Chapter 10

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(r=2\) and \(\theta=1.6,\) find \(s\)

In \(26-33 :\) a. Rewrite each function value in terms of its cofunction. b. Find the exact value of the function value found in a. $$ \sin \left(-\frac{\pi}{4}\right) $$

In \(24-32,\) find the exact value of each expression. $$ \cos \left(\arcsin \left(-\frac{\sqrt{2}}{2}\right)\right) $$

A supporting cable runs from the ground to the top of a tree that is in danger of falling down. The tree is 18 feet tall and the cable makes an angle of \(\frac{2 \pi}{9}\) with the ground. Determine the length of the cable to the nearest tenth of a foot.

In \(26-33 :\) a. Rewrite each function value in terms of its cofunction. b. Find the exact value of the function value found in a. $$ \cos \frac{8 \pi}{3} $$

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \sin \theta=\frac{\sqrt{2}}{2} $$

An airplane climbs at an angle of \(\frac{\pi}{15}\) with the ground. When the airplane has reached an altitude of 500 feet: a. What is the distance in the air that the airplane has traveled? b. What is the horizontal distance that the airplane has traveled?

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(s=16\) and \(\theta=0.4,\) find \(r\)

$$ \tan \left(-\frac{5 \pi}{3}\right) $$

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \cos \theta=\frac{\sqrt{3}}{2} $$

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(r=4.2\) and \(s=21,\) find \(\theta\)

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \tan \theta=1 $$

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(r=6\) and \(\theta=\frac{2 \pi}{3},\) find \(s\)

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \sec \theta=-1 $$

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(s=18\) and \(\theta=\frac{6 \pi}{5},\) find \(r\)

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \csc \theta=-\sqrt{2} $$

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(\theta=6 \pi\) and \(r=1,\) find \(s\)

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \cot \theta=\sqrt{3} $$

Circle \(O\) has a radius of 1.7 inches. What is the length, in inches, of an arc intercepted by a central angle whose measure is 2 radians?

In a circle whose radius measures 5 feet, a central angle intercepts an are of length 12 feet. Find the radian measure of the central angle.

The central angle of circle \(O\) has a measure of 4.2 radians and it intercepts an arc whose length is 6.3 meters. What is the length, in meters, of the radius of the circle?

a. Restrict the domain of the cosecant function to form a one-to-one function that has an inverse function. Justify your domain. b. Is the restricted domain found in a the same as the restricted domain of the sine function? c. Find the range of the restricted cosecant function. d. Find the domain of the inverse cosecant function, that is, the arccosecant function. e. Find the range of the arcosecant function.

Complete the following table, expressing degree measures in radian measure in terms of \(\pi\) $$ \begin{array}{|c|c|c|c|c|c|c|}\hline \text { Degrees } 30^{\circ} & {45^{\circ}} & {60^{\circ}} & {90^{\circ}} & {180^{\circ}} & {270^{\circ}} & {360^{\circ}} \\ \hline \text { Radians } & {} & {} & {} \\\ \hline\end{array} $$

a. Restrict the domain of the cotangent function to form a one-to-one function that has an inverse function. Justify your domain. b. Is the restricted domain found in a the same as the restricted domain of the tangent function? c. Find the range of the restricted cotangent function. d. Find the domain of the inverse cotangent function, that is, the arccotangent function. e. Find the range of the arccotangent function.

The pendulum of a clock makes an angle of 2.5 radians as its tip travels 18 feet. What is the length of the pendulum?

Jennifer lives near the airport. An airplane approaching the airport flies at a constant altitude of 1 mile toward a point, \(P,\) above Jennifer's house. Let \(\theta\) be the measure of the angle of elevation of the plane and \(d\) be the horizontal distance from \(P\) to the airplane. a. Express \(\theta\) in terms of \(d\) . b. Find \(\theta\) when \(d=1\) mile and when \(d=0.5\) mile.

A wheel whose radius measures 16 inches is rotated. If a point on the circumference of the wheel moves through an arc of 12 feet, what is the measure, in radians, of the angle through which a spoke of the wheel travels?

The wheels on a bicycle have a radius of 40 centimeters. The wheels on a cart have a radius of 10 centimeters. The wheels of the bicycle and the wheels of the cart all make one complete revolution. a. Do the wheels of the bicycle rotate through the same angle as the wheels of the cart? Justify your answer. b. Does the bicvcle travel the same distance as the cart? Justify vour answer.

Latitude represents the measure of a central angle with vertex at the center of the earth, its initial side passing through a point on the equator, and its terminal side passing through the given location. (See the figure.) Cities A and \(\mathrm{B}\) are on a north-south line. City \(\mathrm{A}\) is located at \(30^{\circ} \mathrm{N}\) and City \(\mathrm{B}\) is located at \(52^{\circ} \mathrm{N}\) . If the radius of the earth is approximately \(6,400\) kilometers, find \(d\) , the distance between the two cities along the circumference of the earth. Assume that the earth is a perfect sphere.