Chapter 5

Use the function value(s) and the trigonometric identities to evaluate each trigonometric function. \(\sin 30^{\circ}=\frac{1}{2}, \tan 30^{\circ}=\frac{\sqrt{3}}{3}\) (a) \(\csc 30^{\circ}\) (b) \(\cot 60^{\circ}\) (c) \(\cos 30^{\circ}\) (d) \(\cot 30^{\circ}\)

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=315^{\circ}$$

Use the properties of inverse functions to find the exact value of the expression, if possible. tan(arctan 25)

Sketch the graph of the function. Use a graphing utility to verify your sketch. (Include two full periods.) $$y=2 \cos x-3$$

Determine the quadrant in which each angle lies. (The angle measure is given in radians.) (a) \(-\frac{5 \pi}{12}\) (b) \(-\frac{13 \pi}{9}\)

Harmonic Motion Find a model for simple harmonic motion satisfying the specified conditions. Displacement \((t=0)$$\quad\)Amplitude\(\quad\)Period 0\(\quad\)8 centimeters\(\quad\)2 seconds

Use the function value(s) and the trigonometric identities to evaluate each trigonometric function. \(\csc \theta=3, \sec \theta=\frac{3 \sqrt{2}}{4}\) (a) \(\sin \theta\) (b) \(\cos \theta\) (c) \(\tan \theta\) (d) \(\sec \left(90^{\circ}-\theta\right)\)

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-45^{\circ}$$

Sketch the graph of the function. Use a graphing utility to verify your sketch. (Include two full periods.) $$y=\frac{2}{3} \cos \left(\frac{x}{2}-\frac{\pi}{4}\right)$$

Determine the quadrant in which each angle lies. (The angle measure is given in radians.) (a) \(-1\) (b) \(-2\)

Harmonic Motion Find a model for simple harmonic motion satisfying the specified conditions. Displacement \((t=0)$$\quad\)Amplitude\(\quad\)Period 0\(\quad\)3 meters\(\quad\)6 seconds

Use the function value(s) and the trigonometric identities to evaluate each trigonometric function. \(\sec \theta=5, \tan \theta=2 \sqrt{6}\) (a) \(\cos \theta\) (b) \(\cot \theta\) (c) \(\cot \left(90^{\circ}-\theta\right)\) (d) \(\sin \theta\)

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-330^{\circ}$$

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\sin [\arcsin (-0.1)]\)

Sketch the graph of the function. Use a graphing utility to verify your sketch. (Include two full periods.) $$y=-2 \cos (4 \pi x+1)$$

Determine the quadrant in which each angle lies. (The angle measure is given in radians.) (a) \(3.5\) (b) \( 2.25\)

Harmonic Motion Find a model for simple harmonic motion satisfying the specified conditions. Displacement \((t=0)$$\quad\)Amplitude\(\quad\)Period 3 inches\(\quad\)3 inches\(\quad\)1.5 seconds

Use the function value(s) and the trigonometric identities to evaluate each trigonometric function. \(\cot \alpha=4\) (a) \(\tan \alpha\) (b) \(\csc \alpha\) (c) \(\sec \alpha\) (d) \(\tan \left(90^{\circ}-\alpha\right)\)

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. \(\tan \left[\tan ^{-1}(-0.5)\right]\)

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

Sketch each angle in standard position. (a) \(\frac{3 \pi}{2}\) (b) \(-\frac{\pi}{2}\)

Harmonic Motion Find a model for simple harmonic motion satisfying the specified conditions. Displacement \((t=0)$$\quad\)Amplitude\(\quad\)Period -2 feet\(\quad\)2 feet\(\quad\)10 seconds

Use the function value(s) and the trigonometric identities to evaluate each trigonometric function. \(\tan \beta=3\) (a) \(\cot \beta\) (b) \(\cos \beta\) (c) \(\tan \left(90^{\circ}-\beta\right)\) (d) \(\csc \beta\)

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=\frac{3 \pi}{4}$$

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

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-10 \cos \frac{\pi x}{6}$$

Sketch each angle in standard position. (a) \(\frac{3 \pi}{4}\) (b) \(\frac{4 \pi}{3}\)

Harmonic Motion For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=4 \cos 8 \pi t$$

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

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) \(x \rightarrow \frac{\pi^{+}}{2}\left(\text { as } x \text { approaches } \frac{\pi}{2} \text { from the right }\right)\) (b) \(x \rightarrow \frac{\pi^{-}}{2}\left(\text { as } x \text { approaches } \frac{\pi}{2} \text { from the left }\right)\) (c) \(x \rightarrow-\frac{\pi^{+}}{2}\left(\text { as } x \text { approaches }-\frac{\pi}{2} \text { from the right }\right)\) (d) \(x \rightarrow-\frac{\pi^{-}}{2}\left(\text { as } x \text { approaches }-\frac{\pi}{2} \text { from the left }\right)\) \(f(x)=\tan x\)

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-\frac{5 \pi}{6}$$

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\arctan \left(\tan \frac{11 \pi}{6}\right)\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-4+5 \cos \frac{\pi t}{12}$$

Sketch each angle in standard position. (a) \(-\frac{7 \pi}{4}\) (b) \(-\frac{5 \pi}{2}\)

Harmonic Motion For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{2} \cos 20 \pi t$$

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

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) \(x \rightarrow \frac{\pi^{+}}{2}\left(\text { as } x \text { approaches } \frac{\pi}{2} \text { from the right }\right)\) (b) \(x \rightarrow \frac{\pi^{-}}{2}\left(\text { as } x \text { approaches } \frac{\pi}{2} \text { from the left }\right)\) (c) \(x \rightarrow-\frac{\pi^{+}}{2}\left(\text { as } x \text { approaches }-\frac{\pi}{2} \text { from the right }\right)\) (d) \(x \rightarrow-\frac{\pi^{-}}{2}\left(\text { as } x \text { approaches }-\frac{\pi}{2} \text { from the left }\right)\) \(f(x)=\sec x\)

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=\frac{7 \pi}{6}$$

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\arcsin \left(\sin \frac{4 \pi}{3}\right)\)

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

Sketch each angle in standard position. (a) \(\frac{11 \pi}{6}\) (b) \(-\frac{2 \pi}{3}\)

Harmonic Motion For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=-\frac{1}{16} \sin 140 \pi t$$

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

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 \(f(x) \rightarrow\square\). \(f(x)=\csc x\)

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=\frac{11 \pi}{6}$$

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\sin ^{-1}\left(\sin \frac{5 \pi}{2}\right)\)

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

Sketch each angle in standard position. (a) \(5 \pi\) (b) \(-4\)

Harmonic Motion For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{64} \sin 792 \pi t$$