Chapter 5

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=\frac{1}{2} \cos \frac{\pi}{2} x\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=-\frac{1}{3} \cot (3 x-\pi)$$

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\sin \theta=0.6612$$

Added in 1990 and removed in \(1997,\) the highest advertising sign in the world was a large letter I situated at the top of the 73 -story First Interstate World Center building in Los Angeles. At a distance of 200 feet from a point directly below the sign, the angle between the ground and the top of the sign was \(78.87^{\circ} .\) Approximate the height of the top of the sign.

Exer. \(25-28:\) Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$12.864^{\circ}$$

A pilot, flying at an altitude of 5000 feet, wishes to approach the numbers on a runway at an angle of \(10^{\circ} .\) Approximate, to the nearest 100 feet, the distance from the airplane to the numbers at the beginning of the descent.

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=5 \sin \left(3 x-\frac{\pi}{2}\right)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=-\frac{1}{2} \cot \left(\frac{1}{2} x+\frac{\pi}{4}\right)$$

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\tan \theta=3.7$$

Exer. \(25-28:\) Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$310.6215^{\circ}$$

A guy wire is attached to the top of a radio antenna and to a point on horizontal ground that is 40.0 meters from the base of the antenna. If the wire makes an angle of \(58^{\circ} 20^{\prime}\) with the ground, approximate the length of the wire.

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=-4 \cos \left(2 x+\frac{\pi}{3}\right)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=4 \cot \left(\frac{1}{3} x-\frac{\pi}{6}\right)$$

Exer. \(25-28:\) Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$81.7238^{\circ}$$

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\cos \theta=0.8$$

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\sin \theta=0.4217$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=3 \cos \left(\frac{1}{2} x-\frac{\pi}{4}\right)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\sec \left(x-\frac{\pi}{2}\right)$$

Approximate to four decimal places, when appropriate. (a) \(\sin 42^{\circ}\) (b) \(\cos 77^{\circ}\) (c) csc \(123^{\circ}\) (d) sec \(\left(-190^{\circ}\right)\)

To measure the height \(h\) of a cloud cover, a meteorology student directs a spotlight vertically upward from the ground. From a point \(P\) on level ground that is \(d\) meters from the spotlight, the angle of elevation \(\theta\) of the light image on the clouds is then measured (see the figure). (a) Express \(h\) in terms of \(d\) and \(\theta\) (b) Approximate \(h\) if \(d=1000 \mathrm{m}\) and \(\theta=59^{\circ}\) (IMAGE CAN NOT COPY)

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\tan \theta=4.91$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=-2 \sin \left(\frac{1}{2} x+\frac{\pi}{2}\right)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\sec \left(x-\frac{3 \pi}{4}\right)$$

Approximate to four decimal places, when appropriate. (a) \(\tan 282^{\circ}\) (b) \(\cot \left(-81^{\circ}\right)\) (c) \(\sec 202^{\circ}\) (d) \(\sin 97^{\circ}\)

Exer \(29-30:\) If a circular are of the given length \(s\) subtends the central angle \(\theta\) on a circle, find the radius of the circle. $$s=3 \mathrm{km}, \quad \theta=20^{\circ}$$

A rocket is fired at sea level and climbs at a constant angle of \(75^{\circ}\) through a distance of \(10,000\) feet. Approximate its altitude to the nearest foot.

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\sec \theta=4.246$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=-5 \cos \left(\frac{1}{3} x+\frac{\pi}{6}\right)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\sec 2 x$$

Approximate to four decimal places, when appropriate. (a) cot \((\pi / 13)\) (b) csc 1.32 (c) \(\cos (-8.54)\) (d) \(\tan (3 \pi / 7)\)

An airplane takes off at a \(10^{\circ}\) angle and travels at the rate of 250 ft/sec. Approximately how long does it take the airplane to reach an altitude of \(15,000\) feet?

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\csc \theta=11$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=4 \sin \left(\frac{1}{3} x-\frac{\pi}{3}\right)\)

Approximate to four decimal places, when appropriate. (a) \(\sin (-0.11)\) (b) \(\sec \frac{31}{27}\) (c) \(\cos (-8.54)\) (d) \(\tan (3 \pi / 7)\)

Exer. 33-34: (a) Find the radian and degree measures of the central angle \(\boldsymbol{\theta}\) subtended by the given are of length \(s\) on a circle of radius \(r .\) (b) Find the area of the sector determined by \(\theta\) $$s=7 \mathrm{cm}, \quad r=4 \mathrm{cm}$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=4 \sin \left(\frac{1}{3} x-\frac{\pi}{3}\right)\)

Approximate to four decimal places, when appropriate. (a) \(\sin 30^{\circ}\) (b) \(\sin 30\) (c) \(\cos \pi^{\circ}\) (d) \(\cos \pi\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\sec \frac{1}{3} x$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=-2 \sin (2 \pi x+\pi)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\sec 3 x$$

Exer. \(35-36:\) (a) Find the length of the are that subtends the given central angle \(\boldsymbol{\theta}\) on a circle of diameter \(d .\) (b) Find the area of the sector determined by \(\theta\) $$\theta=50^{\circ}, \quad d=16 \mathrm{m}$$

Approximate the angle of elevation \(\alpha\) of the sun if a person 5.0 feet tall casts a shadow 4.0 feet long on level ground (see the figure). (IMAGE CAN NOT COPY)

Approximate, to the nearest \(0.1^{\circ},\) all angles \(\theta\) in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) that satisfy the equation. (a) \(\sin \theta=-0.5640\) (b) \(\cos \theta=0.7490\) (c) \(\tan \theta=2.798\) (d) cot \(\theta=-0.9601\) (e) \(\sec \theta=-1.116\) (f) cse \(\theta=1.485\)

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=-\sqrt{2} \sin \left(\frac{\pi}{2} x-\frac{\pi}{4}\right)\)

Use the Pythagorean Identities to write the expression as an integer. (a) \(\tan ^{2} 4 \beta-\sec ^{2} 4 \beta\) (b) \(4 \tan ^{2} \beta-4 \sec ^{2} \beta\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=2 \sec \left(2 x-\frac{\pi}{2}\right)$$

Exer. \(35-36:\) (a) Find the length of the are that subtends the given central angle \(\boldsymbol{\theta}\) on a circle of diameter \(d .\) (b) Find the area of the sector determined by \(\theta\) $$\theta=2.2, \quad d=120 \mathrm{cm}$$

A builder wishes to construct a ramp 24 feet long that rises to a height of 5.0 feet above level ground. Approximate the angle that the ramp should make with the horizontal.

Approximate, to the nearest \(0.1^{\circ},\) all angles \(\theta\) in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) that satisfy the equation. (a) \(\sin \theta=0.8225\) (b) \(\cos \theta=-0.6604\) (c) \(\tan \theta=-1.5214\) (d) cot \(\theta=1.3752\) (e) \(\sec \theta=1.4291\) (f) csc \(\theta=-2.3179\)

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=\sqrt{3} \cos \left(\frac{\pi}{4} x-\frac{\pi}{2}\right)\)