Chapter 9

Describe the transformation of the graph of \(f\) represented by the function \(g\). \(f(x)=\cos x, g(x)=\cos 6(x-\pi)+9\)

Rewrite each function. Justify your answers. a. Write \(\sin 3 x\) as a function of \(\sin x\). b. Write \(\cos 3 x\) as a function of \(\cos x\). c. Write \(\tan 3 x\) as a function of \(\tan x\).

Your school's marching band is performing at halftime during a football game. In the last formation, the band members form a circle 100 feet wide in the center of the field. You start at a point on the circle 100 feet from the goal line, march \(300^{\circ}\) around the circle, and then walk toward the goal line to exit the field. How far from the goal line are you at the point where you leave the circle?

\(f(x)=4 \csc x\); vertical stretch by a factor of 2 and a reflection in the \(x\)-axis

Graph the function. \(g(x)=-\cos x+3\)

Solve the equation. Check your solution(s). \(1-\frac{9}{x-2}=-\frac{7}{2}\)

Graph the function. \(g(x)=-\sin x-5\)

MODELING WITH MATHEMATICS Katoomba Scenic Railway in Australia is the steepest railway in the world. The railway makes an angle of about 52° with the ground. The railway extends horizontally about 458 feet. What is the height of the railway

Use the interactive unit circle tool at BigIdeasMath.com to describe all values of \(\theta\) for each situation. a. \(\sin \theta>0, \cos \theta<0\), and \(\tan \theta>0\) b. \(\sin \theta>0, \cos \theta<0\), and \(\tan \theta<0\)

Graph the function. \(g(x)=-\sin \frac{1}{2} x-2\)

Solve the equation. Check your solution(s). \(\frac{2 x-3}{x+1}=\frac{10}{x^2-1}+5\)

CRITICAL THINKING Write \(\tan \theta\) as the ratio of two other trigonometric functions. Use this ratio to explain why \(\tan 90^{\circ}\) is undefined but \(\cot 90^{\circ}=0\).

REASONING You are standing on a bridge 140 feet above the ground. You look down at a car traveling away from the underpass. The distance \(d\) (in feet) the car is from the base of the bridge can be modeled by \(d=140 \tan \theta\). Graph the function. Describe what happens to \(\theta\) as \(d\) increases.

Graph the function. \(g(x)=-\cos 2 x+1\)

MODELING WITH MATHEMATICS The Duquesne Incline in Pittsburgh, Pennsylvania, has an angle of elevation of 30°. The track has a length of about 800 feet. Find the height of the incline.

USING TOOLS You use a video camera to pan up the Statue of Liberty. The height \(h\) (in feet) of the part of the Statue of Liberty that can be seen through your video camera after time \(t\) (in seconds) can be modeled by \(h=100 \tan \frac{\pi}{36} t\). Graph the function using a graphing calculator. What viewing window did you use? Explain.

Graph the function. \(g(x)=-\sin (x-\pi)+4\)

MODELING WITH MATHEMATICS You are standing on the Grand View Terrace viewing platform at Mount Rushmore, 1000 feet from the base of the monument. Not drawn to scale 24° b 1000 ft a. You look up at the top of Mount Rushmore at an angle of 24°. How high is the top of the monument from where you are standing? Assume your eye level is 5.5 feet above the platform. b. The elevation of the Grand View Terrace is 5280 feet. Use your answer in part (a) to fi nd the elevation of the top of Mount Rushmore.

A line with slope \(m\) passes through the origin. An angle \(\theta\) in standard position has a terminal side that coincides with the line. Use a trigonometric function to relate the slope of the line to the angle.

MODELING WITH MATHEMATICS You are standing 120 feet from the base of a 260-foot building. You watch your friend go down the side of the building in a glass elevator. a. Write an equation that gives the distance \(d\) (in feet) your friend is from the top of the building as a function of the angle of elevation \(\theta\). b. Graph the function found in part (a). Explain how the graph relates to this situation.

Graph the function. \(g(x)=-\cos (x+\pi)-2\)

WRITING Write a real-life problem that can be solved using a right triangle. Then solve your problem.

Your friend claims that the only solution to the trigonometric equation \(\tan \theta=\sqrt{3}\) is \(\theta=60^{\circ}\). Is your friend correct? Explain your reasoning.

Graph the function. \(g(x)=-4 \cos \left(x+\frac{\pi}{4}\right)-1\)

MATHEMATICAL CONNECTIONS The Tropic of Cancer is the circle of latitude farthest north of the equator where the Sun can appear directly overhead. It lies \(23.5^{\circ}\) north of the equator, as shown. \begin{tabular}{l|l} Tropic of & North Pole \\ Cancer \\ equator & South Pole \end{tabular} a. Find the circumference of the Tropic of Cancer using 3960 miles as the approximate radius of Earth. b. What is the distance between two points on the Tropic of Cancer that lie directly across from each other?

