Chapter 9

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=2 \cos x$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=-2 \sec \left(x+\frac{\pi}{2}\right)$$

The position of a weight attached to a spring is $$s(t)=-4 \cos 10 t$$ inches after \(t\) seconds. (a) What is the maximum height that the weight rises above the equilibrium position? (b) What are the frequency and period? (c) When does the weight first reach its maximum height? (d) Calculate and interpret \(s(1.466)\)

If we are given an acute angle and a side of a right triangle, what unknown part of the triangle requires the least work to find?

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$sec $30^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-8,15)$$

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=3 \sin x$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=-\frac{3}{2} \sec (x-\pi)$$

Spring Motion \(\quad\) A weight attached to a spring is pulled down 3 inches below the equilibrium position. (a) Assuming that the frequency is \(\frac{6}{\pi}\) oscillations per second, find a trigonometric model that gives the position of the weight at time \(t\) seconds. (b) What is the period?

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\csc 30^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(15,-8)$$

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=\frac{2}{3} \sin x$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=\frac{5}{2} \cot \left[\frac{1}{3}\left(x-\frac{\pi}{2}\right)\right]$$

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\csc 45^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(7,-24)$$

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=\frac{3}{4} \cos x$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=-3 \tan \left[\frac{1}{2}\left(x+\frac{\pi}{4}\right)\right]$$

Use a graphing calculator to graph $$\begin{aligned}&y_{1}=e^{-x} \sin x\\\&y_{2}=e^{-x}\\\&\text { and } \quad y_{3}=-e^{-x}\end{aligned}$$ in the viewing window \([0, \pi]\) by \([-0.5,0.5]\) (a) Find the \(x\) -intercepts of the graph of \(y_{1}\). Explain the relationship of these \(x\) -intercepts to those of the graph of \(y=\sin x\) (b) Find the \(x\) -coordinate of any points of intersection of \(y_{1}\) and \(y_{2}\) or \(y_{1}\) and \(y_{3}\)

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$sec $45^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-24,-7)$$

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=-\cos x$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=\frac{1}{2} \sec (2 x+\pi)$$

Explain why the angle of depression \(D A B\) has the same measure as the angle of elevation \(A B C\) in the figure. (Assume that line \(A D\) is parallel to line \(C B\).)

Damped Motion A weight attached to a spring is submerged in water. Suppose that its displacement from its equilibrium position can be modeled by $$D(t)=2 e^{-2 t} \cos 2 \pi t$$ where \(D\) is in inches and \(t \geq 0\) is in seconds. Note that when \(D<0\) the spring is stretched, and when \(D>0\) the spring is compressed. (a) What is the initial displacement of the weight when \(t=0 ?\) (b) What is the frequency? (c) Approximate graphically the time when the spring is compressed \(\frac{1}{e}\) inch.

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cos 45^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(0,2)$$

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=-\sin x$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=-\frac{1}{3} \csc \left(\frac{1}{2} x-\frac{\pi}{2}\right)$$

The voltage in a circuit with alternating current is given by $$V(t)=160 e^{-20 t} \cos 120 \pi t$$ where \(V\) is in volts and \(t \geq 0\) is in seconds. (a) What is the initial voltage when \(t=0 ?\) (b) What is the frequency of the voltage? (c) Solve \(V(t)=100\) graphically.

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cot 45^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(0,5)$$

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=-2 \sin x$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=-1-\tan \left(x+\frac{\pi}{4}\right)$$

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\sin \frac{\pi}{3}$$

Find the measure of each angle. Supplementary angles with measures \(6 x-4\) degrees and \(8 x-12\) degrees.

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=-3 \cos x$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=2+\cot \left(2 x-\frac{\pi}{3}\right)$$

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cos \frac{\pi}{3}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-5,0)$$

Find the measure of each angle. Complementary angles with measures \(9 z+6\) degrees and \(3 z\) degrees.

Graph each function over a two-period interval. Give the period and amplinde. $$y=\sin \frac{1}{2} x$$

Height of a Ladder on a Wall A 13.5 -meter fire-truck ladder is leaning against a wall. Find the distance \(d\) the ladder goes up the wall (above the top of the fire truck) if the ladder makes an angle of \(43^{\circ} 50^{\prime}\) with the horizontal.

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\tan \frac{\pi}{3}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(0,-4)$$

Perform each calculation. $$62^{\circ} 18^{\prime}+21^{\circ} 41^{\prime}$$

Graph each function over a two-period interval. Give the period and amplinde. $$y=\sin \frac{2}{3} x$$

Graph each function over a one-period interval. $$y=\csc x$$

Length of a Guy Wire A weather tower used to measure wind speed has a guy wire attached to it 175 feet above the ground. The angle between the wire and the vertical tower is \(57.0^{\circ} .\) Approximate the length of the guy wire.

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cot \frac{\pi}{3}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(0,-3)$$