Chapter 6

Exer. 63-66: 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 \(A\) ?

Exer. 63-66: 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. $$\cot \theta+\tan \theta=\csc \theta \sec \theta$$

Exer. 67-70: The formula specifies 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 \(t=0\) ). $$ d=10 \sin 6 \pi t $$

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.

Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength \(\lambda\) and the distance between the two radio towers is equal to \(\frac{1}{2} \lambda\), then the intensity \(I\) of the signal in the direction \(\theta\) is given by $$ I=\frac{1}{2} I_{0}[1+\cos (\pi \sin \theta)] $$ where \(I_{0}\) is the maximum intensity. Approximate \(I\) in terms of \(I_{0}\) for each \(\theta\). (a) \(\theta=0\) (b) \(\theta=\pi / 3\) (c) \(\theta=\pi / 7\)

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

Exer. 67-70: The formula specifies 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 \(t=0\) ). $$ d=\frac{1}{3} \cos \frac{\pi}{4} t $$

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

Exer. 67-70: The formula specifies 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 \(t=0\) ). $$ d=4 \cos \frac{3 \pi}{2} t $$

The strength of Earth's magnetic field varies with the depth below the surface. The strength at depth \(z\) and time \(t\) can sometimes be approximated using the damped sine wave $$ S=A_{0} e^{-\alpha z} \sin (k t-\alpha z), $$ where \(A_{0}, \alpha\), and \(k\) are constants. (a) What is the damping factor? (b) Find the phase shift at depth \(z_{0}\). (c) At what depth is the amplitude of the wave one-half the amplitude of the surface strength?

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

Exer. 67-70: The formula specifies 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 \(t=0\) ). $$ d=6 \sin \frac{2 \pi}{3} t $$

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 \(3 \mathrm{sec}-\) onds and an amplitude of 5 centimeters. Express the motion of \(P\) by means of an equation of the form \(d=a \cos \omega t\).

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

A point \(P\) in simple harmonic motion has a frequency of \(\frac{1}{2}\) oscillation per minute and an amplitude of 4 feet. Express the motion of \(P\) by means of an equation of the form \(d=a \sin \omega t\).

Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) 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 \(6 \mathrm{~A} . \mathrm{M}\). (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.

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

Trigonometric functions are used extensively in the design of industrial robots. Suppose that a robot's shoulder joint is motorized so that the angle \(\theta\) increases at a constant rate of \(\pi / 12\) radian per second from an initial angle of \(\theta=0\). Assume that the elbow joint is always kept straight and that the arm has a constant length of 153 centimeters, as shown in the figure. (a) Assume that \(h=50 \mathrm{~cm}\) when \(\theta=0\). Construct a table that lists the angle \(\theta\) and the height \(h\) of the robotic hand every second while \(0 \leq \theta \leq \pi / 2\). (b) Determine whether or not a constant increase in the angle \(\theta\) produces a constant increase in the height of the hand. (c) Find the total distance that the hand moves.

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

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

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

Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and the terminal side of \(\theta\) is in the specified quadrant and satisfies the given condition. I; on a line having slope \(\frac{4}{3}\)

Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and the terminal side of \(\theta\) is in the specified quadrant and satisfies the given condition. III; bisects the quadrant

Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and the terminal side of \(\theta\) is in the specified quadrant and satisfies the given condition. III; parallel to the line \(2 y-7 x+2=0\)

Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and the terminal side of \(\theta\) is in the specified quadrant and satisfies the given condition. II; parallel to the line through \(A(1,4)\) and \(B(3,-2)\)

Find the exact values of the six trigonometric unctions of each angle, whenever possible. (a) \(90^{\circ}\) (b) \(0^{\circ}\) (c) \(7 \pi / 2\) (d) \(3 \pi\)

Find the exact values of the six trigonometric unctions of each angle, whenever possible. (a) \(180^{\circ}\) (b) \(-90^{\circ}\) (c) \(2 \pi\) (d) \(5 \pi / 2\)

Find the quadrant containing \(\theta\) if the given conditions are true. (a) \(\cos \theta>0\) and \(\sin \theta<0\) (b) \(\sin \theta<0\) and \(\cot \theta>0\) (c) \(\csc \theta>0\) and \(\sec \theta<0\) (d) \(\sec \theta<0\) and \(\tan \theta>0\)

Find the quadrant containing \(\theta\) if the given conditions are true. (a) \(\tan \theta<0\) and \(\cos \theta>0\) (b) \(\sec \theta>0\) and \(\tan \theta<0\) (c) \(\csc \theta>0\) and \(\cot \theta<0\) (d) \(\cos \theta<0\) and \(\csc \theta<0\)

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\tan \theta=-\frac{3}{4}\( and \)\sin \theta>0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\cot \theta=\frac{3}{4}\( and \)\cos \theta<0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. \(\sin \theta=-\frac{5}{13}\) and \(\sec \theta>0\)

Use fundamental identities to find the values of the trigonometric functions for the given conditions. \(\cos \theta=\frac{1}{2}\) and \(\sin \theta<0\)

Use fundamental identities to find the values of the trigonometric functions for the given conditions. \(\cos \theta=-\frac{1}{3}\) and \(\sin \theta<0\)

Use fundamental identities to find the values of the trigonometric functions for the given conditions. \(\csc \theta=5\) and \(\cot \theta<0\)

Use fundamental identities to find the values of the trigonometric functions for the given conditions. \(\sec \theta=-4\) and \(\csc \theta>0\)

Use fundamental identities to find the values of the trigonometric functions for the given conditions. \(\sin \theta=\frac{2}{5}\) and \(\cos \theta<0\)

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{\sec ^{2} \theta-1} ; \quad \pi / 2<\theta<\pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{1+\cot ^{2} \theta}, \quad 0<\theta<\pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{1+\tan ^{2} \theta} ; \quad 3 \pi / 2<\theta<2 \pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{\csc ^{2} \theta-1} ; \quad 3 \pi / 2<\theta<2 \pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{\sin ^{2}(\theta / 2)}, \quad 2 \pi<\theta<4 \pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{\cos ^{2}(\theta / 2)} ; \quad 0<\theta<\pi$$