Chapter 10

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.) $$\frac{1}{\sec ^{2} \theta-1}$$

Suppose that A and B are angles in standand position. Use the given information to find (a) \(\sin (A+B)\), (b) \(\sin (A-B)\), (c) \(\tan (A+B)\), (d) \(\tan (A-B)\), (e) the quadrant of \(A+B\), and ( \(f\) ) the quadrant of \(A-B\). Do not use a calculator. $$\sin A=\frac{3}{5}, \sin B=-\frac{12}{13}, \quad 0

Use a calculator to give each value of \(\theta\) in decimal degrees. $$\theta=\cos ^{-1}(-0.13348816)$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\cos \frac{x}{2}, \text { given cot } x=-3 \text { and } \frac{\pi}{2} < x < \pi$$

How can you determine that the equation \((\cos \theta-4)(\sin \theta+5)=0\) has no solutions by simply knowing the ranges of the sine and cosine functions?

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.) $$\cot ^{2} \theta\left(1+\tan ^{2} \theta\right)$$

Solve each problem. Inducing Voltage A coil of wire rotating in a magnetic field induces a voltage $$ V=20 \sin \left(\frac{\pi t}{4}-\frac{\pi}{2}\right) $$ where \(t\) is time in seconds. Find the least positive time required to produce each voltage. (a) 0 (b) \(10 \sqrt{3}\)

Suppose that A and B are angles in standand position. Use the given information to find (a) \(\sin (A+B)\), (b) \(\sin (A-B)\), (c) \(\tan (A+B)\), (d) \(\tan (A-B)\), (e) the quadrant of \(A+B\), and ( \(f\) ) the quadrant of \(A-B\). Do not use a calculator. $$\cos A=-\frac{8}{17}, \cos B=-\frac{3}{5}, \quad \pi

Use a calculator to give each value of \(\theta\) in decimal degrees. $$\theta=\arccos (-0.39876459)$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\cos x, \text { given } \cos 2 x=-\frac{5}{12} \text { and } \frac{\pi}{2 }< x < \pi$$

For individual or group investigation. Write the equation \(\tan ^{3} x=3 \tan x\) with 0 on one side.

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.) $$\sin ^{2} \theta\left(\csc ^{2} \theta-1\right)$$

Solve each problem. Nautical Mile The British nautical mile is defined as the length \(L\) of a minute of arc on any meridian. Since Earth is flatter at its poles, the nautical mile varies with latitude and, in feet, is $$ L=6077-31 \cos 2 \theta $$ where \(\theta\) is the latitude in degrees. (See the figure.) (Figure can't copy) (a) Find the latitude between \(0^{\circ}\) and \(90^{\circ}\) at which the nautical mile is 6074 feet. (b) At what latitude between \(0^{\circ}\) and \(90^{\circ}\) (inclusive) is the nautical mile 6108 feet? (c) In the United States, the nautical mile is defined everywhere as 6080.2 feet. At what latitude between \(0^{\circ}\) and \(90^{\circ}\) does this agree with the British nautical mile?

Suppose that A and B are angles in standand position. Use the given information to find (a) \(\sin (A+B)\), (b) \(\sin (A-B)\), (c) \(\tan (A+B)\), (d) \(\tan (A-B)\), (e) the quadrant of \(A+B\), and ( \(f\) ) the quadrant of \(A-B\). Do not use a calculator. $$\cos A=-\frac{15}{17}, \sin B=\frac{4}{5}, \quad \frac{\pi}{2}

Use a calculator to give each value of \(\theta\) in decimal degrees. $$\theta=\arcsin 0.77900016$$

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\sin x, \text { given } \cos 2 x=\frac{2}{3} \text { and } \pi < x < \frac{3 \pi}{2}$$

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.) $$(\sec \theta-1)(\sec \theta+1)$$

Solve each problem. Ear Pressure from a Pure Tone A pure tone has a constant frequency and amplitude, and it sounds rather dull and uninteresting. The pressures caused by pure tones on the eardrum are sinusoidal. The change in pressure \(P\) in pounds per square foot on a person's eardrum from a pure tone at time \(t\) in seconds can be modeled by the equation $$ P=A \sin (2 \pi f t+\phi) $$ where \(f\) is the frequency in cycles per second and \(\phi\) is the phase angle. When \(P\) is positive, there is an increase in pressure and the eardrum is pushed inward; when \(P\) is negative, there is a decrease in pressure and the eardrum is pushed outward. (a) Middle C has frequency 261.63 cycles per second. Graph this tone with \(A=0.004\) and \(\phi=\frac{\pi}{7}\) in the window \([0,0.005]\) by \([-0.005,0.005]\) (b) Determine analytically the values of \(t\) for which \(P=0\) on \([0,0.005],\) and support your answers graphically. (c) Determine graphically when \(P<0\) on \([0,0.005]\) (d) Would an eardrum hearing this tone be vibrating outward or inward when \(P<0 ?\)

