Question 1

Solve the problem.
-Find the measure of the smaller angle formed by the hands of the clock shown.

Accepted Answer

A)  150° 
B)  110° 
C)  120° 
D)  60° 
A)  150° 
B)  110° 
C)  120° 
D)  60°

Question 2

Draw the given angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure
of two other angles, one positive and one negative, coterminal with the given angle.
-$- 85 ^ { \circ }$

Accepted Answer

A)  $85 ^ { \circ }$ and $- 105 ^ { \circ }$
  
B)  $275 ^ { \circ }$ and $- 445 ^ { \circ }$
  
C)  $185 ^ { \circ }$ and $- 355 ^ { \circ }$
  
D)  $175 ^ { \circ }$ and $- 5 ^ { \circ }$
  
A)  $85 ^ { \circ }$ and $- 105 ^ { \circ }$
  
B)  $275 ^ { \circ }$ and $- 445 ^ { \circ }$
  
C)  $185 ^ { \circ }$ and $- 355 ^ { \circ }$
  
D)  $175 ^ { \circ }$ and $- 5 ^ { \circ }$

Question 3

An equation of the terminal side of an angle $\theta$ in standard position is given along with a restriction on x. Find the
indicated trigonometric function value of $\theta$ . Do not use a calculator.
-$2 x + 3 y = 0 , x \leq 0 ; \quad$ Find $\csc \theta$

Accepted Answer

A)  $- \frac { 3 } { 2 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 2 }$ 
D)  $- \frac { 2 } { 3 }$ 
A)  $- \frac { 3 } { 2 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 2 }$ 
D)  $- \frac { 2 } { 3 }$

Question 4

Convert the angle to decimal degrees and round to the nearest hundredth of a degree.
-$268 ^ { \circ } 50 ^ { \prime } 1 ^ { \prime \prime }$

Accepted Answer

A)  $268.83 ^ { \circ }$ 
B)  $268.89 ^ { \circ }$ 
C)  $268.79 ^ { \circ }$ 
D)  $268.84 ^ { \circ }$ 
A)  $268.83 ^ { \circ }$ 
B)  $268.89 ^ { \circ }$ 
C)  $268.79 ^ { \circ }$ 
D)  $268.84 ^ { \circ }$

Question 5

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
-A formula used by an engineer to determine the safe radius of a curve, $\mathrm { R }$, when designing a particular road is: $\mathrm { R } = \frac { \mathrm { V } ^ { 2 } } { \mathrm {~g} ( \mathrm { f } + \tan \alpha ) }$, where $\alpha$ is the superelevation of the road and $\mathrm { V }$ is the velocity (in feet per second) for which the curve is designed. If $\alpha = 3.5 ^ { \circ } , \mathrm { f } = 0.1 , \mathrm {~g} = 30$, and $\mathrm { R } = 1357.01 \mathrm { ft }$, find V. Round to the nearest foot per second.

Accepted Answer

A)  $V = 84 \mathrm { ft }$ per sec 
B)  $V = 79 \mathrm { ft }$ per sec 
C)  $V = 81 \mathrm { ft }$ per sec 
D)  $\mathrm { V } = 77 \mathrm { ft }$ per sec 
A)  $V = 84 \mathrm { ft }$ per sec 
B)  $V = 79 \mathrm { ft }$ per sec 
C)  $V = 81 \mathrm { ft }$ per sec 
D)  $\mathrm { V } = 77 \mathrm { ft }$ per sec

Question 6

Identify the quadrant for the angle $\theta$ satisfying the following conditions.
-$\sin \theta > 0 \text { and } \cos \theta > 0$

Accepted Answer

A)  Quadrant II 
B)  Quadrant IV 
C)  Quadrant I 
D)  Quadrant III 
A)  Quadrant II 
B)  Quadrant IV 
C)  Quadrant I 
D)  Quadrant III

Question 7

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
-$\cos \theta = 0.53445102$

Accepted Answer

A)  $32.3066886 ^ { \circ }$ 
B)  $57.6933114 ^ { \circ }$ 
C)  $122.306689 ^ { \circ }$ 
D)  $302.306689 ^ { \circ }$ 
A)  $32.3066886 ^ { \circ }$ 
B)  $57.6933114 ^ { \circ }$ 
C)  $122.306689 ^ { \circ }$ 
D)  $302.306689 ^ { \circ }$

Question 8

Solve the right triangle. If two sides are given, give angles in degrees and minutes.
-On a sunny day, a tree and its shadow form the sides of a right triangle. If the hypotenuse is 35 meters long and the tree is 28 meters tall, how long is the shadow?

