Chapter 9

With your calculator in radian mode, work Exercises in order. Let \(n\) represent the number of letters in your last name. Find an approximation for \(\cos (s+2 n \pi)\).

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=\frac{2}{3} \cos \left(x+\frac{\pi}{2}\right)$$

The highest mountain peak in the world is Mt. Everest, located in the Himalayas. The height of this enormous mountain was determined in 1856 by surveyors using trigonometry long before the peak was first climbed in \(1953 .\) At an altitude of 14.545 feet on a different mountain, the straight-line distance to the peak of Mt. Everest is 27.0134 miles and its angle of elevation is \(\theta=5.82^{\circ} .\) (a) Approximate the height (in feet) of Mt. Everest. (b) In the actual measurement, Mt. Everest was over 100 miles away and the curvature of Earth had to be taken into account. Would the curvature of Earth make the peak appear taller or shorter than it actually is?

Give the reference angle for each angle measure. $$\frac{7 \pi}{6}$$

Begin by reproducing the graph in RiGuRE as. Keep in mind that for each of the four points labeled in the figure, \(r=1 .\) For each quadrantal angle, identify the appropriate values of \(x, y,\) and \(r\) to find the indicated function value. If it is undefined, say so. Check your answers with a calculator in degree mode. $$\sec \left(-540^{\circ}\right)$$

Give an expression that generates all angles co terminal with each angle. Let n represent any integer. $$45^{\circ}$$

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=4 \cos \left(\frac{1}{2} x+\frac{\pi}{2}\right)$$

Graph each function over a one-period interval. $$y=-2 \tan \frac{1}{4} x$$

Cloud Ceiling The U.S. Weather Bureau defines a cloud ceiling as the altitude of the lowest clouds that cover more than half the sky. To determine a cloud ceiling. a powerful searchlight projects a circle of light vertically onto the bottom of the cloud. An observer sights the circle of light in the crosshairs of a tube called a clinometer. A pendant hanging vertically from the tube and resting on a protractor gives the angle of elevation. Find the cloud ceiling if the searchlight is located 1000 feet from the observer and the angle of elevation is \(30.0^{\circ}\) as measured with a clinometer at eye height exactly 6 feet. (Assume three significant digits.)

Give the reference angle for each angle measure. $$\frac{4 \pi}{3}$$

Give an expression that generates all angles co terminal with each angle. Let n represent any integer. $$-90^{\circ}$$

With your calculator in radian mode, work Exercises in order. Let \(s\) represent the number of letters in your last name. Find an approximation for sin \(s\).

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=-\cos \left[\frac{2}{3}\left(x-\frac{\pi}{3}\right)\right]$$

Graph each function over a one-period interval. $$y=3 \tan \frac{1}{2} x$$

Give the reference angle for each angle measure. $$-\frac{7 \pi}{6}$$

Give an expression that generates all angles co terminal with each angle. Let n represent any integer. $$-135^{\circ}$$

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=2-\sin \left(3 x-\frac{\pi}{5}\right)$$

Graph each function over a one-period interval. $$y=\frac{1}{2} \cot 4 x$$

Give an expression that generates all angles co terminal with each angle. Let n represent any integer. $$\frac{\pi}{4}$$

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=-1+\frac{1}{2} \cos (2 x-3 \pi)$$

Graph each function over a one-period interval. $$y=-\frac{1}{2} \cot 2 x$$

Highway curves are sometimes banked so that the outside of the curve is slightly elevated or inclined above the inside of the curve, as shown in the figure. This inclination is called the superelevation. It is important that both the curve's radius and its superelevation be correct for a given speed limit. The relationship between a car's velocity \(v\) in feet per second, the safe radius \(r\) of the curve in feet, and the superelevation \(\theta\) in degrees is modeled by $$ r=\frac{v^{2}}{4.5+32.2 \tan \theta} $$ (a) A curve has a speed limit of 66 feet per second \((45 \mathrm{mph})\) and a superelevation of \(\theta=3^{\circ} .\) Approximate the safe radius \(r\) (b) Find \(r\) if \(\theta=5^{\circ}\) and \(v=66\) (c) Make a conjecture about how increasing \(\theta\) with \(v=66\) affects the safe radius \(r\) (d) Find \(v\) if \(r=1150\) and \(\theta=2.1^{\circ}\)

Give an expression that generates all angles coterminal with each angle. Let \(n\) represent any integer. $$\frac{\pi}{6}$$

Explain how the reference angle is used to find values of the trigonometric functions for an angle in quadrant III.

