Deck 6: An Introduction to Trigonometry Via Right Triangles

Question 1

From a point on ground level, you measure the angle of elevation to the top of a mountain to be $37 ^ { \circ }$ . Then you walk $150 \mathrm {~m}$ farther away from the mountain and find that the angle of elevation is now $20 ^ { \circ }$ . Find the height of the mountain.

A) $55$ m
B) $84$ m
C) $81$ m
D) $113$ m
E) $106$ m

Answer 1

$106$ m

Question 2

In $\triangle A C D$ , you are given $$\angle C = 90 ^ { \circ } , \angle A = 60 ^ { \circ }$$ and $A C = 9$ . If $B$ is a point on $\overline { C D }$ and $\angle B A C = 45 ^ { \circ }$ , find $B D$ .

A) $B D = 6$
B) $B D =$ $3 ( 3 - \sqrt { 3 } )$
C) $B D =$ $6 \sqrt { 3 }$
D) $B D = 3$
E) $B D =$ $9 ( \sqrt { 3 } - 1 )$

Answer 2

$B D =$ $9 ( \sqrt { 3 } - 1 )$

Question 3

The radius of the circle in the figure is 2 units. Express the length $D C$ in terms of $\alpha$ .

A) $2 \cos \alpha$
B) $2 \cot \alpha$
C) $2 \sin \alpha$
D) $2 \tan \alpha$
E) $2 \sec \alpha$

Answer 3

$2 \tan \alpha$

Question 4

Find the area of the triangle. Use a calculator and round your final answer to two decimal places.

A) $2.08 \mathrm {~cm} ^ { 2 }$
B) $5.91 \mathrm {~cm} ^ { 2 }$
C) $11.82 \mathrm {~cm} ^ { 2 }$
D) $1.04 \mathrm {~cm} ^ { 2 }$
E) $2.95 \mathrm {~cm} ^ { 2 }$

Question 5

Refer to the figure. If $\angle A = 60 ^ { \circ }$ and $A B = 40 \mathrm {~cm}$ , find $A C$ .

A) $$A C = 40 \sqrt { 3 }$$ cm
B) $$A C = 20$$ cm
C) $A C = 40$ cm
D) $A C = \sqrt { 3 }$ cm
E) $A C = 20 \sqrt { 5 }$ cm

Question 6

Use the definitions to evaluate the six trigonometric functions of $\theta$ . In cases in which a radical occurs in a denominator, rationalize the denominator.

A) $\sin \theta = \frac { \sqrt { 5 } } { 2 } , \tan \theta = \frac { 5 \sqrt { 5 } } { 2 } , \csc \theta = 5$ $\cos \theta = \sqrt { 5 } , \cot \theta = \frac { \sqrt { 5 } } { 2 } , \sec \theta = \frac { 1 } { 5 }$
B) $\sin \theta = \frac { \sqrt { 11 } } { 5 } , \tan \theta = \frac { 2 \sqrt { 11 } } { 7 } , \csc \theta = 2$ $\cos \theta = \sqrt { 11 } , \cot \theta = \frac { \sqrt { 11 } } { 2 } , \sec \theta = \frac { 1 } { 7 }$
C) $\sin \theta = \frac { 4 \sqrt { 5 } } { 5 } , \tan \theta = 4 , \csc \theta = \frac { \sqrt { 5 } } { 4 }$ $\cos \theta = \frac { \sqrt { 5 } } { 2 } , \cot \theta = \frac { 1 } { 4 } , \sec \theta = \sqrt { 5 }$
D) $\sin \theta = \frac { \sqrt { 5 } } { 5 } , \tan \theta = \frac { 1 } { 2 } , \csc \theta = \sqrt { 5 }$ $\cos \theta = \frac { 2 \sqrt { 5 } } { 5 } , \cot \theta = 2 , \sec \theta = \frac { \sqrt { 5 } } { 2 }$
E) $\sin \theta = \frac { \sqrt { 7 } } { 5 } , \tan \theta = \frac { 2 \sqrt { 7 } } { 5 } , \csc \theta = 2$ $\cos \theta = \sqrt { 7 } , \cot \theta = \frac { \sqrt { 7 } } { 2 } , \sec \theta = \frac { 1 } { 2 }$

