Question 1

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

Accepted Answer

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)  $\frac { \sin \theta + 2 } { 6 }$ 
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)  $\frac { \sin \theta + 2 } { 6 }$ D

Question 2

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 $\sin 3$

Accepted Answer

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

Question 3

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

Accepted Answer

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)  $\frac { 1 - \cos ^ { 2 } C } { \cos C }$ 
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)  $\frac { 1 - \cos ^ { 2 } C } { \cos C }$ A

Question 4

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

Accepted Answer

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 
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 5

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 $y = - \frac { 3 } { 4 }$ .

Accepted Answer

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)  $\sin \theta = \frac { 3 } { 4 }$ , $\cos \theta = - \frac { \sqrt { 7 } } { 4 }$ , $\tan \theta = \frac { 3 \sqrt { 7 } } { 7 }$ , $\sec \theta = - \frac { 4 \sqrt { 7 } } { 7 }$ , $\csc \theta = \frac { 4 } { 3 }$ , $\cot \theta = \frac { \sqrt { 7 } } { 3 }$ 
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 6

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.

Accepted Answer

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)  $$\begin{array} { | c | c | c | c | c | c | } 
\hline 210 ^ { \circ } & 225 ^ { \circ } & 240 ^ { \circ } & 300 ^ { \circ } & 315 ^ { \circ } & 330 ^ { \circ } \
\hline \frac { 7 \pi } { 6 } & \frac { 5 \pi } { 4 } & \frac { 4 \pi } { 3 } & \frac { 5 \pi } { 3 } & \frac { 7 \pi } { 4 } & \frac { 11 \pi } { 6 } \
\hline
\end{array}$$ 
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 7

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

Accepted Answer

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)  $( 3 \sec \beta - 4 ) ( \sec \beta + 4 )$ 
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 8

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

Accepted Answer

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. 
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 9

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.

Accepted Answer

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)  $$\begin{array} { | c | c | c | c | c | c | } 
\hline 30 ^ { \circ } & 45 ^ { \circ } & 60 ^ { \circ } & 120 ^ { \circ } & 135 ^ { \circ } & 150 ^ { \circ } \
\hline \frac { 7 \pi } { 6 } & \frac { \pi } { 3 } & \frac { \pi } { 5 } & \frac { 2 \pi } { 3 } & \frac { 3 \pi } { 7 } & \frac { 3 \pi } { 2 } \
\hline
\end{array}$$ 
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 10

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

Accepted Answer

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)  $\sin 2 \pi = 1 , \cos 2 \pi = 0 , \tan 2 \pi \text { is undefined, } \sec 2 \pi \text { is undefined. } \csc 2 \pi = - 1 , \cot 2 \pi = 0$ 
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 11

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

Accepted Answer

A)  $\frac { 1 - \sin \theta } { 1 + \sin \theta }$ 
B)  $2 \cos ^ { 2 } \theta - 1$ 
C)  $2 \cos ^ { 2 } \theta$ 
D)  $1$ 
E)  $2 \csc \theta$ 
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 12

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

Accepted Answer

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)  $\sec \left( \frac { 7 \pi } { 4 } \right) = \sqrt { 5 }$ $\tan \left( \frac { 7 \pi } { 4 } \right) = - 1$ 
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 13

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

Accepted Answer

A)  $\cot ^ { 2 } \beta - \cos ^ { 2 } \beta$ 
B)  $\cos \beta - \cot \beta$ 
C)  $\cos \beta - 1$ 
D)  $\sin \beta - \cos \beta$ 
E)  $\frac { \cos \beta \cot \beta } { \cos \beta + 1 }$ 
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 14

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

Accepted Answer

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 
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 15

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

Accepted Answer

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)  $\sec \theta = 1.32$ $\csc \theta = 2.20$ $\cot \theta = 1.37$ 
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 16

