Chapter 6

Exer. 5-40: 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) \(1^{\prime}\). $$ \tan \theta=3.7 $$

Telescope resolution Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution \(\theta\) (see the figure) can be improved by using a lens with a larger diameter \(D\). The relationship between \(\theta\) in degrees and \(D\) in meters is given by \(\sin \theta=1.22 \lambda / D\), where \(\lambda\) is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of \(\lambda=550 \times 10^{-9}\) meter, its resolution is \(0.00003769^{\circ}\). Approximate the diameter of the lens.

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.

Exer. 5-40: 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. $$ y=4 \cot \left(\frac{1}{3} x-\frac{\pi}{6}\right) $$

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

Moon phases The phases of the moon can be described using the phase angle \(\theta\), determined by the sun, the moon, and Earth, as shown in the figure. Because the moon orbits Earth, \(\theta\) changes during the course of a month. The area of the region \(A\) of the moon, which appears illuminated to an observer on Earth, is given by \(A=\frac{1}{2} \pi R^{2}(1+\cos \theta)\), where \(R=1080 \mathrm{mi}\) is the radius of the moon. Approximate \(A\) for the following positions of the moon: (a) \(\theta=0^{\circ}\) (full moon) (b) \(\theta=180^{\circ}\) (new moon) (c) \(\theta=90^{\circ}\) (first quarter) (d) \(\theta=103^{\circ}\)

Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$81.7238^{\circ}$$

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=3 \cos \left(\frac{1}{2} x-\frac{\pi}{4}\right) $$

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

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) \(\sec \left(-190^{\circ}\right)\)

Exer. 5-40: 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 the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\). $$ \tan \theta=4.91 $$

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

If a circular arc of the given length \(s\) subtends the central angle \(\boldsymbol{\theta}\) 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.

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=-5 \cos \left(\frac{1}{3} x+\frac{\pi}{6}\right) $$

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

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 \(250 \mathrm{ft} / \mathrm{sec}\). Approximately how long does it take the airplane to reach an altitude of 15,000 feet? 33 Designing a drawbridge

Exer. 5-40: 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 the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\). $$ \csc \theta=11 $$

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\sec \frac{1}{2} x $$

Approximate to four decimal places, when appropriate. (a) \(\sin (-0.11)\) (b) \(\sec \frac{31}{27}\) (c) \(\tan \left(-\frac{3}{13}\right)\) (d) \(\cos 2.4 \pi\)

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=3 \cos (\pi x+4 \pi) $$

Approximate to four decimal places. (a) \(\sin 98^{\circ} 10^{r}\) (b) \(\cos 623.7^{\circ}\) (c) \(\tan 3\) (d) \(\cot 231^{\circ} 40^{t}\) (e) sec \(1175.1^{\circ}\) (f) \(\csc 0.82\)

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

(a) Find the radian and degree measures of the central angle \(\theta\) subtended by the given arc of length \(s\) on a circle of radius \(r\). (b) Find the area of the sector determined by \(\theta\). \(s=7 \mathrm{~cm}, \quad r=4 \mathrm{~cm}\)

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=-2 \sin (2 \pi x+\pi) $$

Approximate to four decimal places. (a) \(\sin 496.4^{\circ}\) (b) \(\cos 0.65\) (c) \(\tan 105^{\circ} 40^{\prime}\) (d) \(\cot 1030.2^{\circ}\) (e) \(\sec 1.46\) (f) \(\csc 320^{\circ} 50^{\prime}\)

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

Approximate to four decimal places, when appropriate. (a) \(\sin 45^{\circ}\) (b) \(\sin 45\) (c) \(\cos (3 \pi / 2)^{\circ}\) (d) \(\cos (3 \pi / 2)\)

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).

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=-\sqrt{2} \sin \left(\frac{\pi}{2} x-\frac{\pi}{4}\right) $$

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) \(\csc \theta=1.485\)

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

(a) Find the length of the arc that subtends the given central angle \(\theta\) on a circle of diameter \(d\). (b) Find the area of the sector determined by \(\theta\). \(\theta=50^{\circ}, \quad d=16 \mathrm{~m}\)

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.

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=\sqrt{3} \cos \left(\frac{\pi}{4} x-\frac{\pi}{2}\right) $$

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

Use the Pythagorean identities to write the expression as an integer. (a) \(\csc ^{2} 3 \alpha-\cot ^{2} 3 \alpha\) (b) \(3 \csc ^{2} \alpha-3 \cot ^{2} \alpha\)

(a) Find the length of the arc that subtends the given central angle \(\theta\) on a circle of diameter \(d\). (b) Find the area of the sector determined by \(\theta\). \(\theta=2.2, \quad d=120 \mathrm{~cm}\)

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=-2 \sin (2 x-\pi)+3 $$