Question 7

Evaluate the expression using the concept of a reference angle. $\sin \left( - 150 ^ { \circ } \right)$

A) $\frac { \sqrt { 6 } } { 6 }$
B) $\frac { \sqrt { 6 } } { 2 }$
C) $\frac { 1 } { 2 }$
D) $- \frac { 1 } { 2 }$
E) $- \frac { 1 } { 6 }$

Question 8

Suppose that $\triangle A B C$ is a right triangle with $\angle C = 90 ^ { \circ }$ . If $A B = 3$ and $B C = \frac { 3 \sqrt { 3 } } { 2 }$ , find the quantities. $\cos A , \sin B$

A) $\cos A = \frac { 1 } { 3 } , \sin B = \frac { \sqrt { 3 } } { 2 }$
B) $\cos A = \frac { 1 } { 2 } , \sin B = \frac { 1 } { 2 }$
C) $\cos A = \frac { \sqrt { 3 } } { 2 } , \sin B = \frac { \sqrt { 3 } } { 2 }$
D) $\cos A = \frac { 2 } { 3 } , \sin B = \frac { \sqrt { 3 } } { 2 }$
E) $\cos A = \frac { \sqrt { 3 } } { 2 } , \sin B = \frac { 1 } { 3 }$

Question 9

Use the following information to express the remaining five trigonometric values as functions of $t$ . Assume that $t$ is positive. Rationalize any denominators that contain radicals. $\cos \theta = - \frac { 3 t } { 4 } , 90 ^ { \circ } < \theta < 180 ^ { \circ }$

A) $\begin{array} { l } \tan \theta = - \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \sec \theta = - \frac { 4 } { 3 t } , \quad \sin \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\\cot \theta = - \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \csc \theta = \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } .\end{array}$
B) $\begin{array} { l } \tan \theta = - \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \sec \theta = - \frac { 4 } { 3 t } , \quad \sin \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\\cot \theta = - \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \csc \theta = \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } .\end{array}$
C) $$\begin{array} { l } \tan \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \sec \theta = \frac { 4 } { 3 t } , \quad \sin \theta = - \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\\cot \theta = \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \csc \theta = - \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } .\end{array}$$
D) $\begin{array} { l } \tan \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \sec \theta = \frac { 4 } { 3 t } , \quad \sin \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\\cot \theta = \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \csc \theta = \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } .\end{array}$
E) $\begin{array} { l } \tan \theta = - \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \sec \theta = \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \sin \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\\cot \theta = - \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \csc \theta = - \frac { 4 } { 3 t } .\end{array}$

Question 10

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the angle. $- 900 ^ { \circ }$

A) $\begin{array} { l l l } \sin \left( - 900 ^ { \circ } \right) = 0 & \tan \left( - 900 ^ { \circ } \right) = 0 & \csc \left( - 900 ^ { \circ } \right) = - 1 \\\cos \left( - 900 ^ { \circ } \right) = - 1 & \cot \left( - 900 ^ { \circ } \right) \text { is undefined } & \sec \left( - 900 ^ { \circ } \right) \text { is undefined }\end{array}$
B) $\begin{array} { l l l } \sin \left( - 900 ^ { \circ } \right) = - 1 & \tan \left( - 900 ^ { \circ } \right) = 0 & \csc \left( - 900 ^ { \circ } \right) \text { is unde fined } \\\cos \left( - 900 ^ { \circ } \right) = 0 & \cot \left( - 900 ^ { \circ } \right) \text { is undefined } & \sec \left( - 900 ^ { \circ } \right) = - 1\end{array}$
C) $\begin{array} { l l l } \sin \left( - 900 ^ { \circ } \right) = - 1 & \tan \left( - 900 ^ { \circ } \right) \text { is unde fined } & \csc \left( - 900 ^ { \circ } \right) = - 1 \\\cos \left( - 900 ^ { \circ } \right) = 0 & \cot \left( - 900 ^ { \circ } \right) = 0 & \sec \left( - 900 ^ { \circ } \right) \text { is undefined }\end{array}$
D) $\begin{array} { l l l } \sin \left( - 900 ^ { \circ } \right) = 0 & \tan \left( - 900 ^ { \circ } \right) \text { is undefined } & \csc \left( - 900 ^ { \circ } \right) \text { is unde fined } \\\cos \left( - 900 ^ { \circ } \right) = - 1 & \cot \left( - 900 ^ { \circ } \right) = 0 & \sec \left( - 900 ^ { \circ } \right) = - 1\end{array}$
E) $\begin{array} { l l l } \sin \left( - 900 ^ { \circ } \right) = 0 & \tan \left( - 900 ^ { \circ } \right) = 0 & \csc \left( - 900 ^ { \circ } \right) \text { is undefined } \\\cos \left( - 900 ^ { \circ } \right) = - 1 & \cot \left( - 900 ^ { \circ } \right) \text { is undefined } & \sec \left( - 900 ^ { \circ } \right) = - 1\end{array}$

