Deck 25: Right Triangle Trigonometry

Question 1

Find the exact values of the six trigonometric functions of the angle ? shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle. )

A)sin ? = $$\frac { 85 } { 13 }$$ csc ? = $\frac { 13 } { 85 }$
Cos ? = $\frac { 85 } { 84 }$
Sec ? = $\frac { 84 } { 85 }$
Tan ? = $\frac { 84 } { 13 }$
Cot ? = $\frac { 13 } { 84 }$
?
B)sin ? = $\frac { 84 } { 13 }$ csc ? = $\frac { 13 } { 84 }$
Cos ? = $\frac { 85 } { 84 }$
Sec ? = $\frac { 13 } { 85 }$
Tan ? = $\frac { 85 } { 13 }$
Cot ? = $\frac { 84 } { 85 }$
?
C)sin ? = $\frac { 84 } { 85 }$ csc ? = $\frac { 13 } { 84 }$
Cos ? = $\frac { 13 } { 85 }$
Sec ? = $\frac { 84 } { 13 }$
Tan ? = $\frac { 85 } { 13 }$
Cot ? = $\frac { 85 } { 84 }$
?
D)sin ? = $\frac { 85 } { 84 }$ csc ? = $$\frac { 85 } { 13 }$$
Cos ? = $$\frac { 84 } { 13 }$$
Sec ? = $\frac { 13 } { 85 }$
Tan ? = $\frac { 13 } { 84 }$
Cot ? = $\frac { 84 } { 85 }$
?
E)sin ? = $\frac { 13 } { 85 }$ csc ? = $\frac { 85 } { 13 }$
Cos ? = $\frac { 84 } { 85 }$
Sec ? = $\frac { 85 } { 84 }$
Tan ? = $\frac { 13 } { 84 }$
Cot ? = $\frac { 84 } { 13 }$
?

Answer 1

sin ? = $\frac { 13 } { 85 }$ csc ? = $\frac { 85 } { 13 }$
Cos ? = $\frac { 84 } { 85 }$
Sec ? = $\frac { 85 } { 84 }$
Tan ? = $\frac { 13 } { 84 }$
Cot ? = $\frac { 84 } { 13 }$
?

Question 2

Evaluate each function.Round your answers to four decimal places.? $\sec \left( \frac { \pi } { 2 } - 2 \right)$ and $\cot \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$ ?

A)1.1998 and 0.4463
B)1.2998 and 0.5463
C)1.1498 and 0.3963
D)1.0998 and 0.3463
E)1.2498 and 0.4963

Answer 2

1.0998 and 0.3463

Question 3

Find the exact values of the six trigonometric functions of the angle ? for each of the two triangles. ?

A)sin ? = $$\frac { 15 } { 8 }$$ csc ? = $\frac { 8 } { 17 }$
Cos ? = $\frac { 17 } { 15 }$
Sec ? = $\frac { 8 } { 15 }$
Tan ? = $\frac { 17 } { 8 }$
Cot ? = $\frac { 15 } { 17 }$
?
B)sin ? = $\frac { 17 } { 15 }$ csc ? = $\frac { 17 } { 8 }$
Cos ? = $\frac { 15 } { 8 }$
Sec ? = $\frac { 8 } { 17 }$
Tan ? = $\frac { 8 } { 15 }$
Cot ? = $\frac { 15 } { 17 }$
?
C)sin ? = $\frac { 8 } { 17 }$ csc ? = $\frac { 17 } { 8 }$
Cos ? = $\frac { 15 } { 17 }$
Sec ? = $\frac { 17 } { 15 }$
Tan ? = $\frac { 8 } { 15 }$
Cot ? = $\frac { 15 } { 8 }$
?
D)sin ? = $\frac { 15 } { 8 }$ csc ? = $\frac { 8 } { 15 }$
Cos ? = $\frac { 17 } { 15 }$
Sec ? = $\frac { 15 } { 17 }$
Tan ? = $\frac { 17 } { 8 }$
Cot ? = $\frac { 8 } { 17 }$
?
E)sin ? = $\frac { 15 } { 17 }$ csc ? = $\frac { 8 } { 15 }$
Cos ? = $\frac { 8 } { 17 }$
Sec ? = $\frac { 15 } { 8 }$
Tan ? = $\frac { 17 } { 8 }$
Cot ? = $\frac { 17 } { 15 }$
?

Answer 3

sin ? = $\frac { 8 } { 17 }$ csc ? = $\frac { 17 } { 8 }$
Cos ? = $\frac { 15 } { 17 }$
Sec ? = $\frac { 17 } { 15 }$
Tan ? = $\frac { 8 } { 15 }$
Cot ? = $\frac { 15 } { 8 }$
?

Question 4

Select a right triangle corresponding to the trigonometric function of the acute angle ?.Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of ?. ?
Tan ? = $\frac { 5 } { 12 }$ ?
?
Sin ? = $\frac { 5 } { 13 }$ csc ? = $\frac { 13 } { 5 }$ cos ? = $\frac { 12 } { 13 }$ sec ? = $\frac { 13 } { 12 }$ tan ? = $\frac { 5 } { 12 }$ cot ? = $\frac { 12 } { 5 }$ ?

