Deck 7: The Trigonometric Functions

Question 1

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the given angle: $\frac { 3 \pi } { 2 }$

A) $\cos \frac { 3 \pi } { 2 } = 0$ , $\sin \frac { 3 \pi } { 2 } = - 1$ , $\tan \frac { 3 \pi } { 2 }$ is undefined, $\sec \frac { 3 \pi } { 2 }$ is undefined, $\cot \frac { 3 \pi } { 2 } = 0$ , $\csc \frac { 3 \pi } { 2 } = - 1$ .
B) $\cos \frac { 3 \pi } { 2 } = - 1$ , $\sin \frac { 3 \pi } { 2 } = 1$ , $\tan \frac { 3 \pi } { 2 } = 0$ , $\sec \frac { 3 \pi } { 2 } = - 1$ , $\cot \frac { 3 \pi } { 2 }$ is undefined, $\csc \frac { 3 \pi } { 2 }$ is undefined.
C) $\cos \frac { 3 \pi } { 2 } = 1$ , $\sin \frac { 3 \pi } { 2 } = 1$ , $\tan \frac { 3 \pi } { 2 }$ is undefined, $\sec \frac { 3 \pi } { 2 }$ is undefined, $\cot \frac { 3 \pi } { 2 } = 0$ , $\csc \frac { 3 \pi } { 2 } = - 1$ .
D) $\cos \frac { 3 \pi } { 2 } = - \frac { 1 } { 2 }$ , $\sin \frac { 3 \pi } { 2 } = \frac { \sqrt { 3 } } { 2 }$ , $\tan \frac { 3 \pi } { 2 } = 0$ , $\sec \frac { 3 \pi } { 2 } = - 1$ , $\cot \frac { 3 \pi } { 2 }$ is undefined, $\csc \frac { 3 \pi } { 2 }$ is undefined.
E) $\cos \frac { 3 \pi } { 2 } = - \frac { 1 } { 2 }$ , $\sin \frac { 3 \pi } { 2 } = \frac { \sqrt { 3 } } { 2 }$ , $\tan \frac { 3 \pi } { 2 }$ is undefined, $\sec \frac { 3 \pi } { 2 }$ is undefined, $\cot \frac { 3 \pi } { 2 } = 0$ , $\csc \frac { 3 \pi } { 2 } = - 1$ .

Answer 1

$\cos \frac { 3 \pi } { 2 } = 0$ , $\sin \frac { 3 \pi } { 2 } = - 1$ , $\tan \frac { 3 \pi } { 2 }$ is undefined, $\sec \frac { 3 \pi } { 2 }$ is undefined, $\cot \frac { 3 \pi } { 2 } = 0$ , $\csc \frac { 3 \pi } { 2 } = - 1$ .

Question 2

Rewrite in terms of sine and cosine, and simplify the expression: $\frac { 3 \sin \theta + 6 } { \sin ^ { 2 } \theta - 4 }$

A) $\frac { \sin \theta + 2 } { 4 }$
B) $\frac { \sin \theta - 2 } { 3 }$
C) $\frac { 6 } { \sin \theta + 2 }$
D) $\frac { 3 } { \sin \theta - 2 }$
E) $\frac { \sin \theta + 2 } { 6 }$

Answer 2

$\frac { 3 } { \sin \theta - 2 }$

Question 3

Rewrite in terms of sine and cosine, and simplify the expression: $\sec A \csc A - \tan A - \cot A$

A) $\sec A$
B) $0$
C) $\cot A$
D) $1$
E) $\tan A$

Answer 3

$0$

Question 4

Use the definition $\theta = \frac { s } { r }$ to determine the radian measure of the angle.

A) $\theta = 4.05$ radians
B) $\theta = 3$ radians
C) $\theta = 3.08$ radians
D) $\theta$ radians
E) $\theta = 3.15$ radians

Question 5

Use the definition $\theta = \frac { s } { r }$ to determine the radian measure of the angle in the figure below.

