Chapter 7

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\sin t, \cos t ; \quad\) quadrant II

(a) Prove that if \(f\) is periodic with period \(p,\) then 1\(/ f\) is also periodic with period \(p\) (b) Prove that cosecant and secant each have period 2\(\pi\) .

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\cos t, \sin t ; \quad\) quadrant IV

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\tan t, \sin t ; \quad\) quadrant IV

Lighthouse The beam from a lighthouse completes one rotation every two minutes. At time \(t\) , the distance \(d\) shown in the figure on the next page is $$d(t)=3 \tan \pi t$$ where \(t\) is measured in minutes and \(d\) in miles. (a) Find \(d(0.15), d(0.25),\) and \(d(0.45) .\) (b) Sketch a graph of the function \(d\) for \(0 \leq t<\frac{1}{2}\) (c) What happens to the distance \(d\) as \(t\) approaches \(\frac{1}{2} ?\)

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\tan t, \cos t ; \quad\) quadrant III

Determine an appropriate viewing rectangle for each function, and use it to draw the graph. $$ y=\sqrt{\tan 10 \pi x} $$

Length of a Shadow On a day when the sun passes directly overhead at noon, a six-foot-tall man casts a shadow of length $$S(t)=6\left|\cot \frac{\pi}{12} t\right|$$ where \(S\) is measured in feet and \(t\) is the number of hours since 6 A.M. (a) Find the length of the shadow at \(8 : 00\) A.M., noon, \(2 : 00 \mathrm{P}, \mathrm{M},\) and \(5 : 45 \mathrm{P.M}\) (b) Sketch a graph of the function \(S\) for \(0 < t < 12\). (c) From the graph determine the values of \(t\) at which the length of the shadow equals the man's height. To what time of day does each of these values correspond? (d) Explain what happens to the shadow as the time approaches 6 \(\mathrm{P} . \mathrm{M}\) . (that is, as \(t \rightarrow 12^{-} )\) .

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\sec t, \tan t ; \quad\) quadrant II

Graph \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$ f(x)=x, \quad g(x)=\sin x $$

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\csc t, \cot t ; \quad\) quadrant III

Graph \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$ f(x)=\sin x, \quad g(x)=\sin 2 x $$

Graph the three functions on a common screen. How are the graphs related? $$ y=x^{2}, \quad y=-x^{2}, \quad y=x^{2} \sin x $$

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\sin t, \sec t ; \quad\) quadrant IV

Graph the three functions on a common screen. How are the graphs related? $$ y=x, \quad y=-x, \quad y=x \cos x $$

Graph the three functions on a common screen. How are the graphs related? $$ y=\sqrt{x}, \quad y=-\sqrt{x}, \quad y=\sqrt{x} \sin 5 \pi x $$

Graph the three functions on a common screen. How are the graphs related? $$ y=\frac{1}{1+x^{2}}, \quad y=-\frac{1}{1+x^{2}}, \quad y=\frac{\cos 2 \pi x}{1+x^{2}} $$

Find the values of the trigonometric functions of \(t\) from the given information. \(\sin t=\frac{3}{5},\) terminal point of \(t\) is in quadrant II

Graph the three functions on a common screen. How are the graphs related? $$ y=\cos 3 \pi x, \quad y=-\cos 3 \pi x, \quad y=\cos 3 \pi x \cos 21 \pi x $$

Find the values of the trigonometric functions of \(t\) from the given information. \(\cos t=-\frac{4}{5},\) terminal point of \(t\) is in quadrant III

Graph the three functions on a common screen. How are the graphs related? $$ y=\sin 2 \pi x, \quad y=-\sin 2 \pi x, \quad y=\sin 2 \pi x \sin 10 \pi x $$

Find the values of the trigonometric functions of \(t\) from the given information. \(\sec t=3, \quad\) terminal point of \(t\) is in quadrant IV

Find the maximum and minimum values of the function. $$ y=\sin x+\sin 2 x $$

Find the values of the trigonometric functions of \(t\) from the given information. \(\tan t=\frac{1}{4},\) terminal point of \(t\) is in quadrant III

Find the maximum and minimum values of the function. $$ y=x-2 \sin x, 0 \leq x \leq 2 \pi $$

Find the values of the trigonometric functions of \(t\) from the given information. \(\tan t=-\frac{3}{4}, \quad \cos t>0\)

Find the maximum and minimum values of the function. $$ y=2 \sin x+\sin ^{2} x $$

Find the values of the trigonometric functions of \(t\) from the given information. \(\sec t=2, \quad \sin t<0\)

Find the maximum and minimum values of the function. $$ y=\frac{\cos x}{2+\sin x} $$

Find the values of the trigonometric functions of \(t\) from the given information. \(\sin t=-\frac{1}{4}, \quad \sec t<0\)