A)? sin ? = $$\frac { 5 } { 13 }$$
Csc ? = $\frac { 13 } { 5 }$
Cos ? = $\frac { 12 } { 13 }$
Sec ? = $\frac { 13 } { 12 }$
Tan ? = $\frac { 5 } { 12 }$
Cot ? = $\frac { 12 } { 5 }$
?
B)? sin ? = $$\frac { 5 } { 13 }$$
Csc ? = $\frac { 5 } { 12 }$
Cos ? = $\frac { 12 } { 13 }$
Sec ? = $\frac { 12 } { 5 }$
Tan ? = $\frac { 13 } { 5 }$
Cot ? = $\frac { 13 } { 12 }$
?
C)? sin ? = $$\frac { 13 } { 12 }$$
Csc ? = $\frac { 13 } { 5 }$
Cos ? = $\frac { 12 } { 5 }$
Sec ? = $\frac { 5 } { 13 }$
Tan ? = $\frac { 5 } { 12 }$
Cot ? = $\frac { 12 } { 13 }$
?
D)? sin ? = $$\frac { 13 } { 5 }$$
Csc ? = $\frac { 5 } { 13 }$
Cos ? = $\frac { 13 } { 12 }$
Sec ? = $\frac { 12 } { 13 }$
Tan ? = $\frac { 12 } { 5 }$
Cot ? = $\frac { 5 } { 12 }$
?
E)? sin ? = $$\frac { 12 } { 5 }$$
Csc ? = $\frac { 5 } { 12 }$
Cos ? = $\frac { 13 } { 12 }$
Sec ? = $\frac { 12 } { 13 }$
Tan ? = $\frac { 13 } { 5 }$
Cot ? = $\frac { 5 } { 13 }$
?

Question 5

Evaluate each function.Round your answers to four decimal places. ?
Csc3 and $\tan \frac { 1 } { 5 }$ ?

A)7.0862 and 0.2027
B)7.1862 and 0.3027
C)7.2362 and 0.3527
D)7.1362 and 0.2527
E)7.2862 and 0.4027

Question 6

Construct an appropriate triangle to complete the table. $$\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } , 0 ^ { \circ } \leq \theta \leq \frac { \pi } { 2 } \right)$$ Function $$\begin{array} { | c | c | l | l | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \sin & 45 ^ { \circ } & & \\\hline\end{array}$$

A)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \sin & 45 & \frac { \pi } { 4 } & \frac { \sqrt { 2 } } { 2 } \\\\\hline\end{array}$$
B)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \sin & 45 & \frac { \pi } { 3 } & \frac { \sqrt { 3 } } { 2 } \\\\\hline\end{array}$$
C)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \sin & 45 & 0 & 0 \\\hline\end{array}$$
D)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \sin & 45 & \frac { \pi } { 2 } & 1\end{array}$$
E)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \sin & 45 & \frac { \pi } { 6 } & \frac { 1 } { 2 } \\\hline\end{array}$$

Question 7

Evaluate each function.Round your answers to four decimal places.? $\cot \left( \frac { \pi } { 15 } \right)$ and $\tan \frac { \pi } { 15 }$ ?

A)4.7046 and 0.2126
B)4.7546 and 0.2626
C)4.8546 and 0.3626
D)4.8046 and 0.3126
E)4.9046 and 0.4126

Question 8

Evaluate each function.Round your answers to four decimal places. ​
Cos17°36' and sin67°23'
​

A)1.0032 and 0.9731
B)1.0532 and 1.0231
C)1.1532 and 1.1231
D)1.1032 and 1.0731
E)0.9532 and 0.9231

Question 9

Construct an appropriate triangle to complete the table. $\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } , 0 ^ { \circ } \leq \theta \leq \frac { \pi } { 2 } \right)$ $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \mathrm { csc } & & \frac { \pi } { 6 } & \\& & & \\\hline\end{array}$$

A)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \csc & 30 ^ { \circ } & \frac { \pi } { 6 } & 2 \\\hline\end{array}$$
B)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \csc & 90 ^ { \circ } & \frac { \pi } { 6 } & 1 \\\hline\end{array}$$
C)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \csc & 45 ^ { \circ } & \frac { \pi } { 6 } & \sqrt { 2 } \\\hline\end{array}$$ ?
D)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \mathrm { csc } & 0 ^ { \circ } & \frac { \pi } { 6 } & 0 \\\hline\end{array}$$
E)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \csc & 30 ^ { \circ } & \frac { \pi } { 6 } & \frac { 2 \sqrt { 3 } } { 3 } \\\hline\end{array}$$

Question 10

Evaluate each function.Round your answers to four decimal places. ​
Sec0.74 and cos0.74
​

A)1.4042 and 0.7885
B)1.5542 and 0.9385
C)1.3542 and 0.7385
D)1.4542 and 0.8385
E)1.5042 and 0.8885

Question 11

Find the exact values of the six trigonometric functions of the angle ? for the triangle. ?