A) $\theta = 0.2$ radians
B) $\theta = 0.4$ radians
C) $\theta = 1.25$ radians
D) $\theta = 0.83$ radians
E) $\theta = 0.13$ radians

Question 6

Evaluate the expressions using reference angles. $\sec \left( \frac { 7 \pi } { 4 } \right)$ $\tan \left( \frac { 7 \pi } { 4 } \right)$

A) $\sec \left( \frac { 7 \pi } { 4 } \right) = \sqrt { 3 }$ $\tan \left( \frac { 7 \pi } { 4 } \right) = 2$
B) $\sec \left( \frac { 7 \pi } { 4 } \right) = \sqrt { 2 }$ $\tan \left( \frac { 7 \pi } { 4 } \right) = - 1$
C) $\sec \left( \frac { 7 \pi } { 4 } \right) = \sqrt { 2 }$ $\tan \left( \frac { 7 \pi } { 4 } \right) = - \frac { 1 } { 2 }$
D) $\sec \left( \frac { 7 \pi } { 4 } \right) = \frac { \sqrt { 2 } } { 2 }$ $\tan \left( \frac { 7 \pi } { 4 } \right) = - 4$
E) $\sec \left( \frac { 7 \pi } { 4 } \right) = \sqrt { 5 }$ $\tan \left( \frac { 7 \pi } { 4 } \right) = - 1$

Question 7

Let $P ( x , y )$ denote the point where the terminal side of angle $\theta$ (in standard position) meets the unit circle. Use the information to evaluate the six trigonometric functions of $\theta$ . $P$ is in Quadrant IV and $y = - \frac { 3 } { 4 }$ .

A) $\sin \theta = - \frac { \sqrt { 7 } } { 4 }$ , $\cos \theta = \frac { 3 } { 4 }$ , $\tan \theta = - \frac { \sqrt { 7 } } { 3 }$ , $\sec \theta = \frac { 4 } { 3 }$ , $\csc \theta = \frac { 4 \sqrt { 7 } } { 7 }$ , $\cot \theta = \frac { 3 \sqrt { 7 } } { 7 }$
B) $\sin \theta = - \frac { \sqrt { 7 } } { 4 }$ , $\cos \theta = \frac { 3 } { 4 }$ , $\tan \theta = \frac { \sqrt { 7 } } { 3 }$ , $\sec \theta = \frac { 4 } { 3 }$ , $\csc \theta = - \frac { 4 \sqrt { 7 } } { 7 }$ , $\cot \theta = \frac { 3 \sqrt { 7 } } { 7 }$
C) $\sin \theta = - \frac { 3 } { 4 }$ , $\cos \theta = \frac { \sqrt { 7 } } { 4 }$ , $\tan \theta = - \frac { 3 \sqrt { 7 } } { 7 }$ , $\sec \theta = \frac { 4 \sqrt { 7 } } { 7 }$ , $\csc \theta = - \frac { 4 } { 3 }$ , $\cot \theta = - \frac { \sqrt { 7 } } { 3 }$
D) $\sin \theta = \frac { \sqrt { 7 } } { 4 } \quad \cos \theta = - \frac { 3 } { 4 } \quad \tan \theta = - \frac { \sqrt { 7 } } { 3 }$ $\sec \theta = \frac { 4 } { 3 }$ , $\csc \theta = \frac { \sqrt { 7 } } { 7 }$ , $\cot \theta = - \frac { \sqrt { 7 } } { 7 }$
E) $\sin \theta = \frac { 3 } { 4 }$ , $\cos \theta = - \frac { \sqrt { 7 } } { 4 }$ , $\tan \theta = \frac { 3 \sqrt { 7 } } { 7 }$ , $\sec \theta = - \frac { 4 \sqrt { 7 } } { 7 }$ , $\csc \theta = \frac { 4 } { 3 }$ , $\cot \theta = \frac { \sqrt { 7 } } { 3 }$

Question 8

Refer to the figure, which shows all of the angles from $0 ^ { \circ }$ to $360 ^ { \circ }$ that are multiples of $30 ^ { \circ }$ or $45 ^ { \circ }$ . Relabel the angles in Quadrant III and IV using radian measure.

