Chapter 5

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arctan \(\frac{7}{2}\)

Use a graphing utility to graph the function (include two full periods). Graph the corresponding reciprocal function in the same viewing window. Describe and compare the graphs. \(y=-2 \sec 4 x\)

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\sin 10^{\circ}\) (b) \(\cos 80^{\circ}\)

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to decimal degree form. Round your answer to three decimal places, if necessary. $$85^{\circ} 18^{\prime} 30^{\prime \prime}$$

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. \(\tan ^{-1}\left(-\frac{95}{7}\right)\)

Find the values of the six trigonometric functions of \(\theta\). Constraint \(\theta\) lies in Quadrant II. \(\theta\) lies in Quadrant III. \(\sin \theta < 0\) \(\cot \theta < 0\) \(0 \leq \theta \leq \pi\) \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\) \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\) \(\pi \leq \theta \leq 2 \pi\) Function Value $$\cos \theta=-\frac{4}{5}$$

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\tan 18.5^{\circ}\) (b) \(\cot 71.5^{\circ}\)

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to decimal degree form. Round your answer to three decimal places, if necessary. $$-408^{\circ} 16^{\prime} 25^{\prime \prime}$$

When an airplane leaves the runway, its angle of climb is \(18^{\circ}\) and its speed is 275 feet per second. Find the plane's altitude after 1 minute.

Use a graphing utility to graph the function (include two full periods). Graph the corresponding reciprocal function in the same viewing window. Describe and compare the graphs. \(y=\frac{1}{3} \sec \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)\)

Find the values of the six trigonometric functions of \(\theta\). Constraint \(\theta\) lies in Quadrant II. \(\theta\) lies in Quadrant III. \(\sin \theta < 0\) \(\cot \theta < 0\) \(0 \leq \theta \leq \pi\) \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\) \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\) \(\pi \leq \theta \leq 2 \pi\) Function Value $$\tan \theta=-\frac{15}{8}$$

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\sec 42^{\circ} 12^{\prime}\) (b) \(\csc 48^{\circ} 7^{\prime} 30^{\prime \prime}\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=-4 \sin x \end{aligned}$$

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to decimal degree form. Round your answer to three decimal places, if necessary. $$-125^{\circ} 36^{\prime \prime}$$

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\cos 8^{\circ} 50^{\prime} 25^{\prime \prime}\) (b) \(\sec 56^{\circ} 10^{\prime \prime}\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=\sin \frac{x}{3} \end{aligned}$$

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to decimal degree form. Round your answer to three decimal places, if necessary. $$330^{\circ} 25^{\prime \prime}$$

Compare the graph of the function with the graph of \(f(x)=\arcsin x\) \(g(x)=\arcsin (-x)\)

Solving a Trigonometric Equation Graphically In Exercises \(35-40,\) use a graph of the function to approximate the solution of the equation on the interval \([-2 \pi, 2 \pi]\) \(\tan x=1\)

Find the values of the six trigonometric functions of \(\theta\). Constraint \(\theta\) lies in Quadrant II. \(\theta\) lies in Quadrant III. \(\sin \theta < 0\) \(\cot \theta < 0\) \(0 \leq \theta \leq \pi\) \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\) \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\) \(\pi \leq \theta \leq 2 \pi\) Function Value $$\sec \theta=-2$$

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\cot \frac{\pi}{16}\) (b) \(\tan \frac{\pi}{8}\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{array}{l} f(x)=\cos \pi x \\ g(x)=1+\cos \pi x \end{array}$$

Find the difference of the angles. Write your answer in \(\mathbf{D}^{\circ} \mathbf{M}^{\prime} \mathbf{S}^{\prime \prime}\) form. $$51^{\circ} 22^{\prime} 30^{\prime \prime} \text { and } 38^{\circ} 17^{\prime} 15^{\prime \prime}$$

The height of an outdoor basketball backboard is \(12 \frac{1}{2}\) feet, and the backboard casts a shadow \(17 \frac{1}{3}\) feet long. (a) Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities. (b) Use a trigonometric function to write an equation involving the unknown angle of elevation. (c) Find the angle of elevation of the sun.