MAKING AN ARGUMENT Your friend states that it is not possible to write a cosecant function that has the same graph as \(y=\sec x\). Is your friend correct? Explain your reasoning.

Graph the function. \(g(x)=-5 \sin \left(x-\frac{\pi}{2}\right)+3\)

The latitude of a point on Earth is the degree measure of the shortest arc from that point to the equator. For example, the latitude of point \(P\) in the diagram equals the degree measure of \(\operatorname{arc} P E\). At what latitude \(\theta\) is the circumference of the circle of latitude at \(P\) half the distance around the equator?

Which of the following is a point where the maximum value of the graph of \(y=-4 \cos \left(x-\frac{\pi}{2}\right)\) occurs? (A) \(\left(-\frac{\pi}{2}, 4\right)\) (B) \(\left(\frac{\pi}{2}, 4\right)\) (C) \((0,4)\) (D) \((\pi, 4)\)

Find all real zeros of the polynomial function. \(f(x)=x^4+2 x^3+x^2+8 x-12\)

ABSTRACT REASONING Use trigonometric identities to rewrite \(a \sec b x\) in terms of \(\cos b x\). Use your results to explain the relationship between the local maximums and minimums of the cosine and secant functions.

PROBLEM SOLVING You measure the angle of elevation from the ground to the top of a building as 32°. When you move 50 meters closer to the building, the angle of elevation is 53°. What is the height of the building?

Find all real zeros of the polynomial function. \(f(x)=x^5+4 x^4-14 x^3-14 x^2-15 x-18\)

THOUGHT PROVOKING Use a graphing calculator to graph the function $$ y=\frac{1}{2}\left(\tan \frac{x}{2}+\cot \frac{x}{2}\right) . $$ Use your graph to write a trigonometric identity involving this function. Explain your reasoning.

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\).. f(x)=3 \sin x \text {; translation } 2 \text { units up and } \pi \text { units right }

MAKING AN ARGUMENT Your friend claims it is possible to draw a right triangle so the values of the cosine function of the acute angles are equal. Is your friend correct? Explain your reasoning.

Graph the function. \(f(x)=2(x+3)^2(x-1)\)

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\).. \(f(x)=\cos 2 \pi x\), translation 4 units down and 3 units left

Graph the function. \(f(x)=\frac{1}{3}(x-4)(x+5)(x+9)\)

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\).. \(f(x)=\frac{1}{3} \cos \pi x\); translation 1 unit down, followed by a reflection in the line \(y=-1\)

Graph the function. \(f(x)=x^2(x+1)^3(x-2)\)

5 years to seconds

The height \(h\) (in feet) of a swing above the ground can be modeled by the function \(h=-8 \cos \theta+10\), where the pivot is 10 feet above the ground, the rope is 8 feet long, and \(\theta\) is the angle that the rope makes with the vertical. Graph the function. What is the height of the swing when \(\theta\) is \(45^{\circ}\) ?

12 pints to gallons

5.6 meters to millimeters

The average wind speed \(s\) (in miles per hour) in the Boston Harbor can be approximated by $$ s=3.38 \sin \frac{\pi}{180}(t+3)+11.6 $$ where \(t\) is the time in days and \(t=0\) represents January 1. Use a graphing calculator to graph the function. On which days of the year is the average wind speed 10 miles per hour? Explain your reasoning.

The water depth \(d\) (in feet) for the Bay of Fundy can be modeled by \(d=35-28 \cos \frac{\pi}{6.2} t\), where \(t\) is the time in hours and \(t=0\) represents midnight. Use a graphing calculator to graph the function. At what time(s) is the water depth 7 feet? Explain.

Find the average rate of change of each function over the interval \(0

Consider the functions \(y=\sin (-x)\) and \(y=\cos (-x)\) a. Construct a table of values for each equation using the quadrantal angles in the interval \(-2 \pi \leq x \leq 2 \pi\) b. Graph each function. c. Describe the transformations of the graphs of the parent functions.

You are riding a Ferris wheel that turns for 180 seconds. Your height \(h\) (in feet) above the ground at any time \(t\) (in seconds) can be modeled by the equation $$ h=85 \sin \frac{\pi}{20}(t-10)+90 $$ a. Graph the function. b. How many cycles does the Ferris wheel make in 180 seconds? c. What are your maximum and minimum heights?