is an identity. $$\frac{\cos (A-B)}{\sin (A+B)}=\frac{1+\cot A \cot B}{\cot A+\cot B}$$

Draw by hand the graph of each inverse function. $$y=\csc ^{-1} x$$

Verify that each equation is an identity. $$\frac{1+\cos 2 x}{\sin 2 x}=\cot x$$

If \(\sin \theta=x\) and \(\theta\) is in quadrant IV, find an expression for \(\sec \theta\) in terms of \(x\).

Verify that each equation is an identity. $$\sec ^{2} \frac{x}{2}=\frac{2}{1+\cos x}$$

]Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$2 \sin \theta=1-2 \cos \theta$$

Perform indicated operation and simplify the result. $$\cot \theta+\frac{1}{\cot \theta}$$

Verify that each equation is an identity. $$\sec 2 x=\frac{1+\tan ^{2} x}{1-\tan ^{2} x}$$

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$\sin ^{2} \theta-\cos \theta=0$$

Perform indicated operation and simplify the result. $$\frac{\sec x}{\csc x}+\frac{\csc x}{\sec x}$$

Solve each problem. Back Stress If a person bends at the waist with a straight back, making an angle of \(\theta\) degrees with the horizontal, then the force \(F\) exerted on the back muscles can be modeled by the equation $$F=\frac{0.6 W \sin \left(\theta+90^{\circ}\right)}{\sin 12^{\circ}}$$ where \(W\) is the weight of the person. (Source: Metcalf, H., Topics in Classical Biophysics, Prentice- Hall.) (a) Calculate \(F\) when \(W=170\) pounds and \(\theta=30^{\circ}\) (b) Use an identity to show that \(F\) is approximately equal to \(2.9 \mathrm{W} \cos \theta\) (c) For what value of \(\theta\) is \(F\) maximum?

Verify that each equation is an identity. $$\cos 2 \theta=\frac{2-\sec ^{2} \theta}{\sec ^{2} \theta}$$

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$\cos ^{2} \theta+\sin \theta=0$$

Perform indicated operation and simplify the result. $$\tan s(\cot s+\csc s)$$

Draw by hand the graph of each inverse function. $$y=\cot ^{-1} 2 x$$

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

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$\tan ^{2} \theta+2 \sec \theta-3=0$$

Perform indicated operation and simplify the result. $$\cos \beta(\sec \beta+\csc \beta)$$

Voltage of a Circuit When the two voltages $$V_{1}=30 \sin 120 \pi t$$ and $$V_{2}=40 \cos 120 \pi t$$are applied to the same circuit, the resulting voltage \(V\) will equal their sum. (Source: Bell, D., Fundamentals of Electric Circuits, Second Edition, Reston Publishing Company.) (a) Graph \(V=V_{1}+V_{2}\) over the interval \(0 \leq t \leq 0.05\) (b) Use the graph to estimate values for \(a\) and \(\phi\) so that \(V=a \sin (120 \pi t+\phi)\) (c) Use identities to verify that your expression for \(V\) is valid.

Draw by hand the graph of each inverse function. Explain why attempting to find \(\sin ^{-1} 1.003\) on your calculator will result in an error message.

Verify that each equation is an identity. $$\tan s+\cot s=2 \csc 2 s$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos \theta+1=0$$

Perform indicated operation and simplify the result. $$\frac{1}{\csc ^{2} \theta}+\frac{1}{\sec ^{2} \theta}$$

Sound Waves Sound is a result of waves applying pressure to a person's eardrum. For a pure sound wave radiating outward in a spherical shape, the trigonometric function $$P=\frac{a}{r} \cos \left(\frac{2 \pi r}{\lambda}-c t\right)$$ can be used to model the sound pressure \(P\) at a radius of \(r\) feet from the source, where \(t\) is time in seconds, \(\lambda\) is length of the sound wave in feet, \(c\) is speed of sound in feet per second, and \(a\) is maximum sound pressure at the source measured in pounds per square foot. (Source: Beranek, L.., Noise and Vibration Control, Institute of Noise Control Engineering. Washington, DC.) Let \(\lambda=4.9\) feet and \(c=1026\) feet per second. (IMAGE CANNOT COPY) (a) Let \(a=0.4\) pound per square foot. Graph the sound pressure at a distance \(r=10\) feet from its source over the interval \(0 \leq t \leq 0.05 .\) Describe \(P\) at this distance. (b) Now let \(a=3\) and \(t=10 .\) Graph the sound pressure for \(0 \leq r \leq 20 .\) What happens to the pressure \(P\) as the radius \(r\) increases? (c) Suppose a person stands at a radius \(r\) so that $$r=n \lambda$$ where \(n\) is a positive integer. Use the difference identity for cosine to simplify \(P\) in this situation.

Verify that each equation is an identity. $$\cos x=\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\tan \theta+1=0$$

Perform indicated operation and simplify the result. $$\frac{1}{\sin \alpha-1}-\frac{1}{\sin \alpha+1}$$

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

Verify that each equation is an identity. $$\sin ^{2} \frac{x}{2}=\frac{\tan x-\sin x}{2 \tan x}$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$3 \csc x-2 \sqrt{3}=0$$

Perform indicated operation and simplify the result. $$\frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}$$

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

Verify that each equation is an identity. $$\frac{\cot \alpha-\tan \alpha}{\cot \alpha+\tan \alpha}=\cos 2 \alpha$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cot x+\sqrt{3}=0$$

Perform indicated operation and simplify the result. $$\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}$$

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

Verify that each equation is an identity. $$\cos 2 x=\frac{1-\tan ^{2} x}{1+\tan ^{2} x}$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$6 \sin ^{2} \theta+\sin \theta=1$$

Perform indicated operation and simplify the result. $$(1+\sin t)^{2}+\cos ^{2} t$$

Give the exact real number value of each expression. Do not use a calculator. $$\cos ^{-1}\left(\cos \left(-\frac{\pi}{6}\right)\right)$$