Chapter 5

Graph the equation on the Interval \([-2,2]\), and describe the behavior of \(y\) as \(x \rightarrow 0^{-}\) and as \(x \rightarrow 0^{+}\) \(y=\sin \frac{1}{x}\)

A ship leaves port at 1: 00 P.M. and sails in the direction \(\mathrm{N} 34^{\circ} \mathrm{W}\) at a rate of \(24 \mathrm{mi} / \mathrm{hr}\). Another ship leaves port at 1: 30 p.M. and sails in the direction \(N 56^{\circ} \mathrm{E}\) at a rate of \(18 \mathrm{mi} / \mathrm{hr}\) (a) Approximately how far apart are the ships at 3: 00 P.M.? (b) What is the bearing, to the nearest degree, from the first ship to the second?

Find the intervals between \(-2 \pi\) and \(2 \pi\) on which the given function is (a) increasing or (b) decreasing. secant

Sketch the graph of the equation. $$y=2^{-x} \cos x$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\sec \theta-\cos \theta=\tan \theta \sin \theta$$

Graph the equation on the Interval \([-2,2]\), and describe the behavior of \(y\) as \(x \rightarrow 0^{-}\) and as \(x \rightarrow 0^{+}\) \(y=|x| \sin \frac{1}{x}\)

Sketch the graph of the equation. $$y=e^{x} \sin x$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{\sin \theta+\cos \theta}{\cos \theta}=1+\tan \theta$$

An airplane flying at a speed of \(360 \mathrm{mi} / \mathrm{hr}\) flies from a point \(A\) in the direction \(137^{\circ}\) for 30 minutes and then flies in the direction \(227^{\circ}\) for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to \(A\).

Graph the equation on the Interval \([-2,2]\), and describe the behavior of \(y\) as \(x \rightarrow 0^{-}\) and as \(x \rightarrow 0^{+}\) \(y=\frac{\sin 2 x}{x}\)

Find the intervals between \(-2 \pi\) and \(2 \pi\) on which the given function is (a) increasing or (b) decreasing. tangent

Sketch the graph of the equation. $$y=|x| \sin x$$

Verify the identity by transforming the lefthand side into the right-hand side. $$(\cot \theta+\csc \theta)(\tan \theta-\sin \theta)=\sec \theta-\cos \theta$$

An airplane flying at a speed of \(400 \mathrm{mi} / \mathrm{hr}\) flies from a point \(A\) in the direction \(153^{\circ}\) for 1 hour and then flies in the direction \(63^{\circ}\) for 1 hour. (a) In what direction does the plane need to fly in order to get back to point \(A ?\) (b) How long will it take to get back to point \(A ?\)

Graph the equation on the Interval \([-2,2]\), and describe the behavior of \(y\) as \(x \rightarrow 0^{-}\) and as \(x \rightarrow 0^{+}\) \(y=\frac{1-\cos 3 x}{x}\)

Find the intervals between \(-2 \pi\) and \(2 \pi\) on which the given function is (a) increasing or (b) decreasing. cotangent

Sketch the graph of the equation. $$y=|x| \cos x$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\cot \theta+\tan \theta=\csc \theta \sec \theta$$

The formula specifles the position of a point \(P\) that is moving harmonically on a vertical axis, where \(t\) is in seconds and \(d\) is in centimeters. Determine the amplitude, period, and frequency, and describe the motion of the point during one complete oscillation (starting at \(t=0\) ). $$d=10 \sin 6 \pi t$$

Graph the equation on the Interval \([-20,20]\), and estimate the hortzontal asymptote. \(y=x^{2} \sin ^{2}\left(\frac{2}{x}\right)\)

Practice sketching the graph of the sine function, taking different units of length on the horizontal and vertical axes. Practice sketching graphs of the cosine and tangent functions in the same manner. Continue this practice until you reach the stage at which, if you were awakened from a sound sleep in the middle of the night and asked to sketch one of these graphs, you could do so in less than thirty seconds.

Graph the function \(f\) in the viewing rectangle \([-2 \pi, 2 \pi, \pi / 2]\) by \([-4,4] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same vlewing rectangle. $$f(x)=\tan 0.5 x ; \quad g(x)=\tan \left[0.5\left(x+\frac{\pi}{2}\right)\right]$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\sec ^{2} 3 \theta \csc ^{2} 3 \theta=\sec ^{2} 3 \theta+\csc ^{2} 3 \theta$$

The formula specifles the position of a point \(P\) that is moving harmonically on a vertical axis, where \(t\) is in seconds and \(d\) is in centimeters. Determine the amplitude, period, and frequency, and describe the motion of the point during one complete oscillation (starting at \(t=0\) ). $$d=\frac{1}{3} \cos \frac{\pi}{4} t$$

Graph the equation on the Interval \([-20,20]\), and estimate the hortzontal asymptote. \(y=\frac{1-\cos ^{2}(2 / x)}{\sin (1 / x)}\)