Accepted Answer

A)  $63 \mathrm {~m}$ 
B)  $21 \mathrm {~m}$ 
C)  $45 \mathrm {~m}$ 
D)  $49 \mathrm {~m}$ 
A)  $63 \mathrm {~m}$ 
B)  $21 \mathrm {~m}$ 
C)  $45 \mathrm {~m}$ 
D)  $49 \mathrm {~m}$

Question 9

Determine the signs of the given trigonometric functions of an angle in standard position with the given measure.
-$\cos \left( 386 ^ { \circ } \right)$ and $\sin \left( 386 ^ { \circ } \right)$

Accepted Answer

A)  positive and positive 
B)  negative and positive 
C)  positive and negative 
D)  negative and negative 
A)  positive and positive 
B)  negative and positive 
C)  positive and negative 
D)  negative and negative

Question 10

Find the exact value of the expression.
-$$\sin \theta = \frac { \sqrt { 3 } } { 2 }$$

Accepted Answer

A)  $150 ^ { \circ }$ and $210 ^ { \circ }$ 
B)  $60 ^ { \circ }$ and $120 ^ { \circ }$ 
C)  $60 ^ { \circ }$ and $300 ^ { \circ }$ 
D)  $210 ^ { \circ }$ and $330 ^ { \circ }$ 
A)  $150 ^ { \circ }$ and $210 ^ { \circ }$ 
B)  $60 ^ { \circ }$ and $120 ^ { \circ }$ 
C)  $60 ^ { \circ }$ and $300 ^ { \circ }$ 
D)  $210 ^ { \circ }$ and $330 ^ { \circ }$

Question 11

Provide an appropriate response.
-Find the complement of an angle whose measure is $38 ^ { \circ } 11 ^ { \prime }$ .

Accepted Answer

A)  $51 ^ { \circ } 11 ^ { \prime }$ 
B)  $52 ^ { \circ } 48 ^ { \prime }$ 
C)  $51 ^ { \circ } 49 ^ { \prime }$ 
D)  $52 ^ { \circ } 49 ^ { \prime }$ 
A)  $51 ^ { \circ } 11 ^ { \prime }$ 
B)  $52 ^ { \circ } 48 ^ { \prime }$ 
C)  $51 ^ { \circ } 49 ^ { \prime }$ 
D)  $52 ^ { \circ } 49 ^ { \prime }$

Question 12

Use the appropriate identity to find the indicated function value. Rationalize the denominator, if applicable. If the given
value is a decimal, round your answer to three decimal places.
-$\csc \theta$, given that $\sin \theta = \frac { \sqrt { 7 } } { 6 }$

Accepted Answer

A)  $\frac { \sqrt { 7 } } { 6 }$ 
B)  $\frac { 7 \sqrt { 7 } } { 6 }$ 
C)  $\frac { \sqrt { 7 } } { 7 }$ 
D)  $\frac { 6 \sqrt { 7 } } { 7 }$ 
A)  $\frac { \sqrt { 7 } } { 6 }$ 
B)  $\frac { 7 \sqrt { 7 } } { 6 }$ 
C)  $\frac { \sqrt { 7 } } { 7 }$ 
D)  $\frac { 6 \sqrt { 7 } } { 7 }$

Question 13

Solve the right triangle. If two sides are given, give angles in degrees and minutes.
- 
$$\mathrm { a } = 103.21 \mathrm { ft } , \mathrm { c } = 292 \mathrm { ft }$$
Round the missing side length to two decimal places.