A) $$\begin{array} { | c | c | c | c | c | c | } \hline 210 ^ { \circ } & 225 ^ { \circ } & 240 ^ { \circ } & 300 ^ { \circ } & 315 ^ { \circ } & 330 ^ { \circ } \\\hline \frac { 3 \pi } { 4 } & \frac { 2 \pi } { 5 } & \frac { 4 \pi } { 7 } & \frac { \pi } { 4 } & \frac { 2 \pi } { 3 } & \frac { 5 \pi } { 2 } \\\hline\end{array}$$
B) $$\begin{array} { | c | c | c | c | c | c | } \hline 210 ^ { \circ } & 225 ^ { \circ } & 240 ^ { \circ } & 300 ^ { \circ } & 315 ^ { \circ } & 330 ^ { \circ } \\\hline \frac { 9 \pi } { 4 } & \frac { 7 \pi } { 4 } & \frac { 5 \pi } { 6 } & \frac { 10 \pi } { 3 } & \frac { 11 \pi } { 4 } & \frac { 11 \pi } { 6 } \\\hline\end{array}$$
C) $$\begin{array} { | c | c | c | c | c | c | } \hline 210 ^ { \circ } & 225 ^ { \circ } & 240 ^ { \circ } & 300 ^ { \circ } & 315 ^ { \circ } & 330 ^ { \circ } \\\hline \frac { 7 \pi } { 6 } & \frac { \pi } { 3 } & \frac { \pi } { 5 } & \frac { 2 \pi } { 3 } & \frac { 3 \pi } { 7 } & \frac { 3 \pi } { 2 } \\\hline\end{array}$$
D) $$\begin{array} { | c | c | c | c | c | c | } \hline 210 ^ { \circ } & 225 ^ { \circ } & 240 ^ { \circ } & 300 ^ { \circ } & 315 ^ { \circ } & 330 ^ { \circ } \\\hline \frac { \pi } { 6 } & \frac { \pi } { 4 } & \frac { \pi } { 3 } & \frac { 2 \pi } { 3 } & \frac { 3 \pi } { 4 } & \frac { 5 \pi } { 6 } \\\hline\end{array}$$
E) $$\begin{array} { | c | c | c | c | c | c | } \hline 210 ^ { \circ } & 225 ^ { \circ } & 240 ^ { \circ } & 300 ^ { \circ } & 315 ^ { \circ } & 330 ^ { \circ } \\\hline \frac { 7 \pi } { 6 } & \frac { 5 \pi } { 4 } & \frac { 4 \pi } { 3 } & \frac { 5 \pi } { 3 } & \frac { 7 \pi } { 4 } & \frac { 11 \pi } { 6 } \\\hline\end{array}$$

Question 9

Factor the expression. $3 \sec ^ { 2 } \beta + 8 \sec \beta - 16$

A) $( 3 \sec \beta - 3 ) ( \sec \beta + 5 )$
B) $( 3 \sec \beta - 7 ) ( \sec \beta + 4 )$
C) $( 3 \sec \beta - 7 ) ( \sec \beta + 3 )$
D) $( 3 \sec \beta - 5 ) ( \sec \beta + 6 )$
E) $( 3 \sec \beta - 4 ) ( \sec \beta + 4 )$

Question 10

Factor the expression. $\tan ^ { 2 } \beta + 6 \tan \beta - 7$

A) $( \tan \beta - 3 ) ( \tan \beta + 5 )$
B) $( \tan \beta - 1 ) ( \tan \beta + 7 )$
C) $( \tan \beta - 1 ) ( \tan \beta + 6 )$
D) $( \tan \beta - 4 ) ( \tan \beta + 4 )$
E) $( \tan \beta - 5 ) ( \tan \beta + 4 )$

Question 11

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the given angle: $- 360 ^ { \circ }$

