Chapter 5

The trigonometric function \(y=\tan x\) has period _______ and asymptotes \(x=\) _______. Sketch a graph of this function the interval \((-\pi / 2, \pi / 2)\)

(a) To define the inverse sine function, we restrict the domain of sine to the interval _____________________ . On this interval the sine function is one-to- one, and its inverse function \(\sin ^{-1}\) is defined by \(\sin ^{-1} x=y \Leftrightarrow \sin\) _________ \(=\) __________ . For example, \(\sin ^{-1} \frac{1}{2}=\) __________ because sin ___________ = __________. (b) To define the inverse cosine function we restrict the domain of cosine to the interval _________________. On this interval the cosine function is one- to-one and its inverse function \(\cos ^{-1}\) is defined by \(\cos ^{-1} x=y \Leftrightarrow\) cos ____________ = ___________. For example, \(\cos ^{-1} \frac{1}{2}=\) _____________ because cos __________ = ___________.

For an object in simple harmonic motion with amplitude \(a\) and period \(2 \pi / \omega,\) find an equation that models the displacement \(y\) at time \(t\) if (a) \(y=0\) at time \(t=0: y=\) _______ (b) \(y=a\) at time \(t=0: y=\) _______

The trigonometric functions \(y=\sin x\) and \(y=\cos x\) have amplitude ______ and period ______. Sketch a graph of each function on the interval \([0,2 \pi]\) (Graph can't copy)

The trigonometric function \(y=\csc x\) has period _______ and asymptotes \(x=\) _______. Sketch a graph of this function on the interval \((-\pi, \pi)\)

(a) If we mark off a distance \(t\) along the unit circle, starting at \((1,0)\) and moving in a counterclockwise direction, we arrive at the _____ point determined by \(t.\) (b) The terminal points determined by \(\pi / 2, \pi,-\pi / 2,2 \pi\) are _____, _____, _____ and _____ respectively.

The trigonometric function \(y=3 \sin 2 x\) has amplitude ________ and period _______.

If \(P(x, y)\) is on the unit circle, then \(x^{2}+y^{2}=\) ____ So for all \(t\) we have \(\sin ^{2} t+\cos ^{2} t=\)____.

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1} 1\) (b) \(\sin ^{-1} \frac{\sqrt{3}}{2}\) (c) \(\sin ^{-1} 2\)

Show that the point is on the unit circle. $$\left(\frac{4}{5},-\frac{3}{5}\right)$$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=2 \sin 3 t$$

Graph the function. $$f(x)=1+\cos x$$

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1}(-1)\) (b) \(\sin ^{-1} \frac{\sqrt{2}}{2}\) (c) \(\sin ^{-1}(-2)\)

Show that the point is on the unit circle. $$\left(-\frac{5}{13}, \frac{12}{13}\right)$$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=3 \cos \frac{1}{2} t$$

Graph the function. $$f(x)=3+\sin x$$

Find the exact value of each expression, if it is defined. (a) \(\cos ^{-1}(-1)\) (b) \(\cos ^{-1} \frac{1}{2}\) (c) \(\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{2 \pi}{3}\) (b) \(\cos \frac{2 \pi}{3}\) (c) \(\tan \frac{2 \pi}{3}\)

Show that the point is on the unit circle. $$\left(\frac{7}{25}, \frac{24}{25}\right)$$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=-\cos 0.3 t$$

Graph the function. $$f(x)=-\sin x$$

Find the exact value of each expression, if it is defined. (a) \(\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)\) (b) \(\cos ^{-1} 1\) (c) \(\cos ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\)

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{5 \pi}{6}\) (b) \(\cos \frac{5 \pi}{6}\) (c) \(\tan \frac{5 \pi}{6}\)

Show that the point is on the unit circle. $$\left(-\frac{5}{7},-\frac{2 \sqrt{6}}{7}\right)$$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=2.4 \sin 3.6 t$$

Graph the function. $$f(x)=2-\cos x$$

Find the exact value of each expression, if it is defined. (a) \(\tan ^{-1}(-1)\) (b) \(\tan ^{-1} \sqrt{3}\) (c) \(\tan ^{-1} \frac{\sqrt{3}}{3}\)

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{7 \pi}{6}\) (b) \(\sin \left(-\frac{\pi}{6}\right)\) (c) \(\sin \frac{11 \pi}{6}\)

Show that the point is on the unit circle. $$\left(-\frac{\sqrt{5}}{3}, \frac{2}{3}\right)$$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=-0.25 \cos \left(1.5 t-\frac{\pi}{3}\right)$$

Graph the function. $$f(x)=-2+\sin x$$

Find the exact value of each expression, if it is defined. (a) \(\tan ^{-1} 0\) (b) \(\tan ^{-1}(-\sqrt{3})\) (c) \(\tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right)\)

Find the exact value of the trigonometric function at the given real number. (a) \(\cos \frac{5 \pi}{3}\) (b) \(\cos \left(-\frac{5 \pi}{3}\right)\) (c) \(\cos \frac{7 \pi}{3}\)

Show that the point is on the unit circle. $$\left(\frac{\sqrt{11}}{6}, \frac{5}{6}\right)$$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=-\frac{3}{2} \sin (0.2 t+1.4)$$

Graph the function. $$f(x)=-1+\cos x$$

Find the exact value of each expression, if it is defined. (a) \(\cos ^{-1}\left(-\frac{1}{2}\right)\) (b) \(\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) (c) \(\tan ^{-1} 1\)

Find the period and graph the function. $$y=4 \tan x$$

Find the exact value of the trigonometric function at the given real number. (a) \(\cos \frac{3 \pi}{4}\) (b) \(\cos \frac{5 \pi}{4}\) (c) \(\cos \frac{7 \pi}{4}\)

Find the missing coordinate of \(P,\) using the fact that \(P\) lies on the unit circle in the given quadrant. Coordinates $$P\left(-\frac{3}{5}, \quad\right)$$ Quadrant III

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=5 \cos \left(\frac{2}{3} t+\frac{3}{4}\right)$$

Graph the function. $$g(x)=3 \cos x$$

Find the exact value of each expression, if it is defined. (a) \(\cos ^{-1} 0\) (b) \(\sin ^{-1} 0\) (c) \(\sin ^{-1}\left(-\frac{1}{2}\right)\)

Find the period and graph the function. $$y=-4 \tan x$$

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{3 \pi}{4}\) (b) \(\sin \frac{5 \pi}{4}\) (c) \(\sin \frac{7 \pi}{4}\)

Find the missing coordinate of \(P,\) using the fact that \(P\) lies on the unit circle in the given quadrant. Coordinates $$P\left(\quad,-\frac{7}{25}\right)$$ Quadrant IV

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=1.6 \sin (t-1.8)$$

Graph the function. $$g(x)=2 \sin x$$

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\sin ^{-1} \frac{2}{3}$$

Find the period and graph the function. $$y=-\frac{1}{2} \tan x$$