Verify that each equation is an identity. $$\sin 2 x=2 \sin x \cos x$$

Answer each of the following.Consider the expression \(\tan \left(\frac{\pi}{2}+x\right)\) (a) Why can't we use the identity for \(\tan (A+B)\) to express it as a function of \(x\) alone? (b) Use the identity \(\tan \theta=\frac{\sin \theta}{\cos \theta}\) to rewrite the expression in terms of sine and cosine. (c) Use the result of part (b) to show that $$ \tan \left(\frac{\pi}{2}+x\right)=-\cot x $$

Use a calculator to give each value of \(\theta\) in decimal degrees. $$\theta=\csc ^{-1} 1.9422833$$

For individual or group investigation. Solve \(\tan ^{3} x=3 \tan x\) by first dividing each side by tan \(x\)

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.) $$(1-\cos \theta)(1+\sec \theta)$$

Verify that each equation is an identity. $$\sin \left(210^{\circ}+x\right)-\cos \left(120^{\circ}+x\right)=0$$

Use a calculator to give each value of \(\theta\) in decimal degrees. $$\theta=\cot ^{-1} 1.7670492$$

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.) $$\frac{\cos \theta+\sin \theta}{\sin \theta}$$

Solve each problem. Hearing Beats in Music Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of the slight variation in the frequency. This phenomenon can be seen on a graphing calculator. (a) Consider two tones with frequencies of 220 and \(223 \mathrm{Hz}\) and pressures \(P_{1}=0.005 \sin 440 \pi t \quad\) and \(\quad P_{2}=0.005 \sin 446 \pi t\) respectively. A graph of \(P_{1}+P_{2}\) as \(Y_{3}\) felt by an eardrum over the 1 -second interval \([0.15,1.15]\) is shown here. How many beats are there in 1 second? (Graph can't copy) (b) Repeat part (a) with frequencies of 220 and 216 (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given.

Verify that each equation is an identity. $$\sin (x+y)+\sin (x-y)=2 \sin x \cos y$$

Use a calculator to give each real-number value of \(y .\) $$y=\arctan 1.1111111$$

Use an identity to write each expression as a single trigonometric function value. $$\sqrt{\frac{1-\cos 40^{\circ}}{2}}$$

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 \(\left[0^{\circ}, 360^{\circ}\right)\) $$3 \sin ^{2} x-\sin x=2$$

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.) $$\frac{\cos ^{2} \theta-\sin ^{2} \theta}{\sin \theta \cos \theta}$$

Hearing Different Tones When a musical instrument creates a tone of \(110 \mathrm{Hz}\), it also creates tones at \(220,330\) \(440,550,660, \dots\) Hz. A small speaker cannot reproduce the \(110-\mathrm{Hz}\) vibration, but it can reproduce the higher frequencies, called the upper harmonics. The low tones can still be heard, because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is \(110 \mathrm{Hz}\), and this difference tone will be heard by a listener. We can model this phenomenon with a graphing calculator. (a) In the window \([0,0.03]\) by \([-1,1]\), graph the upper harmonics represented by the pressure $$ \begin{aligned} P=& \frac{1}{2} \sin [2 \pi(220) t]+\frac{1}{3} \sin [2 \pi(330) t] \\ &+\frac{1}{4} \sin [2 \pi(440) t] \end{aligned} $$ (b) Estimate all \(t\) -coordinates where \(P\) is maximum. (c) What does a person hear in addition to the frequencies of \(220,330,\) and \(440 \mathrm{Hz} ?\) (d) Graph the pressure produced by a speaker that can vibrate at \(110 \mathrm{Hz}\) and above in the window \([0,0.03]\) by \([-2,2]\) (Image can't copy)

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

Use a calculator to give each real-number value of \(y .\) $$y=\arcsin 0.81926439$$

Use an identity to write each expression as a single trigonometric function value. $$\sqrt{\frac{1+\cos 76^{\circ}}{2}}$$

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 \(\left[0^{\circ}, 360^{\circ}\right)\) $$9 \sin ^{2} x=6 \sin x+1$$

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.) $$\frac{1-\sin ^{2} \theta}{1+\cot ^{2} \theta}$$

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

Use a calculator to give each real-number value of \(y .\) $$y=\cot ^{-1}(-0.92170128)$$

Use an identity to write each expression as a single trigonometric function value. $$\sqrt{\frac{1-\cos 147^{\circ}}{1+\cos 147^{\circ}}}$$

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 \(\left[0^{\circ}, 360^{\circ}\right)\) $$\tan ^{2} x+4 \tan x+2=0$$

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.) $$\sec \theta-\cos \theta$$

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

Use an identity to write each expression as a single trigonometric function value. $$\sqrt{\frac{1+\cos 165^{\circ}}{1-\cos 165^{\circ}}}$$

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 \(\left[0^{\circ}, 360^{\circ}\right)\) $$3 \cot ^{2} x-3 \cot x=1$$

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.) $$\frac{1+\tan ^{2} \theta}{1+\cot ^{2} \theta}$$

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

Use a calculator to give each real-number value of \(y .\) $$y=\arcsin 0.92837781$$

Use an identity to write each expression as a single trigonometric function value. $$\frac{1-\cos 59.74^{\circ}}{\sin 59.74^{\circ}}$$