Compare the graph of the function with the graph of \(f(x)=\arcsin x\) \(g(x)=-\arcsin x\)

Use a graph of the function to approximate the solution of the equation on the interval \([-2 \pi, 2 \pi]\) \(\cot x=-1\)

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\sec 1.54\) (b) \(\cos 1.25\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{array}{l} f(x)=4 \sin x \\ g(x)=4 \sin x-1 \end{array}$$

Find the difference of the angles. Write your answer in \(\mathbf{D}^{\circ} \mathbf{M}^{\prime} \mathbf{S}^{\prime \prime}\) form. $$120^{\circ} 45^{\prime} 29^{\prime \prime} \text { and } 12^{\circ} 36^{\prime} 3^{\prime \prime}$$

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=-\frac{1}{2} \sin \frac{x}{2}\\\ &g(x)=2 \sin \frac{x}{4} \end{aligned}$$

Find the difference of the angles. Write your answer in \(\mathbf{D}^{\circ} \mathbf{M}^{\prime} \mathbf{S}^{\prime \prime}\) form. $$48^{\circ} 18^{\prime} \text { and } 25^{\circ} 16^{\prime} 59^{\prime \prime}$$

Compare the graph of the function with the graph of \(f(x)=\arccos x\) \(g(x)=\arccos x-5\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=2 \cos 2 x\\\ &g(x)=-\cos 4 x \end{aligned}$$

A park is showing a movie on the lawn. The base of the screen is 6 feet off the ground and the screen is 22 feet high. (See figure.) (a) Find the angles of elevation to the top of the screen from distances of 15 feet and 100 feet. (b) You are lying on the ground and the angle of elevation to the top of the screen is \(42^{\circ} .\) How far are you from the screen?

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (Include two full periods.) Make a conjecture about the functions. $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=\cos \left(x-\frac{\pi}{2}\right) \end{aligned}$$

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to \(\mathbf{D}^{\circ} \mathbf{M}^{\prime} \mathbf{S}^{\prime \prime}\) form. $$280.6^{\circ}$$

Complete the identity. $$\csc \theta=\frac{1}{\square}$$

Compare the graph of the function with the graph of \(f(x)=\arccos x\) \(g(x)=\arccos (-x)-3\)

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (Include two full periods.) Make a conjecture about the functions. $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=\cos \left(x+\frac{3 \pi}{2}\right) \end{aligned}$$

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to \(\mathbf{D}^{\circ} \mathbf{M}^{\prime} \mathbf{S}^{\prime \prime}\) form. $$-115.8^{\circ}$$

Marine Transportation A ship leaves port at noon and has a bearing of \(S 29^{\circ} \mathrm{W}\). The ship sails at 20 knots. How many nautical miles south and how many nautical miles west does the ship travel by 6: 00 p.M.?

Complete the identity. $$\sec \theta=\frac{1}{\square}$$

Compare the graph of the function with the graph of \(f(x)=\arctan x\) \(g(x)=\arctan x+1\)

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (Include two full periods.) Make a conjecture about the functions. $$\begin{aligned} &f(x)=\cos x\\\ &g(x)=-\sin \left(x-\frac{\pi}{2}\right) \end{aligned}$$

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to \(\mathbf{D}^{\circ} \mathbf{M}^{\prime} \mathbf{S}^{\prime \prime}\) form. $$-345.12^{\circ}$$

Aviation An airplane flying at 600 miles per hour has a bearing of \(52^{\circ}\). After flying for 1.5 hours, how far north and how far east has the plane traveled from its point of departure?

Complete the identity. $$\cot \theta=\frac{1}{\square}$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (Include two full periods.) Make a conjecture about the functions. $$\begin{aligned} &f(x)=\cos x\\\ &g(x)=-\cos (x-\pi) \end{aligned}$$

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to \(\mathbf{D}^{\circ} \mathbf{M}^{\prime} \mathbf{S}^{\prime \prime}\) form. $$490.75^{\circ}$$

A ship is 45 miles east and 30 miles south of port. The captain wants to sail directly to port. What bearing should the captain take?