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.$$\frac { 1 } { 2 }$$

Accepted Answer

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)  $\operatorname { Sec } 30 ^ { \circ } \frac { 2 \sqrt { 3 } } { 3 } \csc 30 ^ { \circ } 2 \csc 60 ^ { \circ } \frac { 2 \sqrt { 3 } } { 3 } \sec 60 ^ { \circ } 2$ 
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 17

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

Accepted Answer

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, $\cot \frac { 3 \pi } { 2 } = 0$ , $\csc \frac { 3 \pi } { 2 } = - 1$ . 
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, $\cot \frac { 3 \pi } { 2 } = 0$ , $\csc \frac { 3 \pi } { 2 } = - 1$ .

Question 18

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.

Accepted Answer

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 
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 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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)  $( \tan \beta - 5 ) ( \tan \beta + 4 )$ 
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)  $( \tan \beta - 5 ) ( \tan \beta + 4 )$

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

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

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

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 $y = - \frac { 3 } { 4 }$ .

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

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

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

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

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

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

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

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

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

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

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

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

Graphs of the Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

Filters

Exam 7: The Trigonometric Functions

Rewrite in terms of sine and cosine, and simplify the expression: 3sin⁡θ+6sin⁡2θ−4\frac { 3 \sin \theta + 6 } { \sin ^ { 2 } \theta - 4 }sin2θ−43sinθ+6​

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\cos 3cos3 and sin⁡3\sin 3sin3

Which of the following is equal to: cos⁡2C−sin⁡2C\cos ^ { 2 } C - \sin ^ { 2 } Ccos2C−sin2C

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

Let P(x,y)P ( x , y )P(x,y) denote the point where the terminal side of angle θ\thetaθ (in standard position) meets the unit circle. Use the information to evaluate the six trigonometric functions of θ\thetaθ . PPP is in Quadrant IV and y=−34y = - \frac { 3 } { 4 }y=−43​ .

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

Factor the expression. 3sec⁡2β+8sec⁡β−163 \sec ^ { 2 } \beta + 8 \sec \beta - 163sec2β+8secβ−16

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

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

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the given angle: 2π2 \pi2π

Which of the following is equal to: sin⁡θcsc⁡θ+cos⁡θsec⁡θ\frac { \sin \theta } { \csc \theta } + \frac { \cos \theta } { \sec \theta }cscθsinθ​+secθcosθ​

Evaluate the expressions using reference angles. sec⁡(7π4)\sec \left( \frac { 7 \pi } { 4 } \right)sec(47π​) tan⁡(7π4)\tan \left( \frac { 7 \pi } { 4 } \right)tan(47π​)

Rewrite in terms of sine and cosine, and simplify the expression: cot⁡β−cot⁡βcos⁡β−cos⁡βsin⁡βcos⁡βcot⁡β\frac { \cot \beta - \cot \beta \cos \beta - \cos \beta \sin \beta } { \cos \beta \cot \beta }cosβcotβcotβ−cotβcosβ−cosβsinβ​

When a clock reads 2:002 : 002:00 , what is the radian measure of the (smaller) angle between the hour hand and the minute hand?

Use a calculator to evaluate sec⁡θ\sec \thetasecθ , csc⁡θ\csc \thetacscθ , and cot⁡θ\cot \thetacotθ for the value of θ\thetaθ . Round the answers to two decimal places. θ=27∘\theta = 27 ^ { \circ }θ=27∘

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the given angle: 3π2\frac { 3 \pi } { 2 }23π​

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\sin 6 , \cos 6sin6,cos6 and tan⁡6\tan 6tan6 are positive or negative.

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

Factor the expression. tan⁡2β+6tan⁡β−7\tan ^ { 2 } \beta + 6 \tan \beta - 7tan2β+6tanβ−7

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

Graphs of the Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

Filters

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

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

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

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 $y = - \frac { 3 } { 4 }$ .

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

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

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

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

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

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

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

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

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

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

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