A)? sin ? = $\frac { 3 } { 5 }$
Csc ? = $$\frac { 5 } { 3 }$$
Cos ? = $\frac { 4 } { 5 }$
Sec ? = $\frac { 5 } { 4 }$
Tan ? = $\frac { 3 } { 4 }$
Cot ? = $\frac { 4 } { 3 }$
?
B)sin ? = $\frac { 4 } { 3 }$ csc ? = $\frac { 3 } { 4 }$
Cos ? = $\frac { 5 } { 4 }$
Sec ? = $\frac { 4 } { 5 }$
Tan ? = $\frac { 5 } { 3 }$
Cot ? = $\frac { 3 } { 5 }$
?
C)sin ? = $\frac { 4 } { 3 }$ csc ? = $\frac { 5 } { 3 }$
Cos ? = $\frac { 5 } { 4 }$
Sec ? = $\frac { 3 } { 5 }$
Tan ? = $\frac { 3 } { 4 }$
Cot ? = $\frac { 4 } { 5 }$
?
D)sin ? = $\frac { 4 } { 5 }$ csc ? = $\frac { 3 } { 4 }$
Cos ? = $\frac { 3 } { 5 }$
Sec ? = $\frac { 4 } { 3 }$
Tan ? = $\frac { 5 } { 3 }$
Cot ? = $\frac { 5 } { 4 }$
?
E)sin ? = $\frac { 4 } { 3 }$ csc ? = $\frac { 3 } { 5 }$
Cos ? = $\frac { 5 } { 4 }$
Sec ? = $\frac { 4 } { 5 }$
Tan ? = $\frac { 5 } { 3 }$
Cot ? = $\frac { 3 } { 4 }$

Question 12

Construct an appropriate triangle to complete the table. $\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } , 0 ^ { \circ } \leq \theta \leq \frac { \pi } { 2 } \right)$ $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \tan & & \frac { \pi } { 2 } & \\\hline\end{array}$$
?

A)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \tan & 90 ^ { \circ } & \frac { \pi } { 2 } & \text { Not defined } \\\hline\end{array}$$
B)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \tan & 45 ^ { \circ } & \frac { \pi } { 2 } & 1 \\\hline\end{array}$$
C)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \tan & 60 ^ { \circ } & \frac { \pi } { 2 } & \sqrt { 3 } \\\hline\end{array}$$
D)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \tan & 0 ^ { \circ } & \frac { \pi } { 2 } & 0 \\\hline\end{array}$$
E)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \tan & 30 ^ { \circ } & \frac { \pi } { 2 } & \frac { \sqrt { 3 } } { 3 } \\\hline\end{array}$$

Question 13

Evaluate each function.Round your answers to four decimal places. ​
Sec48°38' and csc31°57'
​

A)1.7131 and 2.0897
B)1.6131 and 1.9897
C)1.5631 and 1.9397
D)1.5131 and 1.8897
E)1.6631 and 2.0397

Question 14

Construct an appropriate triangle to complete the table. $\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } , 0 ^ { \circ } \leq \theta \leq \frac { \pi } { 2 } \right)$ $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \mathrm { sec } & & \frac { \pi } { 3 } & \\\hline\end{array}$$

A)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \text { sec } & 45 ^ { \circ } & \frac { \pi } { 3 } & \sqrt { 2 } \\\hline\end{array}$$
B)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \text { sec } & 90 ^ { \circ } & \frac { \pi } { 3 } & \text { Not defined } \\\hline\end{array}$$
C)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \mathrm { sec } & 60 ^ { \circ } & \frac { \pi } { 3 } & 2 \\\hline\end{array}$$
D)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \mathrm { sec } & 0 ^ { \circ } & \frac { \pi } { 3 } & 1 \\\hline\end{array}$$
E)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \mathrm { sec } & 30 ^ { \circ } & \frac { \pi } { 3 } & \frac { 2 \sqrt { 3 } } { 3 } \\\hline\end{array}$$

Question 15

Find the exact values of the six trigonometric functions of the angle ? shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle. ) ?

A)sin ? = $\frac { 15 } { 8 }$ csc ? = $\frac { 8 } { 15 }$
Cos ? = $\frac { 17 } { 15 }$
Sec ? = $\frac { 15 } { 17 }$
Tan ? = $\frac { 17 } { 8 }$
Cot ? = $\frac { 8 } { 17 }$
?
B)sin ? = $\frac { 17 } { 8 }$ csc ? = $\frac { 8 } { 17 }$
Cos ? = $\frac { 17 } { 15 }$
Sec ? = $\frac { 15 } { 17 }$
Tan ? = $\frac { 15 } { 8 }$
Cot ? = $\frac { 8 } { 15 }$
?
C)sin ? = $\frac { 8 } { 17 }$ csc ? = $\frac { 17 } { 8 }$
Cos ? = $\frac { 15 } { 17 }$
Sec ? = $\frac { 17 } { 15 }$
Tan ? = $\frac { 8 } { 15 }$
Cot ? = $\frac { 15 } { 8 }$
?
D)sin ? = $\frac { 17 } { 15 }$ csc ? = $\frac { 17 } { 8 }$
Cos ? = $\frac { 15 } { 8 }$
Sec ? = $\frac { 8 } { 17 }$
Tan ? = $\frac { 8 } { 15 }$
Cot ? = $\frac { 15 } { 17 }$
?
E)sin ? = $\frac { 15 } { 17 }$ csc ? = $\frac { 8 } { 15 }$
Cos ? = $\frac { 8 } { 17 }$
Sec ? = $\frac { 15 } { 8 }$
Tan ? = $\frac { 17 } { 8 }$
Cot ? = $\frac { 17 } { 15 }$
?