Graph the function \(f\) in the viewing rectangle \([-2 \pi, 2 \pi, \pi / 2]\) by \([-4,4] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same vlewing rectangle. $$f(x)=0.5 \csc 0.5 x, g(x)=0.5 \csc 0.5 x-2$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{1+\cos ^{2} 3 \theta}{\sin ^{2} 3 \theta}=2 \csc ^{2} 3 \theta-1$$

The formula specifles the position of a point \(P\) that is moving harmonically on a vertical axis, where \(t\) is in seconds and \(d\) is in centimeters. Determine the amplitude, period, and frequency, and describe the motion of the point during one complete oscillation (starting at \(t=0\) ). $$d=4 \cos \frac{3 \pi}{2} t$$

Graph the function \(f\) in the viewing rectangle \([-2 \pi, 2 \pi, \pi / 2]\) by \([-4,4] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same vlewing rectangle. $$f(x)=0.5 \sec 0.5 x, \quad g(x)=0.5 \sec \left[0.5\left(x-\frac{\pi}{2}\right)\right]-1$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\log \csc \theta=-\log \sin \theta$$

The formula specifles the position of a point \(P\) that is moving harmonically on a vertical axis, where \(t\) is in seconds and \(d\) is in centimeters. Determine the amplitude, period, and frequency, and describe the motion of the point during one complete oscillation (starting at \(t=0\) ). $$d=6 \sin \frac{2 \pi}{3} t$$

Use a graph to solve the Inequality on the interval \([-\pi, \pi]\) \(\frac{1}{4} \tan \left(\frac{1}{3} x^{2}\right)<\frac{1}{2} \cos 2 x+\frac{1}{5} x^{2}\)

Graph the function \(f\) in the viewing rectangle \([-2 \pi, 2 \pi, \pi / 2]\) by \([-4,4] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same vlewing rectangle. $$f(x)=\tan x-1 ; \quad g(x)=-\tan x+1$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\log \tan \theta=\log \sin \theta-\log \cos \theta$$

A point \(P\) in simple harmonic motion has a period of 3 seconds and an amplitude of 5 centimeters. Express the motion of \(P\) by means of an equation of the form \(d=a \cos \omega t\)

Graph the function \(f\) in the viewing rectangle \([-2 \pi, 2 \pi, \pi / 2]\) by \([-4,4] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same vlewing rectangle. $$f(x)=3 \cos 2 x ; \quad g(x)=|3 \cos 2 x|-1$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and \(P\) is on the terminal side. $$P(4,-3)$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and \(P\) is on the terminal side. $$P(-8,-15)$$

On March \(17,1981,\) in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation $$T(t)=-12 \cos \left(\frac{\pi}{12} t\right)+60$$ while the relative humidity in percent could be expressed by $$H(t)=20 \cos \left(\frac{\pi}{12} t\right)+60$$ where \(t\) is in hours and \(t=0\) corresponds to 6 A.M. (a) Construct a table that lists the temperature and relative humidity every three hours, beginning at midnight. (b) Determine the times when the maximums and minimums occurred for \(T\) and \(H\) (c) Discuss the relationship between the temperature and relative humidity on this day.

Identify the damping factor \(f(x)\) for the damped wave. Sketch graphs of \(y=\pm f(x)\) and the equation on the same coordinate plane for \(-2 \pi \leq x \leq 2 \pi\) $$y=e^{-x^{4}} \sin 4 x$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and \(P\) is on the terminal side. $$P(-2,-5)$$

Identify the damping factor \(f(x)\) for the damped wave. Sketch graphs of \(y=\pm f(x)\) and the equation on the same coordinate plane for \(-2 \pi \leq x \leq 2 \pi\) $$y=3^{-x / 3} \cos 2 x$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and \(P\) is on the terminal side. $$P(-1,2)$$

Graph the equation, and estimate the values of \(x\) in the specified interval that correspond to the given value of \(y\) $$y=\sin \left(x^{2}\right), \quad[-\pi, \pi] ; \quad y=0.5$$

Graph the function \(f\) on \([-\pi, \pi],\) and estimate the high and low points. $$f(x)=\cos 2 x+2 \sin 4 x-\sin x$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. II; on the line \(y=-4 x\)

Graph the equation, and estimate the values of \(x\) in the specified interval that correspond to the given value of \(y\) $$y=\tan (\sqrt{x}), \quad[0,25] ; \quad y=5$$

Graph the function \(f\) on \([-\pi, \pi],\) and estimate the high and low points. $$f(x)=\tan \frac{1}{4} x-2 \sin 2 x$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. IV; on the line \(3 y+5 x=0\)

Graph \(f\) on the Interval \([-2 \pi, 2 \pi],\) and estimate the coordinates of the high and low points. $$f(x)=x \sin x$$