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Exam 5: Trigonometric Functions: Unit Circle Approach

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Question 41

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Find the exact value of the expression, if it is defined. $\tan ^ { - 1 } ( \tan ( - \pi / 4 ) )$

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Question 42

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Find the sign of $\frac { \tan t } { \sec t }$ if the terminal point determined by $\text { t }$ is in quadrant IV.

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Question 43

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The function $y = - \frac { 1 } { 24 } \sin ( 24 \pi t )$ 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 terminal point $P ( x , y )$ on the unit circle determined by the value .

Given that $\operatorname { sect } < 0$ and $\tan t > 0$ , find the quadrant in which the terminal point determined by $\text { t }$ lies.

Find $P ( x , y )$ on the unit circle, given that the $y$ - coordinate of $P$ is $- \frac { 2\sqrt { 6 } } { 5 }$ and $P$ is in quadrant III.

Find the exact value of $\sin \left( \frac { 7 \pi } { 4 } \right)$ and $\cos \left( \frac { 7 \pi } { 4 } \right)$ .

Find the approximate value of $\tan ^ { - 1 } ( - 0.5 )$ correct to $5$ decimal places.

Sketch the graph of $y = 1 - 2 \cos x$ .

Sketch the graph of $y = | 2 \cos x |$ .

Find the value of $\sin ^ { - 1 } \left( - \frac { 5 } { 4 } \right)$ if it is defined.

Find the period of the function $y = \csc \frac { 2 } { \pi } x$ and sketch its graph.

Find the approximate value of $\cos ( 6.1 )$ using a calculator.

Find the sign of $\cos t \sin t$ if the terminal point determined by $\text { t }$ is in quadrant II.

Find all solutions of the equation $\sec x = - x$ that lie in the interval $[ - 2 \pi , 2 \pi ]$ . State each answer correct to two decimal places.

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. (a) $\cos ^ { - 1 } ( 1.34567 )$ (b) $\sin ^ { - 1 } ( 1.34567 )$ (c) $\tan ^ { - 1 } ( 1.34567 )$

Find the values of the trigonometric functions of $\text { t }$ given that $\operatorname { sect } = - 2$ and $\tan t > 0$ .

Find the amplitude, period and phase shift of $y = \pi \sin ( \pi x + \pi )$ and sketch its graph.

Suppose that the terminal point determined by $\text { t }$ is the point $\left( \frac { 5 } { 13 } , \frac { 12 } { 13 } \right)$ on the unit circle. Find the terminal point determined by (a) $\pi + t$ , (b) $- t$ , (c) $2 \pi - t$ , (d) $t - \pi$ .

Find the period of the function $y = 2 \tan x$ and sketch its graph.