Chapter 5

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

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

Earth The distance between two points \(A\) and \(B\) on Earth is measured along a circle having center \(C\) at the center of Earth and radius equal to the distance from \(C\) to the surface (see the figure). If the diameter of Earth is approximately 8000 miles, approximate the distance between \(A\) and \(B\) if angle \(A C B\) has the indicated measure: \(\begin{array}{lll}\text { (a) } 60^{\circ} & \text { (b) } 45^{\circ}\end{array}\) (c) \(30^{\circ} \quad\) (d) \(10^{\circ}\) (e) \(1^{\circ}\)

Approximate, to the nearest 0.01 radian, all angles \(\theta\) in the interval \([0,2 \pi)\) that satisfy the equation. (a) \(\sin \theta=0.4195\) (b) \(\cos \theta=-0.1207\) (c) \(\tan \theta=-3.2504\) (d) cot \(\theta=2.6815\) (e) \(\sec \theta=1.7452\) (f) \(\csc \theta=-4.8521\)

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

Use the Pythagorean Identities to write the expression as an integer. (a) \(5 \sin ^{2} \theta+5 \cos ^{2} \theta\) (b) \(5 \sin ^{2}(\theta / 4)+5 \cos ^{2}(\theta / 4)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=-\frac{1}{3} \sec \left(\frac{1}{2} x+\frac{\pi}{4}\right)$$

\(A\) conveyor belt 9 meters long can be hydraulically rotated up to an angle of \(40^{\circ}\) to unload cargo from airplanes (see the figure). (a) Find, to the nearest degree, the angle through which the conveyor belt should be rotated up to reach a door that is 4 meters above the platform supporting the belt. (b) Approximate the maximum height above the platform that the belt can reach. (IMAGE CAN NOT COPY)

Approximate, to the nearest 0.01 radian, all angles \(\theta\) in the interval \([0,2 \pi)\) that satisfy the equation. (a) \(\sin \theta=-0.0135\) (b) \(\cos \theta=0.9235\) (c) \(\tan \theta=0.42\) (d) \(\cot \theta=-2.731\) (e) \(\sec \theta=-3.51\) (f) \(\csc \theta=1.258\)

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

Use the Pythagorean Identities to write the expression as an integer. (a) \(7 \sec ^{2} y-7 \tan ^{2} y\) (b) \(7 \sec ^{2}(\gamma / 3)-7 \tan ^{2}(\gamma / 3)\)

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

The tallest man-made structure in the world is a television transmitting tower located near Mayville, North Dakota. From a distance of 1 mile on level ground, its angle of elevation is \(21^{\circ} 20^{\prime} 24^{\prime \prime} .\) Determine its height to the nearest foot.

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\sin x=-1$$

Thichness of the azone layer The thickness of the ozone layer can be estimated using the formula $$ \ln I_{0}-\ln I=k x \sec \theta $$ where \(I_{0}\) is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, \(I\) is the intensity of the same wavelength after passing through a layer of ozone \(x\) centimeters thick, \(k\) is the absorption constant of ozone for that wavelength, and \(\theta\) is the acute angle that the sunlight makes with the vertical. Suppose that for a wavelength of \(3055 \times 10^{-8}\) centimeter with \(k=1.88, I_{0} / I\) is measured as 1.72 and \(\theta=12^{\circ} .\) Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=5 \cos (2 x+2 \pi)+2\)

Simplify the expression. $$\frac{\sin ^{3} \theta+\cos ^{3} \theta}{\sin \theta+\cos \theta}$$

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

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\sin x=1$$

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

Simplify the expression. $$\frac{\cot ^{2} \alpha-4}{\cot ^{2} \alpha-\cot \alpha-6}$$

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

The Pentagon is the largest office building in the world in terms of ground area. The perimeter of the building has the shape of a regular pentagon with each side of length 921 feet. Find the area enclosed by the perimeter of the building.

