Chapter 5

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. $$\sin \theta=\frac{\sqrt{3}}{2}$$

Evaluate the given expressions to four decimal places with a calculator. $$\cot ^{-1}(-3.2)$$

Convert each angle from radians to degrees. $$3 \pi$$

Toy sales at a department store \(t\) months after the month of December can be modeled by the function \(s(t)=210+150 \cos \frac{\pi}{6} t,\) where \(s\) is in thousands of dollars. What is the value of \(s(4),\) and what does it represent? Find the period of this function.

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. $$\sin \theta=\frac{\sqrt{2}}{2}$$

Evaluate the given expressions to four decimal places with a calculator. $$\cot ^{-1}(-1.8)$$

Fill in the given table with the missing information. Approximate all nonexact anstoers to four decimal places. $$\begin{array}{|c|c|c|c|c|} \hline \text { Quadrant } & \sin t & \cos t & \tan t \\ \hline II & & \ & -1 \\ \hline \end{array}$$

Convert each angle from radians to degrees. $$-\frac{5 \pi}{4}$$

Waves Find the frequency of an AM radio station whose wave form is given by \(f(t)=A \sin \left(1.2 \times 10^{6} \pi t\right),\) where \(A\) is some positive constant.

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. $$\cos \theta=\frac{1}{2}$$

In this set of exercises, you will use inverse trigonometric functions to study real-world problems. Round all answers to four decimal places. A surveyor finds that \(\tan \theta=\frac{19}{5},\) where \(\theta\) is the angle that a straight line from the ground makes with the top of a platform. Find \(\theta\) in degrees. (Picture cant copy)

Fill in the given table with the missing information. Approximate all nonexact anstoers to four decimal places. $$\begin{array}{|c|c|c|c|c|} \hline \text { Quadrant } & \sin t & \cos t & \tan t \\ \hline II & & -\frac{1}{2} & \\ \hline \end{array}$$

Convert each angle from radians to degrees. $$\frac{\pi}{180}$$

The voltage in an electrical circuit is given by the function $$V(t)=\sin \left(3 t-\frac{\pi}{2}\right)$$ What is the smallest non-negative value of \(t\) at which the voltage is equal to \(0 ?\)

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. Let \(\alpha\) be an acute angle with \(\sin \alpha=a\). Find \(\csc \alpha\) and \(\cos \left(90^{\circ}-\alpha\right)\) in terms of \(a\)

Convert each angle from radians to degrees. $$\frac{\pi}{45}$$

The charge on an electrical capacitor is given by the function $$q(t)=Q \cos \left(3 t+\frac{\pi}{12}\right)$$ where \(Q\) is a constant.What is the smallest positive value of \(t\) at which the charge is equal to \(q(0) ?\)

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. Let \(\beta\) be an acute angle with \(\cos \beta=b .\) Find \(\sec \beta\) and \(\sin \left(90^{\circ}-\beta\right)\) in terms of \(b\)

In this set of exercises, you will use inverse trigonometric functions to study real-world problems. Round all answers to four decimal places. A 15 -foot pole is to be stabilized by two wires of equal length, one on each side of the pole. One end of each wire is to be attached to the top of the pole; the other end is to be staked to the ground at an acute angle \(\theta\) with respect to the horizontal. Because of considerations, the ratio of the length of either wire to the height of the pole is to be no more than \(\frac{4}{3} .\) What is the limiting value of \(\theta\) in degrees? Is this limiting value a maximum value of \(\theta\) or a minimum value of \(\theta ?\) Explain.

Fill in the given table with the missing information. Approximate all nonexact anstoers to four decimal places. $$\begin{array}{|c|c|c|c|c|} \hline \text { Quadrant } & \sin t & \cos t & \tan t \\ \hline IV &-0.6 && \\ \hline \end{array}$$

Convert each angle from radians to degrees. $$-\frac{2 \pi}{5}$$

The form of a sound wave is given by the function $$f(x)=25 \sin (4 x+\pi)$$ Find the amplitude, period, and frequency of the wave.

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. A 20 -foot-long piece of wire is attached to the top of a pole at one end and nailed to the ground at the other end. If the wire makes an angle of \(30^{\circ}\) with the ground, find the height of the pole.

The form of a light wave is given by the function \(f(x)=3 \cos \left(5 x-\frac{\pi}{2}\right)+4\) What are the minimum and maximum values of this function, and what is the smallest positive value of \(x\) at which the function attains its minimum value?

Convert each angle from radians to degrees. $$-\frac{\pi}{5}$$

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. A pole 10 feet high is supported by a taut wire staked to the ground at a \(35^{\circ}\) angle. How long is the wire?

Fill in the given table with the missing information. Approximate all nonexact anstoers to four decimal places. $$\begin{array}{|c|c|c|c|c|} \hline \text { Quadrant } & \sin t & \cos t & \tan t \\ \hline II & \ &-\frac{5}{13}\\\ \hline \end{array}$$

A plane approaching an airport is told to maintain a holding pattern before being given clearance to land. The formula $$d(t)=80 \sin (0.75 t)+200$$ can be used to determine the distance of the plane in miles from the airport at time \(t .\) To what maximum distance from the airport does the plane travel while it is in the holding pattern?