Find all solutions of the equation that lie in the interval \([0, \pi] .\) State each answer correct to two decimal places. $$ \cos x=0.4 $$

Find the values of the trigonometric functions of \(t\) from the given information. \(\tan t=-4, \quad \csc t>0\)

Find all solutions of the equation that lie in the interval \([0, \pi] .\) State each answer correct to two decimal places. $$ \tan x=2 $$

Determine whether the function is even, odd, or neither. $$ f(x)=x^{2} \sin x $$

Find all solutions of the equation that lie in the interval \([0, \pi] .\) State each answer correct to two decimal places. $$ \csc x=3 $$

Determine whether the function is even, odd, or neither. $$ f(x)=x^{2} \cos 2 x $$

Determine whether the function is even, odd, or neither. $$ f(x)=\sin x \cos x $$

A function \(f\) is given. (a) Is \(f\) even, odd, or neither? (b) Find the \(x\) -intercepts of the graph of \(f\) (c) Graph \(f\) in an appropriate viewing rectangle. (d) Describe the behavior of the function as \(x \rightarrow \pm \infty\) (e) Notice that \(f(x)\) is not defined when \(x=0 .\) What happens as \(x\) approaches 0\(?\) $$ f(x)=\frac{1-\cos x}{x} $$

Determine whether the function is even, odd, or neither. $$ f(x)=\sin x+\cos x $$

A function \(f\) is given. (a) Is \(f\) even, odd, or neither? (b) Find the \(x\) -intercepts of the graph of \(f\) (c) Graph \(f\) in an appropriate viewing rectangle. (d) Describe the behavior of the function as \(x \rightarrow \pm \infty\) (e) Notice that \(f(x)\) is not defined when \(x=0 .\) What happens as \(x\) approaches 0\(?\) $$ f(x)=\frac{\sin 4 x}{2 x} $$

Determine whether the function is even, odd, or neither. $$ f(x)=|x| \cos x $$

Height of a Wave As a wave passes by an offshore piling, the height of the water is modeled by the function $$h(t)=3 \cos \left(\frac{\pi}{10} t\right)$$ where \(h(t)\) is the height in feet above mean sea level at time \(t\) seconds. (a) Find the period of the wave. (b) Find the wave height, that is, the vertical distance between the trough and the crest of the wave.

Determine whether the function is even, odd, or neither. $$ f(x)=x \sin ^{3} x $$

Sound Vibrations \(\quad\) A tuning fork is struck, producing a pure tone as its tines vibrate. The vibrations are modeled by the function $$ v(t)=0.7 \sin (880 \pi t) $$ where \(v(t)\) is the displacement of the tines in millimeters at time \(t\) seconds. (a) Find the period of the vibration. (b) Find the frequency of the vibration, that is, the number of times the fork vibrates per second. (c) Graph the function \(v\) .

Determine whether the function is even, odd, or neither. $$ f(x)=x^{3}+\cos x $$

Blood Pressure Each time your heart beats, your bloo pressure first increases and then decreases as the heart rests between beats. The maximum and minimum blood pressures are called the systolic and diastolic pressures, respectively. Your blood pressure reading is written as systolic/diastolic. A reading of 120\(/ 80\) is considered normal. A certain person's blood pressure is modeled by the function $$ p(t)=115+25 \sin (160 \pi t) $$ where \(p(t)\) is the pressure in mmHg, at time \(t\) measured in minutes. (a) Find the period of \(p\) . (b) Find the number of heartbeats per minute. (c) Graph the function \(p\) . (d) Find the blood pressure reading. How does this compare to normal blood pressure?

Determine whether the function is even, odd, or neither. $$ f(x)=\cos (\sin x) $$

Variable Stars Variable stars are ones whose brightness varies periodically. One of the most visible is R Leonis; its brightness is modeled by the function $$b(t)=7.9-2.1 \cos \left(\frac{\pi}{156} t\right)$$ where \(t\) is measured in days. (a) Find the period of R Leonis. (b) Find the maximum and minimum brightness. (c) Graph the function \(b\) .

Harmonic Motion The displacement from equilibrium of an oscillating mass attached to a spring is given by \(y(t)=4 \cos 3 \pi t\) where \(y\) is measured in inches and \(t\) in seconds. Find the displacement at the times indicated in the table.

Compositions Involving Trigonometric Functions This exercise explores the effect of the inner function \(g\) on a composite function \(y=f(g(x)) .\) (a) Graph the function \(y=\sin \sqrt{x}\) using the viewing rectangle \([0,400]\) by \([-1.5,1.5] .\) In what ways does this graph differ from the graph of the sine function? (b) Graph the function \(y=\sin \left(x^{2}\right)\) using the viewing rectangle \([-5,5]\) by \([-1.5,1.5] .\) In what ways does this graph differ from the graph of the sine function?