Question 11

Determine whether the equation is correct by evaluating each side. Do not use a calculator. Note: Notation such as $\sin ^ { 2 } \theta$ stands for $( \sin \theta ) ^ { 2 }$ . $1 - \tan ^ { 2 } 60 ^ { \circ } = \sec ^ { 2 } 60 ^ { \circ }$

Question 12

An observer in a lighthouse is $62 \mathrm { ft }$ above the surface of the water. The observer sees a ship and finds the angle of depression to be $0.1 ^ { \circ }$ . Estimate the distance of the ship from the base of the lighthouse.

A) $35,545$ ft
B) $35,480$ ft
C) $35,505$ ft
D) $35,570$ ft
E) $35,525$ ft

Question 13

Determine whether the equation is correct by evaluating each side. Do not use a calculator. $\tan 30 ^ { \circ } = \frac { \sin 60 ^ { \circ } } { 1 + \sin 30 ^ { \circ } }$

Question 14

Evaluate the expression using the concept of a reference angle. $\cos \left( - 675 ^ { \circ } \right)$

A) $- \frac { \sqrt { 2 } } { 2 }$
B) $\frac { \sqrt { 2 } } { 2 }$
C) $\frac { \sqrt { 2 } } { 6 }$
D) $- \frac { \sqrt { 6 } } { 2 }$
E) $\frac { \sqrt { 6 } } { 6 }$

Question 15

Use the following information to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals. $\cos \theta = - \frac { 4 } { 7 } , 90 ^ { \circ } < \theta < 180 ^ { \circ }$

A) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = - \frac { \sqrt { 33 } } { 4 } , \csc \theta = \frac { 7 \sqrt { 33 } } { 33 } , \cot \theta = - \frac { 4 \sqrt { 33 } } { 33 } , \sec \theta = - \frac { 7 } { 4 }$
B) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = - \frac { \sqrt { 33 } } { 4 } , \csc \theta = \frac { 7 \sqrt { 33 } } { 33 } , \cot \theta = - \frac { 4 \sqrt { 33 } } { 33 } , \sec \theta = \frac { 7 } { 4 }$
C) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = - \frac { \sqrt { 33 } } { 4 } , \csc \theta = - \frac { 7 } { 4 } , \cot \theta = - \frac { 4 \sqrt { 33 } } { 33 } , \sec \theta = \frac { 7 \sqrt { 33 } } { 33 }$
D) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = \frac { 4 \sqrt { 33 } } { 33 } , \csc \theta = - \frac { 7 } { 4 } , \cot \theta = \frac { \sqrt { 33 } } { 4 } , \sec \theta = \frac { 7 \sqrt { 33 } } { 33 }$
E) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = \frac { 4 \sqrt { 33 } } { 33 } , \csc \theta = \frac { 7 \sqrt { 33 } } { 33 } , \cot \theta = \frac { \sqrt { 33 } } { 4 } , \sec \theta = - \frac { 7 } { 4 }$

Question 16

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the angle. $900 ^ { \circ }$

