Chapter 5

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=-2 \tan 2 x\)

For which of the quadrant angles \(0, \pi / 2, \pi,\) and \(3 \pi / 2\) is the sine function equal to \(0 ?\)

Find the period and amplitude. $$y=3 \sin 2 x$$

Find the exact value of each expression, if possible, without using a calculator. (a) \(\arccos (-\sqrt{3})\) (b) \(\arcsin \frac{\sqrt{2}}{2}\)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=-4 \tan \frac{x}{3}\)

Is the value of \(\cos 170^{\circ}\) equal to the value of \(\cos 10^{\circ} ?\)

Find the period and amplitude. $$y=2 \cos 3 x$$

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\frac{1}{2} \cot \frac{x}{2}\)

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\sin \theta=\frac{5}{6}$$

Find the period and amplitude. $$y=5 \cos \frac{x}{2}$$

Determine the quadrant in which each angle lies. (a) \(55^{\circ}\) (b) \(215^{\circ}\)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=3 \cot \pi x\)

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\cot \theta=5$$

Find the period and amplitude. $$y=-3 \sin \frac{x}{3}$$

Determine the quadrant in which each angle lies. (a) \(121^{\circ}\) (b) \(181^{\circ}\)

Consider the function \(y=\arcsin x\) (a) Use a graphing utility to complete the table. (b) Plot the points from the table in part (a) and graph the function. (Do not use a graphing utility.) (c) Use the graphing utility to graph the inverse sine function and compare the result with your handdrawn graph in part (b). (d) Determine any intercepts and symmetry of the graph.

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=-\frac{1}{2} \sec x\)

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\sec \theta=4$$

Find the period and amplitude. $$y=\frac{2}{3} \sin \pi x$$

Determine the quadrant in which each angle lies. (a) \(-150^{\circ}\) (b) \(282^{\circ}\)

Consider the function \(y=\arctan x\) (a) Use a graphing utility to complete the table. (b) Plot the points from the table in part (a) and graph the function. (Do not use a graphing utility.) (c) Use the graphing utility to graph the inverse tangent function and compare the result with your hand drawn graph in part (b). (d) Determine the horizontal asymptotes of the graph.

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\frac{1}{4} \sec x\)

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\cos \theta=\frac{3}{7}$$

Find the period and amplitude. $$y=\frac{3}{2} \cos \frac{\pi x}{2}$$

Determine the quadrant in which each angle lies. (a) \(87.9^{\circ}\) (b) \(-8.5^{\circ}\)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=3 \csc \frac{x}{2}\)

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\tan \theta=3$$

Find the period and amplitude. $$y=-2 \sin x$$

Determine the quadrant in which each angle lies. (a) \(132^{\circ} 50^{\prime}\) (b) \(-336^{\circ} 30^{\prime}\)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=-\csc \frac{x}{3}\)

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\csc \theta=\frac{17}{4}$$

Find the period and amplitude. $$y=-\cos \frac{2 x}{5}$$

Determine the quadrant in which each angle lies. (a) \(-245.25^{\circ}\) (b) \(12.35^{\circ}\)

A ladder that is 20 feet long leans against the side of a house. The angle of elevation of the ladder is \(75^{\circ} .\) Find the height from the top of the ladder to the ground.

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arcsin 0.45

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\sec \pi x-3\)

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(7,24)$$

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\cot \theta=\frac{3}{2}$$

Find the period and amplitude. $$y=\frac{1}{4} \cos \frac{4 x}{3}$$

Sketch each angle in standard position. (a) \(45^{\circ}\) (b) \(90^{\circ}\)

An electrician is running wire from the electric box on a house to a utility pole 75 feet away. The angle of elevation to the connection on the pole is \(16^{\circ} .\) How much wire does the electrician need?

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\sec \pi x-3\)

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(8,15)$$

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta .\) Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of \(\theta\). $$\sin \theta=\frac{3}{8}$$

Find the period and amplitude. $$y=\frac{5}{2} \cos \frac{x}{4}$$

Sketch each angle in standard position. (a) \(60^{\circ}\) (b) \(180^{\circ}\)

A cadet rappelling down a cliff on a rope needs help. A cadet on the ground pulls tight on the end of the rope that hangs down from the rappelling cadet to lock the cadet in place. The length of the rope between the two cadets is 120 feet, and the angle of elevation of the rope is \(66^{\circ} .\) The cadet on the ground is holding the rope at a height of 4 feet. How high above the ground is the cadct on the rope?

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. \(\tan ^{-1} 0.75\)

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=2 \tan \frac{\pi x}{4}\)

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(5,-12)$$