Chapter 11

Find the phase shift of each function. \(y=\cos (2 x-\pi)\)

In \(27-32,\) for each of the given inverse trigonometric function values, find the exact function value. \(\sin (\arccos 1)\)

Sketch one cycle of each function. \(y=\sin x\)

In \(27-32,\) for each of the given inverse trigonometric function values, find the exact function value. \(\cos (\arcsin 1)\)

Sketch one cycle of each function. \(y=\cos x\)

In \(27-32,\) for each of the given inverse trigonometric function values, find the exact function value. \(\tan (\arctan 1)\)

Sketch one cycle of each function. \(y=\sin 2 x\)

In \(27-32,\) for each of the given inverse trigonometric function values, find the exact function value. \(\sin \left(\arccos -\frac{\sqrt{3}}{2}\right)\)

Sketch one cycle of each function. \(y=\sin \frac{1}{2} x\)

In \(27-32,\) for each of the given inverse trigonometric function values, find the exact function value. \(\sin (\arctan -1)\)

Sketch one cycle of each function. \(y=\cos 3 x\)

In \(27-32,\) for each of the given inverse trigonometric function values, find the exact function value. \(\cos \left(\arccos -\frac{1}{2}\right)\)

Sketch one cycle of each function. \(y=3 \cos x\)

a. On the same set of axes, sketch the graph of \(y=\arcsin x\) and of its inverse function. b. What are the domain and range of each of the functions graphed in part a?

Sketch one cycle of each function. \(y=4 \sin 3 x\)

a. On the same set of axes, sketch the graph of \(y=\arccos x\) and of its inverse function. b. What are the domain and range of each of the functions graphed in part a?

Sketch one cycle of each function. \(y=\frac{1}{2} \cos \frac{1}{3} x\)

a. On the same set of axes, sketch the graph of \(y=\arctan x\) and of its inverse function. b. What are the domain and range of each of the functions graphed in part a?

Sketch one cycle of each function. \(y=-\sin 2 x\)

Sketch one cycle of each function. \(y=-\cos \frac{1}{2} x\)

Sketch one cycle of each function. \(y=\sin \left(x+\frac{\pi}{2}\right)\)

Sketch one cycle of each function. \(y=\frac{1}{2} \cos \left(x-\frac{\pi}{4}\right)\)

Show that the graph of \(y=\sin x\) is the graph of \(y=\cos \left(x-\frac{\pi}{2}\right)\)

As stated in the Chapter Opener, sound can be thought of as vibrating air. Simple sounds can be modeled by a function h \((t)\) of the form $$\mathrm{h}(t)=\sin (2 \pi f t)$$ where the frequency \(f\) is in kilohertz \((\mathrm{kHz})\) and \(t\) is time. a. The frequency of "middle \(\mathrm{C}^{\prime \prime}\) is approximately 0.261 \(\mathrm{kHz}\). Graph two cycles of \(\mathrm{h}(t)\) for middle \(\mathrm{C} .\) b. The frequency of \(C_{3},\) or the \(C\) note that is one octave lower than middle \(C\) , is approximately 0.130 \(\mathrm{kHz}\) . On the same set of axes, graph two cycles of \(\mathrm{h}(t)\) for \(\mathrm{C}_{3}\). c. Based on the graphs from parts a and b, the periods of each function appear to be related in what way?