Exam 5: Trigonometric Functions: Unit Circle Approach

Suppose that the terminal point determined by $\text { t }$ is the point $\left( \frac { \sqrt { 21 } } { 5 } , - \frac { 2 } { 5 } \right)$ . Find the terminal point determined by each of the following: (a) $2 \pi - t$ , (b) $- t$ , (c) $t - \pi$ , (d) $4 \pi + t$ .

Find the exact value of $\tan \left( \frac { 7 \pi } { 2 } \right)$ and .

Find the value of $\tan ^ { - 1 } \frac { 1 } { \sqrt { 3 } }$ if it is defined.

Find the exact value of the expression, if it is defined. $\cos \left( \tan ^ { - 1 } ( - \sqrt { 3 } ) \right)$

Find the exact value of the expression, if it is defined. $\sin \left( \sin ^ { - 1 } ( 2 / 3 ) \right)$

Find the exact value of the expression, if it is defined. $\sin ^ { - 1 } ( \sin ( - \pi / 3 ) )$

Find the terminal point $P ( x , y )$ on the unit circle determined by the value $t = - \frac {3 \pi } { 2 }$ .

If the terminal point determined by $\text { t }$ is $\left( \frac { 1 } { 2 } , - \frac { \sqrt{3} } { 2 } \right)$ , find $\sin$ $\text { t }$ , $\cos$ $\text { t }$ and $\tan$ $\text { t }$ .

Find the reference number for $t = - \frac { 26 \pi } { 5 }$ .

Graph the three functions $y = \frac { - 1 } { \sqrt { 1 + x ^ { 2 } } }$ , $y = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$ , and $y = \frac { \sin 2 \pi x } { \sqrt { 1 + x ^ { 2 } } }$ on a common screen. Explain how the graphs are related.

Find the exact value of each expression, if it is defined. $\sin ^ { - 1 } ( - 1 )$ $\cos ^ { - 1 } ( - \sqrt { 3 } / 2 )$ $\tan ^ { - 1 } ( \sqrt { 3 } / 3 )$

Sketch the graph of $y = 3 + \cos x$ .

Find $P ( x , y )$ on the unit circle, given that the $x$ - coordinate of $P$ is $\frac { 1 } { 3 }$ and $P$ is in quadrant IV.

Find the exact value of $\sin \left( \frac { 11 \pi } { 6 } \right)$ and $\cos \left( - \frac { 11 \pi } { 6 } \right)$ .

If the terminal point determined by $\text { t }$ is $\left( \frac { 12 } { 13 } , - \frac { 5 } { 13 } \right)$ , find $\sin t$ , $\operatorname { cost }$ and $\tan t$ .

Find the maximum and minimum values of the function $y = \cos x - \frac { 1 } { 2 } \cos 2 x$ .

If the terminal point determined by $\text { t }$ is $\left( \frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } \right)$ , find $\sin$ $\text { t }$ , $\cos$ $\text { t }$ and $\tan$ $\text { t }$ .

Find the value of $\cos \left( \cos ^ { - 1 } \left( - \frac { 3 } { 4 } \right) \right)$ if it is defined.

The function $y = - 2 \cos \left( \frac { t } { 3 } \right)$ models the displacement of an object moving in simple harmonic motion, where y is measured in inches and t in seconds. Find the amplitude, period, and frequency of motion and sketch a graph of the function over one complete period.

Find the amplitude, period and phase shift of $y = 2 \cos \left( 4 x - \frac { \pi } { 2 } \right)$ and sketch its graph.