A) $\begin{array} { l l l } \sin \left( 900 ^ { \circ } \right) = 1 & \tan \left( 900 ^ { \circ } \right) = 0 & \csc \left( 900 ^ { \circ } \right) = - 1 \\\cos \left( 900 ^ { \circ } \right) = 0 & \cot \left( 900 ^ { \circ } \right) \text { is undefined } & \sec \left( 900 ^ { \circ } \right) \text { is undefined }\end{array}$
B) $\begin{array} { l l l } \sin \left( 900 ^ { \circ } \right) = - 1 & \tan \left( 900 ^ { \circ } \right) = 0 & \csc \left( 900 ^ { \circ } \right) = - 1 \\\cos \left( 900 ^ { \circ } \right) = 0 & \cot \left( 900 ^ { \circ } \right) \text { is undefined } & \sec \left( 900 ^ { \circ } \right) \text { is undefined }\end{array}$
C) $\begin{array} { l l l } \sin \left( 900 ^ { \circ } \right) = 0 & \tan \left( 900 ^ { \circ } \right) = 0 & \csc \left( 900 ^ { \circ } \right) = 0 \\\cos \left( 900 ^ { \circ } \right) = - 1 & \cot \left( 900 ^ { \circ } \right) = - 1 & \sec \left( 900 ^ { \circ } \right) = - 1\end{array}$
D) $\begin{array} { l l l } \sin \left( 900 ^ { \circ } \right) = 0 & \tan \left( 900 ^ { \circ } \right) = 0 & \csc \left( 900 ^ { \circ } \right) \text { is undefined } \\\cos \left( 900 ^ { \circ } \right) = - 1 & \cot \left( 900 ^ { \circ } \right) \text { is undefined } & \sec \left( 900 ^ { \circ } \right) = - 1\end{array}$
E) $\begin{array} { l l l } \sin \left( 900 ^ { \circ } \right) = - 1 & \tan \left( 900 ^ { \circ } \right) \text { is undefined } & \csc \left( 900 ^ { \circ } \right) = - 1 \\\cos \left( 900 ^ { \circ } \right) = 0 & \cot \left( 900 ^ { \circ } \right) = 0 & \sec \left( 900 ^ { \circ } \right) \text { is undefined }\end{array}$

Question 17

The accompanying figure shows two ships at points $P$ and $Q$ , which are in the same vertical plane as an airplane at point $$R$$ . When the height of the airplane is $3,100 \mathrm { ft }$ , the angle of depression to $P$ is $35 ^ { \circ }$ and that to $Q$ is $30 ^ { \circ }$ .Find the distance between the two ships.

A) $9,800$ ft
B) $3,960$ ft
C) $380$ ft
D) $60,250$ ft
E) $41,420$ ft

Question 18

Use the following formation to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals. $\sec B = - \frac { 9 } { 4 } , 180 ^ { \circ } < B < 270 ^ { \circ }$

A) $\sin B = - \frac { 4 } { 9 } , \tan B = \frac { 4 \sqrt { 65 } } { 65 } , \csc B = - \frac { 9 \sqrt { 65 } } { 65 } , \cot B = \frac { \sqrt { 65 } } { 4 } , \cos B = - \frac { \sqrt { 65 } } { 9 }$
B) $\sin B = - \frac { \sqrt { 65 } } { 9 } , \tan B = \frac { \sqrt { 65 } } { 4 } , \csc B = - \frac { 9 \sqrt { 65 } } { 65 } , \cot B = \frac { 4 \sqrt { 65 } } { 65 } , \cos B = - \frac { 4 } { 9 }$
C) $\sin B = \frac { \sqrt { 65 } } { 9 } , \tan B = - \frac { \sqrt { 65 } } { 4 } , \csc B = \frac { 9 \sqrt { 65 } } { 65 } , \cot B = - \frac { 4 \sqrt { 65 } } { 65 } , \cos B = \frac { 4 } { 9 }$
D) $\sin B = \frac { \sqrt { 65 } } { 9 } , \tan B = \frac { \sqrt { 65 } } { 4 } , \csc B = \frac { 9 \sqrt { 65 } } { 65 } , \cot B = \frac { 4 \sqrt { 65 } } { 65 } , \cos B = \frac { 4 } { 9 }$
E) $\sin B = - \frac { \sqrt { 65 } } { 9 } , \tan B = - \frac { \sqrt { 65 } } { 4 } , \csc \theta = - \frac { 9 \sqrt { 65 } } { 65 } , \cot B = - \frac { 4 \sqrt { 65 } } { 65 } , \cos B = - \frac { 4 } { 9 }$