A) $\sin \left( - 360 ^ { \circ } \right) = 0$ , $\cos \left( - 360 ^ { \circ } \right) = 1$ , $\tan \left( - 360 ^ { \circ } \right) = 0$ , $\sec \left( - 360 ^ { \circ } \right) = 1$ , $\csc \left( - 360 ^ { \circ } \right)$ is undefined, $\cot \left( - 360 ^ { \circ } \right)$ is undefined.
B) $\sin \left( - 360 ^ { \circ } \right) = 0$ , $\cos \left( - 360 ^ { \circ } \right) = 1$ , $\tan \left( - 360 ^ { \circ } \right)$ is undefined, $\sec \left( - 360 ^ { \circ } \right)$ is undefined. $$\csc \left( - 360 ^ { \circ } \right) = 0$$ , $$\cot \left( - 360 ^ { \circ } \right) = 1$$ .
C) $\sin \left( - 360 ^ { \circ } \right) = 0$ , $\cos \left( - 360 ^ { \circ } \right) = - 1$ , $\tan \left( - 360 ^ { \circ } \right) = 0$ , $\sec \left( - 360 ^ { \circ } \right) = - 1$ , $\csc \left( - 360 ^ { \circ } \right)$ is undefined, $\cot \left( - 360 ^ { \circ } \right)$ is undefined.
D) $\sin \left( - 360 ^ { \circ } \right) = 1$ , $\cos \left( - 360 ^ { \circ } \right) = 1$ , $\tan \left( - 360 ^ { \circ } \right)$ is undefined, $\sec \left( - 360 ^ { \circ } \right)$ is undefined. $\csc \left( - 360 ^ { \circ } \right) = 0$ , $\cot \left( - 360 ^ { \circ } \right) = 0$ .
E) $\sin \left( - 360 ^ { \circ } \right) = 1$ , $\cos \left( - 360 ^ { \circ } \right) = - 1$ , $\tan \left( - 360 ^ { \circ } \right) = 1$ , $\sec \left( - 360 ^ { \circ } \right) = - 1$ , $\csc \left( - 360 ^ { \circ } \right)$ is undefined, $\cot \left( - 360 ^ { \circ } \right)$ is undefined.

Question 12

You are given the rate of rotation of a wheel as well as its radius. Determine the following: (a) the angular speed, in units of radians/sec; (b) the linear speed, in units of cm/sec, of a point on the circumference of the wheel; (c) and the linear speed, in cm/sec, of a point halfway between the center of the wheel and the circumference. $$520$$ rpm; $r = 45$ cm

A) (a) 156 $\pi$ radian/sec,
(b) 780 $\pi$ cm/sec,
(c) 390 $\pi$ cm/sec
B) (a) $166 ~\pi ~\mathrm { radian } / \mathrm { sec }$
(b) 1,278 $\pi$ cm/sec,
(c) 400 $\pi$ cm/sec
C) (a) $\frac { 52 \pi } { 11 }$ radian/sec,
(b) 1,248 $\pi$ cm/sec,
(c) 624 $\pi$ cm/sec
D) (a) 520 $\pi$ radian/sec,
(b) 780 $\pi$ cm/sec,
(c) 1,560 $\pi$ cm/sec
E) (a) $\frac { 52 \pi } { 3 }$ radian/sec,
(b) 780 $\pi$ cm/sec,
(c) 390 $\pi$ cm/sec

Question 13

Use a calculator to evaluate $\sec \theta$ , $\csc \theta$ , and $\cot \theta$ for the value of $\theta$ . Round the answers to two decimal places. $\theta = 27 ^ { \circ }$

A) $\sec \theta = 1.12$ $\csc \theta = 2.20$ $\cot \theta = 1.96$
B) $\sec \theta = 1.32$ $\csc \theta = 1.87$ $\cot \theta = 1.67$
C) $\sec \theta = 1.60$ $\csc \theta = 3.15$ $\cot \theta = 1.37$
D) $\sec \theta = 1.12$ $\csc \theta = 3.15$ $\cot \theta = 1.67$
E) $\sec \theta = 1.32$ $\csc \theta = 2.20$ $\cot \theta = 1.37$

Question 14

Refer to the figure, which indicates radian measure on the unit circle for angles in standard position. Use the figure (and the unit circle definitions) to determine whether $\sin 6 , \cos 6$ and $\tan 6$ are positive or negative.

A) $\sin 6$ is positive, $\cos 6$ is positive, $\tan 6$ is negative
B) $\sin 6$ is negative, $\cos 6$ is positive, $\tan 6$ is negative
C) $\sin 6$ is positive, $\cos 6$ is positive, $\tan 6$ is positive
D) $\sin 6$ is positive, $\cos 6$ is negative, $\tan 6$ is positive
E) $\sin 6$ is negative, $\cos 6$ is negative, $\tan 6$ is positive

Question 15

When a clock reads $2 : 00$ , what is the radian measure of the (smaller) angle between the hour hand and the minute hand?