Question 16

Construct an appropriate triangle to complete the table. $$\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } , 0 ^ { \circ } \leq \theta \leq \frac { \pi } { 2 } \right)$$ $$\begin{array} { | c | c | l | l | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cos & 30 ^ { \circ } & & \\\hline\end{array}$$

A)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cos & 30 & \frac { \pi } { 3 } & \frac { \sqrt { 3 } } { 2 } \\\hline\end{array}$$
B)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \text { rad } ) & \text { Function Value } \\\hline \cos & 30 & \frac { \pi } { 4 } & \frac { \sqrt { 2 } } { 2 } \\\hline\end{array}$$
C)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cos & 30 & 0 & 0 \\\hline\end{array}$$
D)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cos & 30 & \frac { \pi } { 2 } & 1 \\\hline\end{array}$$
?E)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cos & 30 & \frac { \pi } { 6 } & \frac { 1 } { 2 } \\\hline\end{array}$$

Question 17

Construct an appropriate triangle to complete the table. $\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } , 0 ^ { \circ } \leq \theta \leq \frac { \pi } { 2 } \right)$ $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cot & & \frac { \pi } { 2 } & \\\hline\end{array}$$
?

A)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \text { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cot & 30 ^ { \circ } & \frac { \pi } { 2 } & \sqrt { 3 } \\\hline\end{array}$$
B)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cot & 45 ^ { \circ } & \frac { \pi } { 2 } & 1 \\\hline\end{array}$$
C)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \text { cot } & 90 ^ { \circ } & \frac { \pi } { 2 } & 0 \\\hline\end{array}$$
D)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \cot & 0 ^ { \circ } & \frac { \pi } { 2 } & \text { Not defined } \\\hline\end{array}$$
E)? $$\begin{array} { | c | c | c | c | } \hline \text { Function } & \theta ( \mathrm { deg } ) & \theta ( \mathrm { rad } ) & \text { Function Value } \\\hline \\\cot & 60 ^ { \circ } & \frac { \pi } { 2 } & \frac { \sqrt { 3 } } { 3 } \\\\\hline\end{array}$$

Question 18

Find the exact values of the six trigonometric functions of the angle ? shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle. ) ?

A)sin ? = $\frac { 5 } { 3 }$ csc ? = $$\frac { 3 } { 5 }$$
Cos ? = $\frac { 5 } { 4 }$
Sec ? = $\frac { 4 } { 5 }$
Tan ? = $\frac { 4 } { 3 }$
Cot ? = $\frac { 3 } { 4 }$
?
B)sin ? = $\frac { 4 } { 3 }$ ? csc ? = $\frac { 3 } { 4 }$ ?
Cos ? = $\frac { 5 } { 4 }$ ?
Sec ? = $\frac { 4 } { 5 }$ ?
Tan ? = $\frac { 5 } { 3 }$ ?
Cot ? = $\frac { 3 } { 5 }$ ?
?
C)sin ? = $\frac { 4 } { 5 }$ ? csc ? = $\frac { 3 } { 4 }$ ?
Cos ? = $\frac { 3 } { 5 }$
Sec ? = $\frac { 4 } { 3 }$
Tan ? = $\frac { 5 } { 3 }$
Cot ? = $\frac { 5 } { 4 }$
?
D)sin ? = $\frac { 3 } { 5 }$ csc ? = $\frac { 5 } { 3 }$
Cos ? = $\frac { 4 } { 5 }$
Sec ? = $\frac { 5 } { 4 }$
Tan ? = $\frac { 3 } { 4 }$
Cot ? = $\frac { 4 } { 3 }$
?
E)sin ? = $\frac { 4 } { 3 }$ csc ? = $\frac { 5 } { 3 }$
Cos ? = $\frac { 5 } { 4 }$
Sec ? = $\frac { 3 } { 5 }$
Tan ? = $\frac { 3 } { 4 }$
Cot ? = $\frac { 4 } { 5 }$
?

Question 19

Evaluate each function.Round your answers to four decimal places. ​
Sin22.3° and csc22.3°
​

A)0.5795 and 2.8354
B)0.3795 and 2.6354
C)0.4795 and 2.7354
D)0.4295 and 2.6854
E)0.5295 and 2.7854

Question 20

Evaluate each function.Round your answers to four decimal places. ​
Tan15.7° and cot74.3°
​

A)0.4811 and 0.4811
B)0.3311 and 0.3311
C)0.2811 and 0.2811
D)0.3811 and 0.3811
E)0.4311 and 0.4311

Question 21

Use the given function value(s),and trigonometric identities (including the cofunction identities),to find the indicated trigonometric functions. ?sin 30° = $\frac { 1 } { 2 }$ ,cos 30° = $\frac { \sqrt { 3 } } { 2 }$ ?
?sin 60° and cos 60°

A)?sin 60° = 2 and cos 60° = $\frac { 1 } { 2 }$
B)?sin 60° = $\sqrt { 3 }$ and cos 60° = $\frac { 1 } { 2 }$
C)?sin 60° = $\frac { \sqrt { 3 } } { 2 }$ and cos 60° = $\sqrt { 2 }$
D)?sin 60° = $\frac { \sqrt { 3 } } { 2 }$ and cos 60° = $\frac { 1 } { 2 }$
E)?sin 60° = $\frac { \sqrt { 3 } } { 2 }$ and cos 60° = 2

Question 22

Use a graphing utility to complete the table. $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 20 ^ { \circ } & 30 ^ { \circ } & 40 ^ { \circ } & 50 ^ { \circ } & 60 ^ { \circ } \\\hline \cos \theta & & & & & \\\hline \sin \left( 90 ^ { \circ } - \theta \right) & & & & & \\\hline\end{array}$$

A) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 20 ^ { \circ } & 30 ^ { \circ } & 40 ^ { \circ } & 50 ^ { \circ } & 60 ^ { \circ } \\\hline \cos \theta & 0.94 & 0.5 & 0.77 & 0.87 & 0.5 \\\hline \sin \left( 90 ^ { \circ } - \theta \right) & 0.94 & 0.5 & 0.77 & 0.87 & 0.5 \\\hline\end{array}$$
B) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 20 ^ { \circ } & 30 ^ { \circ } & 40 ^ { \circ } & 50 ^ { \circ } & 60 ^ { \circ } \\\hline \cos \theta & 0.5 & 0.64 & 0.77 & 0.87 & 0.94 \\\hline \sin \left( 90 ^ { \circ } - \theta \right) & 0.94 & 0.87 & 0.77 & 0.64 & 0.5 \\\hline\end{array}$$
C) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 20 ^ { \circ } & 30 ^ { \circ } & 40 ^ { \circ } & 50 ^ { \circ } & 60 ^ { \circ } \\\hline \cos \theta & 0.94 & 0.87 & 0.77 & 0.64 & 0.5 \\\hline \sin \left( 90 ^ { \circ } - \theta \right) & 0.94 & 0.87 & 0.77 & 0.64 & 0.5 \\\hline\end{array}$$
D) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 20 ^ { \circ } & 30 ^ { \circ } & 40 ^ { \circ } & 50 ^ { \circ } & 60 ^ { \circ } \\\hline \cos \theta & 0.94 & 0.87 & 0.77 & 0.64 & 0.5 \\\hline \sin \left( 90 ^ { \circ } - \theta \right) & 0.5 & 0.64 & 0.77 & 0.87 & 0.94 \\\hline\end{array}$$
E) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 20 ^ { \circ } & 30 ^ { \circ } & 40 ^ { \circ } & 50 ^ { \circ } & 60 ^ { \circ } \\\hline \cos \theta & 0 & 0.87 & 0.77 & 0.64 & 0.5 \\\hline \sin \left( 90 ^ { \circ } - \theta \right) & 0 & 0.87 & 0.77 & 0.64 & 0.5 \\\hline\end{array}$$

Question 23

Use a compass to sketch a quarter of a circle of radius 10 centimeters.Using a protractor, construct an angle of $a = 10 ^ { \circ }$ in standard position (see figure).Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle.By actual measurement,calculate the coordinates $( x , y )$ of the point of intersection and use these measurements to approximate the six trigonometric functions of a $\text { "a" }$ angle.(Round your answer to two decimal places. )? ?

A)sin10° ? 0.98,cos10° ? 0.17,tan10° ? 5.76, csc10° ? 0.18,sec10° ? 1.02,cot10° ? 5.67
B)sin10° ? 5.67,cos10° ? 1.02,tan10° ? 5.76, csc10° ? 0.18,sec10° ? 0.98,cot10° ? 5.67
?
C)sin10° ? 0.17,cos10° ? 0.98,tan10° ? 0.18, csc10° ? 5.76,sec10° ? 1.02,cot10° ? 5.67
?
D)sin10° ? 0.18,cos10° ? 0.98,tan10° ? 0.17, csc10° ? 10,sec10° ? 0.98,cot10° ? 5.76
?
E)sin10° ? 0.17,cos10° ? 0.98,tan10° ? 0.18, csc10° ? 5.76,sec10° ? 5.67,cot10° ? 1.02

Question 24

Use the given function value(s),and trigonometric identities (including the cofunction identities),to find the indicated trigonometric functions.? $\cos \beta = \frac { \sqrt { 5 } } { 4 }$ ? $\sec \beta \text { and } \sin \left( 90 ^ { \circ } - \beta \right)$ ?

A) $\sec \beta = \frac { 4 \sqrt { 5 } } { 5 }$ and $\sin \left( 90 ^ { \circ } - \beta \right) = \frac { \sqrt { 5 } } { 4 }$
B)? $\sec \beta = \sqrt { 5 }$ and $\sin \left( 90 ^ { \circ } - \beta \right) = 4$
C)? $\sec \beta = \frac { \sqrt { 5 } } { 4 }$ and $\sin \left( 90 ^ { \circ } - \beta \right) = \frac { \sqrt { 5 } } { 4 }$
D)? $\sec \beta = \frac { \sqrt { 5 } } { 4 }$ and $\sin \left( 90 ^ { \circ } - \beta \right) = \frac { 4 \sqrt { 5 } } { 5 }$
E)? $\sec \beta = \frac { 4 \sqrt { 5 } } { 5 }$ and $\sin \left( 90 ^ { \circ } - \beta \right) = \frac { 4 \sqrt { 5 } } { 5 }$

Question 25

Find the values of ? in degrees $\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } \right)$ and radians $\left( 0 < \theta < \frac { \pi } { 2 } \right)$ without the aid of a calculator. ?sin ? = 1 and csc ? = 1
?
?

A)90° = $\frac { \pi } { 2 }$
B)?30° = $\frac { \pi } { 6 }$
C)?0° = 0
D)?60° = $\frac { \pi } { 3 }$
E)?45° = $\frac { \pi } { 4 }$

Question 26

A biologist wants to know the width w of a river so that instruments for studying the pollutants in the water can be set properly.From point A the biologist walks downstream $x = 140$ feet and sights to point C (see figure).From this sighting,it is determined that ? = 54°.How wide is the river?? (Round your answer to three decimal places. ) ?