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$25^{\circ} 45^{\prime} 15^{\prime \prime}$$

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. From the basket of a hot-air balloon 100 feet above the ground, the angle of depression of a point \(A\) on the ground is \(10.5^{\circ} .\) What is the distance from point \(A\) to the basket of the balloon?

In this set of exercises, you will use inverse trigonometric functions to study real-world problems. Round all answers to four decimal places. The pitch of a roof is its slope, which is given as \(\frac{\text { rise }}{\text { run }}\). If the pitch of a roof is \(\frac{2}{5},\) what acute angle does it make with the horizontal? Express your answer in radians.

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$32^{\circ} 30^{\prime} 30^{\prime \prime}$$

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. The angle of depression of a small boat near the coast with respect to the top of a lighthouse is \(8^{\circ} .\) If the lighthouse is 120 feet high, what is the distance from the top of the lighthouse to the boat?

The position of a block that is attached to one end of a spring oscillates according to the formula \(d=5 \sin 2 t\) for \(t\) in the interval \(\left[-\frac{\pi}{4}, \frac{\pi}{4}\right] .\) Express \(t\) as a function of \(d\), and state the domain of your function.

For Exercises \(61-72,\) fill in the given table with the missing information. A pproximate all nonexact answers to four decimal places. $$ \begin{array}{|r|c|c|c|c|} \hline & \text { Quadrant } & \sin t & \cos t & \tan t \\ \hline 61 . & \mathrm{I} & \frac{1}{2} & & \\ \hline 62 . & \mathrm{IV} & & \frac{1}{2} & \\ \hline 63 . & \mathrm{III} & & & 1 \\ \hline 64 . & \mathrm{II} & & & -1 \\ \hline 65 . & \mathrm{II} & & -\frac{1}{2} & \\ \hline 66 . & \mathrm{II} & & -\frac{\sqrt{3}}{2} & \\ \hline 67 . & \mathrm{IV} & -0.6 & & \\ \hline 68 . & \mathrm{III} & -0.8 & & \\ \hline 69 . & \mathrm{II} & & -\frac{5}{13} & \\ \hline 70 . & \mathrm{IV} & & \frac{12}{13} & \\ \hline 71 . & \mathrm{IV} & & & -2 \\ \hline 72 . & \mathrm{II} & & & \\ \hline \end{array} $$

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$16^{\circ} 24^{\prime} 45^{\prime \prime}$$

The function $$P(t)=50 \sin \frac{2 \pi}{23} t+50$$ is used in biorhythm theory to predict an individual's physical potential (as a percentage of the maximum) on a particular day, with \(t=0\) corresponding to birth. (a) What is the period of the function? (b) What is an individual's physical potential on her or his third birthday (day \(1095) ?\)

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. The angle of elevation of the top of a tower with respect to a certain point on the ground is \(38^{\circ}\) From a point 15 feet closer to the tower, the angle of elevation is \(42^{\circ} .\) Find the height of the tower.

A sound wave has the form \(y=2 \cos \left(3 x-\frac{\pi}{4}\right)\) for \(x\) in the interval \(\left[\frac{\pi}{12}, \frac{5 \pi}{12}\right] .\) Express \(x\) as a function of \(y\) and state the domain of your function.

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$40^{\circ} 20^{\prime} 30^{\prime \prime}$$

How can you show graphically that \(\cos \left(\frac{\pi}{2}-x\right)=\sin x ?\)

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. From a certain point on a 10 -foot-high platform, the angle of depression of the base of a building is \(15^{\circ},\) and the angle of elevation of the top of the building is \(55^{\circ} .\) How high is the building?

The horizontal range of a projectile that is fired with an initial velocity \(v_{0}\) at an acute angle \(\theta\) with respect to the horizontal is given by $$R=\frac{\left(v_{0}\right)^{2} \sin 2 \theta}{g}$$ where \(g\) is the gravitational constant, 9.8 meters per second per second \(\left(\mathrm{m} / \mathrm{sec}^{2}\right) .\) If \(v_{0}=30\) meters per second, find the angle at which the projectile must be fired if it is to have a horizontal range of 80 meters. Express your answers in degrees.

Use the negative-angle identities to compute the exact value of each of the given trigonometric functions. $$\sin \left(-\frac{2 \pi}{3}\right)$$

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$120^{\circ} 50^{\prime} 15^{\prime \prime}$$

Using appropriate graphs, show that \(\cos (-x)=\cos x\)

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. A 16 -foot-long ladder leans against a vertical wall. The base of the ladder makes an angle of \(68^{\circ}\) with the lawn on which the foot of the ladder rests. How high above the surface of the lawn is the top of the ladder?

Use the negative-angle identities to compute the exact value of each of the given trigonometric functions. $$\cos \left(-\frac{10 \pi}{3}\right)$$

Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$150^{\circ} 40^{\prime} 20^{\prime \prime}$$

Using appropriate graphs, show that \(\sin (-x)=-\sin x\)