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\sin x=\frac{1}{2}$$

The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula $$ R=R_{0} \cos \theta \sin \phi $$ where \(R_{0}\) is the maximum solar radiation possible, \(\theta\) is the angle that the sun makes with the horizontal, and \(\phi\) is the direction of the sun in the sky, with \(\phi=90^{\circ}\) when the sun is in the east and \(\phi=0^{\circ}\) when the sun is in the south. (a) When does the maximum solar radiation \(R_{0}\) strike the wall? (b)What percentage of \(R_{0}\) is striking the wall when \(\theta\) is equal to \(60^{\circ}\) and the sun is in the southeast?

Simplify the expression. $$\frac{2-\tan \theta}{2 \csc \theta-\sec \theta}$$

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

A tomado's core \(A\) simple model of the core of a tornado is a right circular cylinder that rotates about its axis. If a tornado has a core diameter of 200 feet and maximum wind speed of \(180 \mathrm{mi} / \mathrm{hr}\) (or \(264 \mathrm{ft} / \mathrm{sec}\) ) at the perimeter of the core, approximate the number of revolutions the core makes each minute.

A regular octagon is inscribed in a circle of radius 12.0 centimeters. Approximate the perimeter of the octagon.

In the mid-latitudes it is sometimes possible to estimate the distance between consecutive regions of low pressure. If \(\phi\) is the latitude (in degrees), \(R\) is Earth's radius (in kilometers), and \(v\) is the horizontal wind velocity (in \(\mathrm{km} / \mathrm{hr}\) ), then the distance \(d\) (in kilometers) from one low pressure area to the next can be estimated using the formula $$ d=2 \pi\left(\frac{v R}{0.52 \cos \phi}\right)^{1 / 3} $$ (a) At a latitude of \(48^{\circ}\). Earth's radius is approximately 6369 kilometers. Approximate \(d\) if the wind speed is \(45 \mathrm{km} / \mathrm{hr}\) (b) If \(v\) and \(R\) are constant, how does \(d\) vary as the latitude increases?

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\sin x=-\sqrt{2} / 2$$

Simplify the expression. $$\frac{\csc \theta+1}{\left(1 / \sin ^{2} \theta\right)+\csc \theta}$$

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

Earth's rotation Earth rotates about its axis once every 23 hours, 56 minutes, and 4 seconds. Approximate the number of radians Earth rotates in one second.

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\cos x=1$$

Robot's arm Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 18 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at \((18,0)\) and then rotates through an angle of \(60^{\circ}\) what is the new location of the hand?

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\cot \theta, \sin \theta$$

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

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\cos x=-1$$

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\tan \theta, \cos \theta$$

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

Exer. \(45-46:\) A wheel of the given radius is rotating at the indicated rate. (a) Find the angular speed (in radians per minute). (b) Find the linear speed of a point on the circumference (In \(\mathrm{ft} / \mathrm{min}\) ). radius 5 in., 40 rpm

From a point \(P\) on level ground, the angle of elevation of the top of a tower is \(26^{\circ} 50^{\prime} .\) From a point 25.0 meters closer to the tower and on the same line with \(P\) and the base of the tower, the angle of elevation of the top is \(53^{\circ} 30^{\prime} .\) Approximate the height of the tower.

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\cos x=\sqrt{2} / 2$$

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\sec \theta, \sin \theta$$

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

Exer. \(45-46:\) A wheel of the given radius is rotating at the indicated rate. (a) Find the angular speed (in radians per minute). (b) Find the linear speed of a point on the circumference (In \(\mathrm{ft} / \mathrm{min}\) ). radius 9 in., 2400 rpm

A ladder 20 feet long leans against the side of a building, and the angle between the ladder and the building is \(22^{\circ}\) (a) Approximate the distance from the bottom of the ladder to the building. (b) If the distance from the bottom of the ladder to the building is increased by 3.0 feet, approximately how far does the top of the ladder move down the building?

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\cos x=-\frac{1}{2}$$

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\csc \theta, \cos \theta$$