A)212.693
B)197.693
C)207.693
D)202.693
E)192.693

Question 27

You are skiing down a mountain with a vertical height of 1250 feet.The distance from the top of the mountain to the base is 2500 feet.What is the angle of elevation from the base to the top of the mountain? ?

A)60° = $\frac { \pi } { 3 }$
B)30° = $\frac { \pi } { 6 }$
C)0° = 0
D)90° = $\frac { \pi } { 2 }$
E)45° = $\frac { \pi } { 4 }$

Question 28

In traveling across flat land,you notice a mountain directly in front of you.Its angle of elevation (to the peak)is 3.5°.After you drive $x = 23$ miles closer to the mountain,the angle of elevation is 9°.Approximate the height of the mountain.(Round your answer upto one decimal place. )? ?

A)10.3 mi
B)2.3 mi
C)4.3 mi
D)8.3 mi
E)6.3 mi

Question 29

Find the values of ? in degrees $\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } \right)$ and radians $\left( 0 < \theta < \frac { \pi } { 2 } \right)$ without the aid of a calculator. cos ? = 0 and tan ? = Not defined
?
?

A)90° = $\frac { \pi } { 2 }$
B)?60° = $\frac { \pi } { 3 }$
C)?0° = 0
D)?30° = $\frac { \pi } { 6 }$
E)?45° = $\frac { \pi } { 4 }$

Question 30

Use the given function value(s),and trigonometric identities (including the cofunction identities),to find the indicated trigonometric functions. ​
​sin 90° = 1,tan 90° = Not defined
Sin 0° and tan 0°
​

A)​sin 0° = 0 and tan 0° = 0
B)sin 0° = 1 and tan 0° = 1
C)​sin 0° = Not defined and tan 0° = Not defined
D)​sin 0° = Not defined and tan 0° = 1
E)​sin 0° = 1 and tan 0° = Not defined

Question 31

A 20-meter line is used to tether a helium-filled balloon.Because of a breeze,the line makes an angle of approximately 85° with the ground.Use a trigonometric function to write an equation involving the unknown quantity. ?

A) $$\sin 85 ^ { \circ } = \frac { h } { 20 }$$ (where h is height. )
B) $\cos 85 ^ { \circ } = \frac { h } { 20 }$ (where h is height. )
C) $\cos 85 ^ { \circ } = \frac { 20 } { h }$ (where h is height. )
D) $\sin 85 ^ { \circ } = \frac { 20 } { h }$ (where h is height. )
E) $\tan 85 ^ { \circ } = \frac { h } { 20 }$ (where h is height. )

Question 32

Find the values of ? in degrees $\left( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } \right)$ and radians $\left( 0 < \theta < \frac { \pi } { 2 } \right)$ without the aid of a calculator. sec ? = $\sqrt { 2 }$ and cot ? = 1
?
?

A)45° = $\frac { \pi } { 4 }$
B)?90° = $\frac { \pi } { 2 }$
C)?0° = 0
D)?60° = $\frac { \pi } { 3 }$
E)?30° = $\frac { \pi } { 6 }$

Question 33

Complete the table.
$$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 0.8 & 0.9 & 1 & 1.1 & 1.2 \\\hline \sin \theta & & & & & \\\hline\end{array}$$
(Round your answer to four decimal places. )

A) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 0.8 & 0.9 & 1 & 1.1 & 1.2 \\\hline \sin \theta & 0.7174 & 0.7833 & 0.8415 & 0.8912 & 0.932 \\\hline\end{array}$$
B) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 0.8 & 0.9 & 1 & 1.1 & 1.2 \\\hline \sin \theta & 0 & 1 & 2 & 3 & 4 \\\hline\end{array}$$
C) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 0.8 & 0.9 & 1 & 1.1 & 1.2 \\\hline \sin \theta & 0.932 & 0.8912 & 0.8415 & 0.7833 & 0.7174 \\\hline\end{array}$$
D) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 0.8 & 0.9 & 1 & 1.1 & 1.2 \\\hline \sin \theta & 0.7174 & 0.7174 & 0.8415 & 0.7174 & 0.932 \\\hline\end{array}$$
E) $$\begin{array} { | c | c | c | c | c | c | } \hline \theta & 0.8 & 0.9 & 1 & 1.1 & 1.2 \\\hline \sin \theta & 0.8 & 0.9 & 1 & 1.1 & 1.2 \\\hline\end{array}$$

Question 34

A 10-meter line is used to tether a helium-filled balloon.Complete the table,which shows the heights (in meters)of the balloon for decreasing angle measures $\theta$ .(Round your answers to one decimal places. ) $$\begin{array} { | c | c | c | c | c | c | } \hline \text { Angle, } \theta & 50 ^ { \circ } & 40 ^ { \circ } & 30 ^ { \circ } & 20 ^ { \circ } & 10 ^ { \circ } \\\hline \text { Height } & & & & & \\\hline\end{array}$$