Question 19

Evaluate the expression using the concept of a reference angle. $\cot \left( - 600 ^ { \circ } \right)$

A) $- \frac { \sqrt { 3 } } { 3 }$
B) $\frac { \sqrt { 3 } } { 3 }$
C) $- \frac { \sqrt { 3 } } { 5 }$
D) $- \frac { 1 } { 5 }$
E) $\frac { \sqrt { 5 } } { 5 }$

Question 20

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the angle. $- 720 ^ { \circ }$

A) $\begin{array} { l l l } \sin \left( - 720 ^ { \circ } \right) = 1 & \tan \left( - 720 ^ { \circ } \right) \text { is undefined } & \csc \left( - 720 ^ { \circ } \right) = 1 \\\cos \left( - 720 ^ { \circ } \right) = 0 & \cot \left( - 720 ^ { \circ } \right) = 0 & \sec \left( - 720 ^ { \circ } \right) \text { is undefined }\end{array}$
B) $\begin{array} { l l l } \sin \left( - 720 ^ { \circ } \right) = - 1 & \tan \left( - 720 ^ { \circ } \right) \text { is undefined } & \csc \left( - 720 ^ { \circ } \right) = - 1 \\\cos \left( - 720 ^ { \circ } \right) = 0 & \cot \left( - 720 ^ { \circ } \right) = 0 & \sec \left( - 720 ^ { \circ } \right) \text { is undefined }\end{array}$
C) $\begin{array} { l l l } \sin \left( - 720 ^ { \circ } \right) = 0 & \tan \left( - 720 ^ { \circ } \right) = 0 & \csc \left( - 720 ^ { \circ } \right) \text { is undefined } \\\cos \left( - 720 ^ { \circ } \right) = - 1 & \cot \left( - 720 ^ { \circ } \right) \text { is undefined } & \sec \left( - 720 ^ { \circ } \right) = - 1\end{array}$
D) $\begin{array} { l l l } \sin \left( - 720 ^ { \circ } \right) \text { is undefined } & \tan \left( - 720 ^ { \circ } \right) \text { is undefined } & \csc \left( - 720 ^ { \circ } \right) \text { is undefined } \\\cos \left( - 720 ^ { \circ } \right) = 1 & \cot \left( - 720 ^ { \circ } \right) = 1 & \sec \left( - 720 ^ { \circ } \right) = 1\end{array}$
E) $\begin{array} { l l l } \sin \left( - 720 ^ { \circ } \right) = 0 & \tan \left( - 720 ^ { \circ } \right) = 0 & \csc \left( - 720 ^ { \circ } \right) \text { is undefined } \\\cos \left( - 720 ^ { \circ } \right) = 1 & \cot \left( - 720 ^ { \circ } \right) \text { is undefined } & \sec \left( - 720 ^ { \circ } \right) = 1\end{array}$

Question 21

Determine the answer that establishes an identity. $\csc ^ { 2 } A + \sec ^ { 2 } A = ?$

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

Question 22

Determine the answer that establishes an identity. $\frac { \sin \theta } { \csc \theta } + \frac { \cos \theta } { \sec \theta } = ?$

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

Question 23

Use the following information to express the remaining five trigonometric values as functions of $u$ . Assume that $$u$$ is positive. Rationalize any denominators that contain radicals. $\cos \theta = \frac { u } { \sqrt { 10 } } , 0 ^ { \circ } < \theta < 90 ^ { \circ }$