Accepted Answer

A)  $\mathrm { A } = 18 ^ { \circ } 42 ^ { \prime } , \mathrm { B } = 71 ^ { \circ } 18 ^ { \prime } , \mathrm { b } = 286.81 \mathrm { ft }$ 
B)  $A = 20 ^ { \circ } 42 ^ { \prime } , B = 69 ^ { \circ } 18 ^ { \prime } , b = 254.03 \mathrm { ft }$ 
C)  $\mathrm { A } = 21 ^ { \circ } 42 ^ { \prime } , \mathrm { B } = 68 ^ { \circ } 18 ^ { \prime } , \mathrm { b } = 273.15 \mathrm { ft }$ 
D)  $\mathrm { A } = 20 ^ { \circ } 42 ^ { \prime } ; \mathrm { B } = 69 ^ { \circ } 18 ^ { \prime } ; \mathrm { b } = 273.15 \mathrm { ft }$ 
A)  $\mathrm { A } = 18 ^ { \circ } 42 ^ { \prime } , \mathrm { B } = 71 ^ { \circ } 18 ^ { \prime } , \mathrm { b } = 286.81 \mathrm { ft }$ 
B)  $A = 20 ^ { \circ } 42 ^ { \prime } , B = 69 ^ { \circ } 18 ^ { \prime } , b = 254.03 \mathrm { ft }$ 
C)  $\mathrm { A } = 21 ^ { \circ } 42 ^ { \prime } , \mathrm { B } = 68 ^ { \circ } 18 ^ { \prime } , \mathrm { b } = 273.15 \mathrm { ft }$ 
D)  $\mathrm { A } = 20 ^ { \circ } 42 ^ { \prime } ; \mathrm { B } = 69 ^ { \circ } 18 ^ { \prime } ; \mathrm { b } = 273.15 \mathrm { ft }$

Question 14

Identify the quadrant for the angle $\theta$ satisfying the following conditions.
-$\tan \theta < 0 \text { and } \sin \theta < 0$

Accepted Answer

A)  Quadrant IV 
B)  Quadrant II 
C)  Quadrant I 
D)  Quadrant III 
A)  Quadrant IV 
B)  Quadrant II 
C)  Quadrant I 
D)  Quadrant III

Question 15

Find the angle of least positive measure coterminal with the given angle.
-$869 ^ { \circ }$

Accepted Answer

A)  $509 ^ { \circ }$ 
B)  $499 ^ { \circ }$ 
C)  $149 ^ { \circ }$ 
D)  $139 ^ { \circ }$ 
A)  $509 ^ { \circ }$ 
B)  $499 ^ { \circ }$ 
C)  $149 ^ { \circ }$ 
D)  $139 ^ { \circ }$

Question 16

Solve the problem.
-A surveyor recording data for a new subdivision measured an angle as $16.77 ^ { \circ }$. The next day, a different surveyor measured the same angle as $16 ^ { \circ } 34 ^ { \prime }$. Find the difference between these measurements (i) to the nearest minute and (ii) to the nearest hundredth of a degree.

Accepted Answer

A)  (i) $35 ^ { \prime }$(ii) $0.39 ^ { \circ }$ 
B)  (i) $80 ^ { \prime }$(ii) $1.34 ^ { \circ }$ 
C)  (i) $12 ^ { \prime }$(ii) $0.20 ^ { \circ }$ 
D)  (i) $43 ^ { \prime }$(ii) $0.43 ^ { \circ }$ 
A)  (i) $35 ^ { \prime }$(ii) $0.39 ^ { \circ }$ 
B)  (i) $80 ^ { \prime }$(ii) $1.34 ^ { \circ }$ 
C)  (i) $12 ^ { \prime }$(ii) $0.20 ^ { \circ }$ 
D)  (i) $43 ^ { \prime }$(ii) $0.43 ^ { \circ }$

Question 17

Solve the problem.
-In one area, the lowest angle of elevation of the sun in winter is $28 ^ { \circ } 39 ^ { \prime }$. Find the minimum distance $x$ that a plant needing full sun can be placed from a fence that is $4.5$ feet high. Round your answer to the tenths place when necessary.