A) $\frac { 3 \pi } { 13 }$ radians
B) $\frac { 2 \pi } { 9 }$ radians
C) $\frac { \pi } { 3 }$ radians
D) $\frac { \pi } { 4 }$ radians
E) $\frac { 2 \pi } { 7 }$ radians

Question 16

Use the figure to approximate the trigonometric values to within successive tenths. Then use a calculator to compute the values to the nearest hundredth. $\cos 3$ and $\sin 3$

A) $$\begin{array} { | c | c | c | } \hline & \text { Approximate } & \text { Calculator } \\\hline \cos 3 & 0.9 < \cos 3 < 1 & 0.99 \\\hline \sin 3 & - 0.2 < \sin 3 < - 0.1 & - 0.14 \\\hline\end{array}$$
B) $$\begin{array} { | c | c | c | } \hline & \text { Approximate } & \text { Calculator } \\\hline \cos 3 & 0.1 < \cos 3 < 0.2 & 0.14 \\\hline \sin 3 & - 1 < \sin 3 < - 0.9 & - 0.99 \\\hline\end{array}$$
C) $$\begin{array} { | c | c | c | } \hline & \text { Approximate } & \text { Calculator } \\\hline \cos 3 & 0.9 < \cos 3 < 1 & 0.99 \\\hline \sin 3 & 0.1 < \sin 3 < 0.2 & 0.14 \\\hline\end{array}$$
D) $$\begin{array} { | c | c | c | } \hline & \text { Approximate } & \text { Calculator } \\\hline \cos 3 & - 0.2 < \cos 3 < - 0.1 & - 0.14 \\\hline \sin 3 & 0.9 < \sin 3 < 1 & 0.99 \\\hline\end{array}$$
E) $$\begin{array} { | c | c | c | } \hline & \text { Approximate } & \text { Calculator } \\\hline \cos 3 & - 1 < \cos 3 < - 0.9 & - 0.99 \\\hline \sin 3 & 0.1 < \sin 3 < 0.2 & 0.14 \\\hline\end{array}$$

Question 17

Refer to the figure, which shows all of the angles from $$0 ^ { \circ }$$ to $360 ^ { \circ }$ that are multiples of $30 ^ { \circ }$ or $45 ^ { \circ }$ . Relabel the angles in Quadrant I and II using radian measure.

A) $$\begin{array} { | c | c | c | c | c | c | } \hline 30 ^ { \circ } & 45 ^ { \circ } & 60 ^ { \circ } & 120 ^ { \circ } & 135 ^ { \circ } & 150 ^ { \circ } \\\hline \frac { \pi } { 4 } & \frac { \pi } { 3 } & \frac { \pi } { 6 } & \frac { 3 \pi } { 2 } & \frac { 6 \pi } { 7 } & \frac { 5 \pi } { 9 } \\\hline\end{array}$$
B) $$\begin{array} { | c | c | c | c | c | c | } \hline 30 ^ { \circ } & 45 ^ { \circ } & 60 ^ { \circ } & 120 ^ { \circ } & 135 ^ { \circ } & 150 ^ { \circ } \\\hline \frac { 7 \pi } { 6 } & \frac { 5 \pi } { 4 } & \frac { 4 \pi } { 3 } & \frac { 5 \pi } { 3 } & \frac { 7 \pi } { 4 } & \frac { 11 \pi } { 6 } \\\hline\end{array}$$
C) $$\begin{array} { | c | c | c | c | c | c | } \hline 30 ^ { \circ } & 45 ^ { \circ } & 60 ^ { \circ } & 120 ^ { \circ } & 135 ^ { \circ } & 150 ^ { \circ } \\\hline \frac { 3 \pi } { 4 } & \frac { 2 \pi } { 5 } & \frac { 4 \pi } { 7 } & \frac { \pi } { 4 } & \frac { 2 \pi } { 3 } & \frac { 5 \pi } { 2 } \\\hline\end{array}$$
D) $$\begin{array} { | c | c | c | c | c | c | } \hline 30 ^ { \circ } & 45 ^ { \circ } & 60 ^ { \circ } & 120 ^ { \circ } & 135 ^ { \circ } & 150 ^ { \circ } \\\hline \frac { \pi } { 6 } & \frac { \pi } { 4 } & \frac { \pi } { 3 } & \frac { 2 \pi } { 3 } & \frac { 3 \pi } { 4 } & \frac { 5 \pi } { 6 } \\\hline\end{array}$$
E) $$\begin{array} { | c | c | c | c | c | c | } \hline 30 ^ { \circ } & 45 ^ { \circ } & 60 ^ { \circ } & 120 ^ { \circ } & 135 ^ { \circ } & 150 ^ { \circ } \\\hline \frac { 7 \pi } { 6 } & \frac { \pi } { 3 } & \frac { \pi } { 5 } & \frac { 2 \pi } { 3 } & \frac { 3 \pi } { 7 } & \frac { 3 \pi } { 2 } \\\hline\end{array}$$