A) $$\begin{array} { | c | c | c | c | c | c | } \hline \text { Angle, } \theta & 50 ^ { \circ } & 40 ^ { \circ } & 30 ^ { \circ } & 20 ^ { \circ } & 10 ^ { \circ } \\\hline \text { Height } & 5.7 & 6.4 & 5 & 3.4 & 1.7 \\\hline\end{array}$$
B) $$\begin{array} { | c | c | c | c | c | c | } \hline \text { Angle } , \theta & 50 ^ { \circ } & 40 ^ { \circ } & 30 ^ { \circ } & 20 ^ { \circ } & 10 ^ { \circ } \\\hline \text { Height } & 7.7 & 6.4 & 5 & 3.4 & 1.7 \\\hline\end{array}$$
C) $$\begin{array} { | c | c | c | c | c | c | } \hline \text { Angle, } \theta & 50 ^ { \circ } & 40 ^ { \circ } & 30 ^ { \circ } & 20 ^ { \circ } & 10 ^ { \circ } \\\hline \text { Height } & 4.7 & 6.4 & 5 & 3.4 & 1.7 \\\hline\end{array}$$
D) $$\begin{array} { | c | c | c | c | c | c | } \hline \text { Angle, } \theta & 50 ^ { \circ } & 40 ^ { \circ } & 30 ^ { \circ } & 20 ^ { \circ } & 10 ^ { \circ } \\\hline \text { Height } & 3.7 & 6.4 & 5 & 3.4 & 1.7 \\\hline\end{array}$$
E) $$\begin{array} { | c | c | c | c | c | c | } \hline \text { Angle, } \theta & 50 ^ { \circ } & 40 ^ { \circ } & 30 ^ { \circ } & 20 ^ { \circ } & 10 ^ { \circ } \\\hline \text { Height } & 6.7 & 6.4 & 5 & 3.4 & 1.7 \\\hline\end{array}$$

Question 35

A tapered shaft has a diameter of $x = 9$ centimeters at the small end and is 15 centimeters long (see figure).The taper is 3°.Find the diameter d of the large end of the shaft.(Round your answer to two decimal places. )? ? ?

A)10.57 cm
B)14.57 cm
C)12.57 cm
D)18.57 cm
E)16.57 cm

Question 36

Use the given function value(s),and trigonometric identities (including the cofunction identities),to find the indicated trigonometric functions.? $\cos \theta = \frac { 1 } { 5 }$ ? sec ? and csc(90° - ?)
?

A)? $\sec \theta = \frac { 1 } { 5 }$ and $\csc \left( 90 ^ { \circ } - \theta \right) = \frac { 1 } { 5 }$
B) $\sec \theta = 5$ and $\csc \left( 90 ^ { \circ } - \theta \right) = 5$
C)? $\sec \theta = \frac { 1 } { 5 }$ and $\csc \left( 90 ^ { \circ } - \theta \right) = 5$
D)? $\sec \theta = 5$ and $\csc \left( 90 ^ { \circ } - \theta \right) = \frac { 1 } { 5 }$
E)? $\sec \theta = - 5$ and $\csc \left( 90 ^ { \circ } - \theta \right) = - 5$

Question 37

A guy wire runs from the ground to a cell tower.The wire is attached to the cell tower $x = 100$ feet above the ground.The angle formed between the wire and the ground is 43° (see figure).How long is the guy wire. ?? ?

A)166.6 ft
B)?146.6 ft
C)?156.6 ft
D)?161.6 ft
E)?151.6 ft

Question 38

Use the given function value(s),and trigonometric identities (including the cofunction identities),to find the indicated trigonometric functions.? $\sec \theta = 9$ ? cos ?
?

A) $\cos \theta = 9$
B)? $\cos \theta = \frac { 1 } { 9 }$
C)? $\cos \theta = - \frac { 1 } { 9 }$
D)? $\cos \theta = - 9$
E)? $\cos \theta = 0$

Question 39

A 50-meter line is used to tether a helium-filled balloon.Because of a breeze,the line makes an angle of approximately 55° with the ground.What is the height of the balloon? (Round the answer to one decimal place. ) ​

A)45 m
B)41 m
C)43 m
D)49 m
E)47 m

Question 40

Use the given function value(s),and trigonometric identities (including the cofunction identities),to find the indicated trigonometric functions. ?? $\cot \alpha = 3$ ? $\tan \alpha \text { and } \cot \left( 90 ^ { \circ } - \alpha \right)$ ?

A)? $\tan \alpha = - \frac { 1 } { 3 }$ and $\cot \left( 90 ^ { \circ } - \alpha \right) = - \frac { 1 } { 3 }$
B)? $\tan \alpha = \frac { 1 } { 3 }$ and $\cot \left( 90 ^ { \circ } - \alpha \right) = 3$
C) $\tan \alpha = \frac { 1 } { 3 }$ and $\cot \left( 90 ^ { \circ } - \alpha \right) = \frac { 1 } { 3 }$
D)? $\tan \alpha = 3$ and $\cot \left( 90 ^ { \circ } - \alpha \right) = 3$
E)? $\tan \alpha = 3$ and $\cot \left( 90 ^ { \circ } - \alpha \right) = \frac { 1 } { 3 }$

Question 41

If $\sin \theta = \frac { \sqrt { 2 } } { 2 }$ ,find the value of ? in degrees $\left( 0 ^ { \circ } < \theta < 90 ^ { \circ } \right)$ without the aid of a calculator.

A)? = 30°
B)?? = 45°
C)??? = 90°
D)??? = 75°
E)??? = 15°

Question 42

Use a calculator to evaluate tan 49°34´.Round your answer to four decimal places.

A)-1.3283
B)0.0423
C)-0.8403
D)1.1643
E)1.1736

Question 43

Use a calculator to evaluate tan 68°23'.Round your answer to four decimal places.