Begin by reproducing the graph in RiGuRE as. Keep in mind that for each of the four points labeled in the figure, \(r=1 .\) For each quadrantal angle, identify the appropriate values of \(x, y,\) and \(r\) to find the indicated function value. If it is undefined, say so. Check your answers with a calculator in degree mode. $$\cot 1800^{\circ}$$

Graph each function over a two-period interval. $$y=\tan (2 x-\pi)$$

Give an expression that generates all angles coterminal with each angle. Let \(n\) represent any integer. $$-\frac{3 \pi}{4}$$

Find exact values of the six trigonometric functions for each angle by hand. Do not use a calculator. $$300^{\circ}$$

Graph each function over a two-period interval. $$y=\tan \left(\frac{x}{2}+\pi\right)$$

A solar cell converts the energy of sunlight directly into electrical energy. The amount of energy a cell produces depends on its area. Suppose a solar cell is hexagonal, as shown in the figure. Express its area in terms of \(\sin \theta\) and any side \(x .\) (Hint: Consider one of the six equilateral triangles from the hexagon. See the figure.)

Give an expression that generates all angles coterminal with each angle. Let \(n\) represent any integer. $$-\frac{7 \pi}{6}$$

Find exact values of the six trigonometric functions for each angle by hand. Do not use a calculator. $$315^{\circ}$$

Graph each function over a two-period interval. $$y=\cot \left(3 x+\frac{\pi}{4}\right)$$

Convert each degree measure to radians. Leave answers as rational multiples of \(\pi .\) $$60^{\circ}$$

Find exact values of the six trigonometric functions for each angle by hand. Do not use a calculator. $$405^{\circ}$$

Use the appropriate reciprocal identity to find each firnction value. Rarionalize denominators when applicable. $$\sec \theta, \text { given that } \cos \theta=\frac{2}{3}$$

Graph each function over a two-period interval. $$y=\cot \left(2 x-\frac{3 \pi}{2}\right)$$

Convert each degree measure to radians. Leave answers as rational multiples of \(\pi .\) $$90^{\circ}$$

Find exact values of the six trigonometric functions for each angle by hand. Do not use a calculator. $$420^{\circ}$$

Use the appropriate reciprocal identity to find each firnction value. Rarionalize denominators when applicable. $$\sec \theta, \text { given that } \cos \theta=\frac{5}{8}$$

Graph each function over a two-period interval. $$y=1+\tan x$$

Convert each degree measure to radians. Leave answers as rational multiples of \(\pi .\) $$150^{\circ}$$

Find exact values of the six trigonometric functions for each angle by hand. Do not use a calculator. $$\frac{11 \pi}{6}$$

Use the appropriate reciprocal identity to find each firnction value. Rarionalize denominators when applicable. $$\csc \theta, \text { given that } \sin \theta=-\frac{3}{7}$$

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\csc \theta, \text { given that } \sin \theta=-\frac{8}{43}$$

Graph each function over a two-period interval. $$y=-2+\tan x$$

Convert each degree measure to radians. Leave answers as rational multiples of \(\pi .\) $$270^{\circ}$$

Find exact values of the six trigonometric functions for each angle by hand. Do not use a calculator. $$\frac{5 \pi}{3}$$

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\cot \theta, \text { given that } \tan \theta=5$$

Graph each function over a two-period interval. $$y=1-\cot x$$