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 . Then you walk farther away from the mountain and find that the angle of elevation is now . Find the height of the mountain.

A) m
B) m
C) m
D) m
E) m

Answer 1

m

Question 2

In , you are given and . If is a point on and , find .

A)
B)
C)
D)
E)

Answer 2

Question 3

The radius of the circle in the figure is 2 units. Express the length in terms of .

A)
B)
C)
D)
E)

Answer 3

Question 4

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

A)
B)
C)
D)
E)

Question 5

Refer to the figure. If and , find .

A) cm
B) cm
C) cm
D) cm
E) cm

Question 6

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

A)
B)
C)
D)
E)

Question 7

Evaluate the expression using the concept of a reference angle.

A)
B)
C)
D)
E)

Question 8

Suppose that is a right triangle with . If and , find the quantities.

A)
B)
C)
D)
E)

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.

A)
B)
C)
D)
E)

Question 11

Determine whether the equation is correct by evaluating each side. Do not use a calculator. Note: Notation such as stands for .

Question 12

An observer in a lighthouse is above the surface of the water. The observer sees a ship and finds the angle of depression to be . Estimate the distance of the ship from the base of the lighthouse.

A) ft
B) ft
C) ft
D) ft
E) ft

Question 13

Determine whether the equation is correct by evaluating each side. Do not use a calculator.

Question 14

Evaluate the expression using the concept of a reference angle.

A)
B)
C)
D)
E)

Question 15

Use the following information to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals.

A)
B)
C)
D)
E)

Question 16

Use the definitions (not a calculator) to evaluate the six trigonometric functions of the angle.

A)
B)
C)
D)
E)

Question 17

The accompanying figure shows two ships at points and , which are in the same vertical plane as an airplane at point . When the height of the airplane is , the angle of depression to is and that to is .Find the distance between the two ships.

A) ft
B) ft
C) ft
D) ft
E) ft

Question 18

Use the following formation to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals.

A)
B)
C)
D)
E)

Question 19

Evaluate the expression using the concept of a reference angle.

A)
B)
C)
D)
E)

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.

A)
B)
C)
D)
E)

Question 22

Determine the answer that establishes an identity.

A)
B)
C)
D)
E)

Question 23

Use the following information to express the remaining five trigonometric values as functions of . Assume that is positive. Rationalize any denominators that contain radicals.

A)
B)
C)
D)
E)

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.

A)
B)
C)
D)
E)