A)2.5236
B)2.5040
C)-1.2222
D)-0.8980
E)0.7269

Question 44

Find the exact value of csc ?,using the triangle shown in the figure below,if a = 3 and b = 4.

A) $$\frac { 5 } { 4 }$$
B)? $\frac { 5 } { 3 }$
C)? $\frac { 3 } { 4 }$
D)? $$\frac { 4 } { 5 }$$
E)? $\frac { 3 } { 5 }$

Question 45

Use a calculator to evaluate $\csc \left( \frac { \pi } { 10 } \right)$ .Round your answer to four decimal places.

A)-1.8382
B)-24.0997
C)0.3090
D)3.2361
E)-0.0415

Question 46

If $\sin \theta = \frac { \sqrt { 2 } } { 2 }$ ,find the value of ? in degrees $\left( 0 < \theta < 90 ^ { \circ } \right)$ without the aid of a calculator.

A)?? = 45°
B)?? = 30°
C)?? = 15°
D)?? = 90°
E)?? = 75°

Question 47

Will Barrow wanted to know how tall the flagpole was in front of his school.To find its height,he drove a stake into the ground at the tip of the flagpole's shadow and recorded the angle of elevation at two different times during the day.He then measured the distance between the stakes.Will's data is below: Stake
Time
Angle of Elevation
A
2:00 PM
82°
B
3:00 PM
​61°
Distance between stakes A & B
10 feet Determine the height of the flagpole.Round your answer to nearest foot.

A)22 feet
B)24 feet
C)20 feet
D)26 feet
E)18 feet

Question 48

Given $\sec \theta = \sqrt { 10 }$ and tan ? = 3,determine the following. ?
Cos(90° - ?)
?

A)?cos(90° - ?)= $\frac { 1 } { 3 }$
B)?cos(90° - ?)= $\frac { 3 \sqrt { 10 } } { 10 }$
C)?cos(90° - ?)= $\frac { \sqrt { 10 } } { 10 }$
D)?cos(90° - ?)= 3
E)undefined

Question 49

Find the exact value of csc ?,using the triangle shown in the figure below,if $a = 4$ and $b = 3$ .

A)? $\frac { 3 } { 5 }$
B)? $\frac { 4 } { 3 }$
C)? $\frac { 5 } { 4 }$
D)? $\frac { 4 } { 5 }$
E)? $\frac { 5 } { 3 }$

Question 50

If ? is an acute angle and $\cot \theta = \frac { 1 } { 4 }$ ,determine sin ?.

A) $\sin \theta = \frac { \sqrt { 17 } } { 4 }$
B)? $\sin \theta = \sqrt { 17 }$
C)? $\sin \theta = 4$
D)? $\sin \theta = \frac { \sqrt { 17 } } { 17 }$
E)? $\sin \theta = \frac { 4 \sqrt { 17 } } { 17 }$

Question 51

Given $\sin 30 ^ { \circ } = \frac { 1 } { 2 }$ and $\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$ ,determine the following: ?sec 30°
?

A)?sec 30° = 1
B)?sec 30° = $\sqrt { 3 }$
C)?sec 30° = $\frac { 2 \sqrt { 3 } } { 3 }$
D)?sec 30° = $\frac { \sqrt { 2 } } { 2 }$
E)undefined

Question 52

Using the figure below,if ? = 26° and y = 6,determine the exact value of x.

A)? $x = \frac { 6 } { \cot 26 ^ { \circ } }$
B)? $x = \frac { 26 } { \csc 6 ^ { \circ } }$
C) $x = \frac { 6 } { \tan 26 ^ { \circ } }$
D)? $x = \frac { 3 } { \sin 13 ^ { \circ } }$
E)? $x = \frac { 13 } { \tan 3 ^ { \circ } }$

Question 53

Given sin 30° = $\frac { 1 } { 2 }$ and cos 30° = $\frac { \sqrt { 3 } } { 2 }$ ,determine the following: ?cot 30°

A)?cot 30° = $\frac { \sqrt { 2 } } { 2 }$
B)undefined
C)?cot 30° = $\frac { \sqrt { 3 } } { 3 }$
D)?cot 30° = $\sqrt { 3 }$
E)?cot 30° = 2

Question 54

Using the figure below,if ? = 33° and y = 9,determine the exact value of x.

A) $x = \frac { 3 } { \sin 11 ^ { \circ } }$
B)? $x = \frac { 9 } { \tan 33 ^ { \circ } }$
C)? $x = \frac { 11 } { \tan 3 ^ { \circ } }$
D)? $x = \frac { 9 } { \cot 33 ^ { \circ } }$
E)? $x = \frac { 33 } { \csc 9 ^ { \circ } }$

Question 55

Using the figure below,if ? = 20° and y = 16,determine the exact value of r.

A) $$r = \frac { 5 } { \cos 4 ^ { \circ } }$$
B)? $r = \frac { 16 } { \cos 20 ^ { \circ } }$
C)? $r = \frac { 4 } { \csc 5 ^ { \circ } }$
D)? $r = \frac { 16 } { \tan 20 ^ { \circ } }$
E)? $r = \frac { 16 } { \sin 20 ^ { \circ } }$

Question 56

Using trigonometric identities,determine which of the following is equivalent to the following expression. ?
Tan ? + cot ? ?

A)?1
B)?csc ? sec ?
C) $\mathrm { csc }$ ? ? + $\sin$ ?
D)?cos ? + sec ?
E)?sec ? + csc ?