Question 18

Evaluate the expressions using reference angles. $\csc \left( - 840 ^ { \circ } \right)$ $\cot \left( - 840 ^ { \circ } \right)$

A) $\csc \left( - 840 ^ { \circ } \right) = - \frac { 2 \sqrt { 2 } } { 3 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 2 } } { 2 }$
B) $\csc \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 3 } } { 3 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 3 } } { 3 }$
C) $\csc \left( - 840 ^ { \circ } \right) = - \frac { \sqrt { 2 } } { 2 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 2 } } { 3 }$
D) $\csc \left( - 840 ^ { \circ } \right) = - \frac { 1 } { 3 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { 1 } { 3 }$
E) $\csc \left( - 840 ^ { \circ } \right) = - \frac { 2 \sqrt { 3 } } { 3 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 3 } } { 3 }$

Question 19

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the given angle: $2 \pi$

A) $\sin 2 \pi = 0 , \cos 2 \pi = 1 , \tan 2 \pi = 0 , \sec 2 \pi = 1 , \csc 2 \pi \text { is undefined, } \cot 2 \pi \text { is undefined. }$
B) $\sin 2 \pi = 1 , \cos 2 \pi = 1 , \tan 2 \pi = 1 , \sec 2 \pi = 1 , \csc 2 \pi = 1 , \cot 2 \pi = 1$
C) $\sin 2 \pi = 0 , \cos 2 \pi = 1 , \tan 2 \pi \text { is undefined, } \sec 2 \pi \text { is undefined. } \csc 2 \pi = 0 , \cot 2 \pi = 1$
D) $\sin 2 \pi = - 1 \cdot \cos 2 \pi = 1 \cdot \tan 2 \pi = - 1 \cdot \sec 2 \pi = 1 \cdot \csc 2 \pi = 1 \cdot \cot 2 \pi = 1$
E) $\sin 2 \pi = 1 , \cos 2 \pi = 0 , \tan 2 \pi \text { is undefined, } \sec 2 \pi \text { is undefined. } \csc 2 \pi = - 1 , \cot 2 \pi = 0$

Question 20

Match an appropriate value from the right-hand column with each expression in the left-hand column.
a. $$\sec 30 ^ { \circ }$$
A. $$\sqrt { 3 }$$
b. $$\csc 30 ^ { \circ }$$
B. $$\sqrt { 2 }$$
c. $$\csc 60 ^ { \circ }$$
C. $$\frac { 2 } { \sqrt { 3 } }$$
d. $$\sec 60 ^ { \circ }$$
D. $$\frac { \sqrt { 2 } } { 2 }$$
E. $$2$$
F. $$\frac { 1 } { 2 }$$

A) $\sec 30 ^ { \circ }$ $\sqrt { 3 }$ $\csc 30 ^ { \circ }$ $\sqrt { 2 }$ $\csc 60 ^ { \circ }$ $\frac { 1 } { 2 }$ $\sec 60 ^ { \circ }$ $\frac { 2 \sqrt { 3 } } { 3 }$
B) $\sec 30 ^ { \circ } \sqrt { 3 } \csc 30 ^ { \circ } \sqrt { 3 } \csc 60 ^ { \circ } \frac { 2 \sqrt { 3 } } { 3 } \sec 60 ^ { \circ } 2$
C) $\operatorname { Sec } 30 ^ { \circ } \frac { \sqrt { 2 } } { 2 } \csc 30 ^ { \circ } 2 \csc 60 ^ { \circ } 2 \frac { \sqrt { 3 } } { 3 } \sec 60 ^ { \circ } 2$
D) $\operatorname { Sec } 30 ^ { \circ } \frac { \sqrt { 2 } } { 2 } \csc 30 ^ { \circ } \quad \sqrt { 2 } \csc 60 ^ { \circ } \frac { \sqrt { 2 } } { 2 } \sec 60 ^ { \circ } \quad \sqrt { 2 }$
E) $\operatorname { Sec } 30 ^ { \circ } \frac { 2 \sqrt { 3 } } { 3 } \csc 30 ^ { \circ } 2 \csc 60 ^ { \circ } \frac { 2 \sqrt { 3 } } { 3 } \sec 60 ^ { \circ } 2$

