Chapter 5

Use trigonometric identities to transform one side of the equation into the other \((0<{\theta}<\pi /2)\). $$\cot \theta \sin \theta=\cos \theta$$

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\) (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\square\). (b) As \(x \rightarrow 0^{-}\), the value of \(f(x) \rightarrow\square\). (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\square\). (d) \(\mathrm{As} x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\square\). \(f(x)=\cot x\)

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-\frac{5 \pi}{3}$$

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\cos ^{-1}\left(\cos \frac{3 \pi}{2}\right)\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=\frac{5}{2} \cos (6 x+\pi)$$

A point on the end of a tuning fork moves in the simple harmonic motion described by \(d=a \sin \omega t\) A tuning fork for middle \(C\) has a frequency of 264 vibrations per second. Find \(\omega.\)

Use trigonometric identities to transform one side of the equation into the other \((0<{\theta}<\pi /2)\). $$(1+\cos \theta)(1-\cos \theta)=\sin ^{2} \theta$$

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=208^{\circ}$$

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-2 \sin (4 x+\pi)$$

Rewrite each angle in radian measure as a multiple of \(\pi .\) (Do not use a calculator.) (a) \(30^{\circ}\) (b) \(150^{\circ}\)

A buoy oscillates in simple harmonic motion as waves go past. The buoy moves a total of 3.5 feet from its high point to its low point (see figure), and it returns to its high point every 10 seconds. Write an equation that describes the motion of the buoy, where the high point corresponds to the time \(t=0.\)

Use trigonometric identities to transform one side of the equation into the other \((0<{\theta}<\pi /2)\). $$(\csc \theta+\cot \theta)(\csc \theta-\cot \theta)=1$$

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=322^{\circ}$$

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\cos ^{-1}\left(\tan \frac{3 \pi}{4}\right)\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-4 \sin \left(\frac{2}{3} x-\frac{\pi}{3}\right)$$

Rewrite each angle in radian measure as a multiple of \(\pi .\) (Do not use a calculator.) (a) \(315^{\circ}\) (b) \(120^{\circ}\)

A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by \(y=\frac{1}{4} \cos 16 t, \quad t>0.\) where \(y\) is measured in feet and \(t\) is the time in seconds. (a) Use a graphing utility to graph the function. (b) What is the period of the oscillations? (c) Determine the first time the ball passes the point of equilibrium \((y=0).\)

Use trigonometric identities to transform one side of the equation into the other \((0<{\theta}<\pi /2)\). $$\frac{\sec \theta-\cos \theta}{\sec \theta}=\sin ^{2} \theta$$

An object weighing \(W\) pounds is suspended from a ceiling by a steel spring. The weight is pulled downward (positive direction) from its equilibrium position and released (see figure). The resulting motion of the weight is described by the function \(y=\frac{1}{2} e^{-t / 4} \cos 4 t,\) where \(y\) is the distance in feet and \(t\) is the time in seconds \((t>0)\). (a) Use a graphing utility to graph the function. (b) Describe the behavior of the displacement function for increasing values of time \(t\).

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-292^{\circ}$$

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\sin \left(\arctan \frac{4}{3}\right)\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=\cos \left(2 \pi x-\frac{\pi}{2}\right)+1$$

Rewrite each angle in radian measure as a multiple of \(\pi .\) (Do not use a calculator.) (a) \(18^{\circ}\) (b) \(-240^{\circ}\)

The numbers of hours \(H\) of daylight in Denver, Colorado, on the 15th of each month are given by ordered pairs of the form \((t, H(t)),\) where \(t=1\) represents January. A model for the data is \(H(t)=12.18+2.81 \sin \left(\frac{\pi t}{6}-\frac{\pi}{2}\right).\) (Spreadsheet at LarsonPrecalculus.com) (Source: United States Navy) (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem? Explain.

Use trigonometric identities to transform one side of the equation into the other \((0<{\theta}<\pi /2)\). $$\frac{\tan \theta+\cot \theta}{\tan \theta}=\csc ^{2} \theta$$

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-165^{\circ}$$

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\cos \left(\text { arcsin } \frac{24}{25}\right)\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2$$

Rewrite each angle in radian measure as a multiple of \(\pi .\) (Do not use a calculator.) (a) \(-330^{\circ}\) (b) \(144^{\circ}\)

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\sin \theta=\frac{1}{2}\) (b) \(\csc \theta=2\)

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-\frac{11 \pi}{5}$$

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\sec \left(\arcsin \frac{4}{5}\right)\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=5 \sin (\pi-2 x)+10$$

Rewrite each angle in degree measure. (Do not use a calculator.) (a) \(\frac{3 \pi}{2}\) (b) \(-\frac{7 \pi}{6}\)

Irrigation Engineering The cross sections of an irrigation canal are isosceles trapezoids, where the lengths of three of the sides are 8 feet (see figure). The objective is to find the angle \(\theta\) that maximizes the area of the cross sections. [Hint: The area of a trapezoid is given by \(\left.(h / 2)\left(b_{1}+b_{2}\right) .\right]\) (a) Complete seven rows of the table. $$\begin{array}{|c|c|c|c|} \hline \text {Base } I & \text {Base 2} & \text {Altitude} & \text {Area} \\ \hline 8 & 8+16 \cos 10^{\circ} & 8 \sin 10^{\circ} & 22.06 \\ \hline 8 & 8+16 \cos 20^{\circ} & 8 \sin 20^{\circ} & 42.46 \\ \hline \end{array}$$ (b) Use the table feature of a graphing utility to generate additional rows of the table. Use the table to estimate the maximum cross-sectional area. (c) Write the area \(A\) as a function of \(\theta.\) (d) Use the graphing utility to graph the function. Use the graph to estimate the maximum cross-sectional area. How does your estimate compare with that in part (b)?

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\cos \theta=\frac{\sqrt{2}}{2}\) (b) \(\tan \theta=1\)

Determine whether the statement is true or false. Justify your answer. The graph of \(y=-\frac{1}{8} \tan \left(\frac{x}{2}+\pi\right)\) has an asymptote at \(x=-7 \pi\).

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=\frac{10 \pi}{3}$$

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\csc \left[\arctan \left(-\frac{12}{5}\right)\right]\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=5 \cos (\pi-2 x)+6$$

Rewrite each angle in degree measure. (Do not use a calculator.) (a) \(-4 \pi\) (b) \(3 \pi\)

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\sec \theta=2\) (b) \(\cot \theta=1\)

Determine whether the statement is true or false. Justify your answer. For the graph of \(y=2^{x} \sin x,\) as \(x\) approaches \(-\infty\) \(y\) approaches \(0 .\)

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\boldsymbol{\theta}=1.8$$

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\sin \left[\arccos \left(-\frac{2}{3}\right)\right]\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=\frac{1}{100} \sin 120 \pi t$$

Rewrite each angle in degree measure. (Do not use a calculator.) (a) \(\frac{7 \pi}{3}\) (b) \(-\frac{13 \pi}{60}\)

True or False Determine whether the statement is true or false. Justify your answer. Simple harmonic motion does not involve a damping factor.

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\csc \theta=\frac{2 \sqrt{3}}{3}\) (b) \(\sin \theta=\frac{1}{2}\)