A) $\begin{array} { l } \tan \theta = - \frac { \sqrt { 1 - u ^ { 2 } } } { u } , \quad \sec \theta = \frac { \sqrt { 10 } } { u } , \quad \sin \theta = - \frac { \sqrt { 10 - 10 u ^ { 2 } } } { 10 } , \\\cot \theta = - \frac { u \sqrt { 1 - u ^ { 2 } } } { 1 - u ^ { 2 } } , \quad \csc \theta = - \frac { \sqrt { 10 - 10 u ^ { 2 } } } { 1 - u ^ { 2 } } .\end{array}$
B) $\begin{array} { l } \tan \theta = \frac { u \sqrt { 1 - u ^ { 2 } } } { 1 - u ^ { 2 } } , \quad \sec \theta = \frac { \sqrt { 10 - 10 u ^ { 2 } } } { 1 - u ^ { 2 } } , \quad \sin \theta = \frac { \sqrt { 10 - 10 u ^ { 2 } } } { 10 } , \\\cot \theta = \frac { \sqrt { 1 - u ^ { 2 } } } { u } , \quad \csc \theta = \frac { \sqrt { 10 } } { u } .\end{array}$
C) $\begin{array} { l } \tan \theta = \frac { \sqrt { 10 - u ^ { 2 } } } { u } , \quad \sec \theta = \frac { \sqrt { 10 } } { u } , \quad \sin \theta = \frac { \sqrt { 100 - 10 u ^ { 2 } } , } { 10 } , \\\cot \theta = \frac { u \sqrt { 10 - u ^ { 2 } } , } { 10 - u ^ { 2 } } , \quad \csc \theta = \frac { \sqrt { 100 - 10 u ^ { 2 } } } { 10 - u ^ { 2 } } .\end{array}$
D) $\begin{array} { l } \tan \theta = \frac { u \sqrt { 1 - u ^ { 2 } } } { 1 - u ^ { 2 } } , \quad \sec \theta = \frac { \sqrt { 10 } } { u } , \quad \sin \theta = \frac { \sqrt { 10 - 10 u ^ { 2 } } } { 10 } , \\\cot \theta = \frac { \sqrt { 1 - u ^ { 2 } } } { u } , \quad \csc \theta = \frac { \sqrt { 10 - 10 u ^ { 2 } } } { 1 - u ^ { 2 } } .\end{array}$
E) $\begin{array} { l } \tan \theta = \frac { \sqrt { 1 - u ^ { 2 } } } { u } , \quad \sec \theta = \frac { \sqrt { 10 - 10 u ^ { 2 } } } { 1 - u ^ { 2 } } , \quad \sin \theta = \frac { \sqrt { 10 - 10 u ^ { 2 } } } { 10 } , \\\cot \theta = \frac { u \sqrt { 1 - u ^ { 2 } } } { 1 - u ^ { 2 } } , \quad \csc \theta = \frac { \sqrt { 10 } } { u } .\end{array}$

Question 24

Use the following information to express the remaining five trigonometric values as functions of $u$ . Assume that $u$ is positive. Rationalize any denominators that contain radicals. $\sin \theta = - u ^ { 2 } , 270 ^ { \circ } < \theta < 360 ^ { \circ }$

A) $\begin{array} { l } \tan \theta = - \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \sec \theta = - \frac { 1 } { u ^ { 2 } } , \quad \cos \theta = \sqrt { 1 - u ^ { 4 } } , \\\cot \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , \quad \csc \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } .\end{array}$
B) $\begin{array} { l } \tan \theta = - \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \sec \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \cos \theta = \sqrt { 1 + u ^ { 4 } } , \\\cot \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , \quad \csc \theta = - \frac { 1 } { u ^ { 2 } } .\end{array}$
C) $\begin{array} { l } \tan \theta = - \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \sec \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \cos \theta = \sqrt { 1 - u ^ { 4 } } , \\\cot \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , \quad \csc \theta = - \frac { 1 } { u ^ { 2 } } .\end{array}$
D) $\begin{array} { l } \tan \theta = \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \sec \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \cos \theta = - \sqrt { 1 - u ^ { 4 } } , \\\cot \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , \quad \csc \theta = \frac { 1 } { u ^ { 2 } } .\end{array}$
E) $\begin{array} { l l } \tan \theta = \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , & \sec \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \cos \theta = \sqrt { 1 - u ^ { 4 } } , \\\cot \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , & \csc \theta = \frac { 1 } { u ^ { 2 } } .\end{array}$

Question 25

Determine the answer that establishes an identity. $\frac { \sin B } { 1 + \cos B } + \frac { 1 + \cos B } { \sin B } = ?$

A) $1$
B) $2 \cos B$
C) $2 \sec B$
D) $2 \sin B$
E) $2 \csc B$