Question 21

Simplify the expression and enter the answer in terms of sine and cosine. $( \sec B + \tan B ) ( \sec B - \tan B )$

A) $\sec ^ { 2 } B + \tan ^ { 2 } B$
B) $- 2 \cos B \sin B$
C) $0$
D) $1$
E) $1 - \cos ^ { 2 } B$

Question 22

Suppose that $A B C$ is a right triangle with $\angle C = 90 ^ { \circ }$ . If $A C = 12$ and $B C = 6$ , find the quantities. $\cos A , \sin A , \tan A , \sec B , \csc B , \cot B$ .

A) $\cos A = \frac { 2 \sqrt { 5 } } { 5 }$ , $\sec B = 5$ , $\tan A = \frac { 1 } { 2 }$ , $\sin A = \frac { \sqrt { 5 } } { 5 }$ , $\csc B = \frac { 1 } { 5 }$ , $\cot B = \frac { 1 } { 2 }$
B) $\cos A = \frac { 2 \sqrt { 5 } } { 5 }$ , $\sec B = \sqrt { 5 }$ , $\tan A = 2$ , $\sin A = \frac { \sqrt { 5 } } { 5 }$ , $\csc B = \frac { \sqrt { 5 } } { 2 }$ , $\cot B = \frac { 1 } { 2 }$
C) $\cos A = \frac { 2 \sqrt { 5 } } { 5 }$ , $\sec B = \frac { \sqrt { 5 } } { 2 }$ , $\tan A = \frac { 1 } { 2 }$ , $\sin A = \frac { \sqrt { 5 } } { 5 }$ , $\csc B = \sqrt { 5 }$ , $\cot B = 2$
D) $\cos A = \frac { 2 \sqrt { 5 } } { 5 }$ , $\sec B = \sqrt { 5 }$ , $\tan A = \frac { 1 } { 2 }$ , $\sin A = \frac { \sqrt { 5 } } { 5 }$ , $\csc B = \frac { \sqrt { 5 } } { 2 }$ , $\cot B = \frac { 1 } { 2 }$
E) $\cos A = \frac { 2 \sqrt { 5 } } { 5 }$ , $\sec B = \sqrt { 5 }$ , $\tan A = \frac { 1 } { 2 }$ , $\sin A = \frac { \sqrt { 5 } } { 5 }$ , $\csc B = \frac { \sqrt { 5 } } { 2 }$ , $\cot B = 2$

Question 23

Rewrite in terms of sine and cosine, and simplify the expression: $\frac { \cot \beta - \cot \beta \cos \beta - \cos \beta \sin \beta } { \cos \beta \cot \beta }$

A) $\cot ^ { 2 } \beta - \cos ^ { 2 } \beta$
B) $\cos \beta - \cot \beta$
C) $\cos \beta - 1$
D) $\sin \beta - \cos \beta$
E) $\frac { \cos \beta \cot \beta } { \cos \beta + 1 }$

Question 24

Which of the following is equal to: $\frac { \sin \theta } { \csc \theta } + \frac { \cos \theta } { \sec \theta }$

A) $\frac { 1 - \sin \theta } { 1 + \sin \theta }$
B) $2 \cos ^ { 2 } \theta - 1$
C) $2 \cos ^ { 2 } \theta$
D) $1$
E) $2 \csc \theta$

Question 25

Which of the following is equal to: $\cos ^ { 2 } C - \sin ^ { 2 } C$

A) $1 - 2 \sin ^ { 2 } c$
B) $\tan C \sin C$
C) $2 - \sec C \csc C$
D) $\csc ^ { 2 } C \sec ^ { 2 } C$
E) $\frac { 1 - \cos ^ { 2 } C } { \cos C }$