Accepted Answer

A)  $8.4 \mathrm { ft }$ 
B)  $11.6 \mathrm { ft }$ 
C)  $7.8 \mathrm { ft }$ 
D)  $8.2 \mathrm { ft }$ 
A)  $8.4 \mathrm { ft }$ 
B)  $11.6 \mathrm { ft }$ 
C)  $7.8 \mathrm { ft }$ 
D)  $8.2 \mathrm { ft }$

Question 18

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
-$\csc \theta = 1.7208729$

Accepted Answer

A)  $35.5279790 ^ { \circ }$ 
B)  $59.8391099 ^ { \circ }$ 
C)  $54.4720210 ^ { \circ }$ 
D)  $30.1608900 ^ { \circ }$ 
A)  $35.5279790 ^ { \circ }$ 
B)  $59.8391099 ^ { \circ }$ 
C)  $54.4720210 ^ { \circ }$ 
D)  $30.1608900 ^ { \circ }$

Question 19

Solve the right triangle. If two sides are given, give angles in degrees and minutes.
-A tunnel is to be dug from point A to point B. Both A and B are visible from point C. If AC is 250 miles and BC is 519 miles, and if angle C is 90°, find the measure of angle B. Round your answer to
The tenths place.

Accepted Answer

A)  18.7° 
B)  25.7° 
C)  31.4° 
D)  34.1° 
A)  18.7° 
B)  25.7° 
C)  31.4° 
D)  34.1°

Question 20

Find the reference angle for the given angle.
-407°

Accepted Answer

A)  43° 
B)  133° 
C)  137° 
D)  47° 
A)  43° 
B)  133° 
C)  137° 
D)  47°

Solve the problem. -Find the measure of the smaller angle formed by the hands of the clock shown.

Draw the given angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, coterminal with the given angle. - $- 85 ^ { \circ }$

An equation of the terminal side of an angle $\theta$ in standard position is given along with a restriction on x. Find the indicated trigonometric function value of $\theta$ . Do not use a calculator. - $2 x + 3 y = 0 , x \leq 0 ; \quad$ Find $\csc \theta$

Convert the angle to decimal degrees and round to the nearest hundredth of a degree. - $268 ^ { \circ } 50 ^ { \prime } 1 ^ { \prime \prime }$

Identify the quadrant for the angle $\theta$ satisfying the following conditions. - $\sin \theta > 0 \text { and } \cos \theta > 0$

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary. - $\cos \theta = 0.53445102$

Solve the right triangle. If two sides are given, give angles in degrees and minutes. -On a sunny day, a tree and its shadow form the sides of a right triangle. If the hypotenuse is 35 meters long and the tree is 28 meters tall, how long is the shadow?

Determine the signs of the given trigonometric functions of an angle in standard position with the given measure. - $\cos \left( 386 ^ { \circ } \right)$ and $\sin \left( 386 ^ { \circ } \right)$

Find the exact value of the expression. - $\sin \theta = \frac { \sqrt { 3 } } { 2 }$

Provide an appropriate response. -Find the complement of an angle whose measure is $38 ^ { \circ } 11 ^ { \prime }$ .

Use the appropriate identity to find the indicated function value. Rationalize the denominator, if applicable. If the given value is a decimal, round your answer to three decimal places. - $\csc \theta$ , given that $\sin \theta = \frac { \sqrt { 7 } } { 6 }$

Identify the quadrant for the angle $\theta$ satisfying the following conditions. - $\tan \theta < 0 \text { and } \sin \theta < 0$

Find the angle of least positive measure coterminal with the given angle. - $869 ^ { \circ }$

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary. - $\csc \theta = 1.7208729$

Find the reference angle for the given angle. -407°

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 6: Trigonometric Functions

Solve the problem. -Find the measure of the smaller angle formed by the hands of the clock shown.

Draw the given angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, coterminal with the given angle. - −85∘- 85 ^ { \circ }−85∘

An equation of the terminal side of an angle θ\thetaθ in standard position is given along with a restriction on x. Find the indicated trigonometric function value of θ\thetaθ . Do not use a calculator. - 2x+3y=0,x≤0;2 x + 3 y = 0 , x \leq 0 ; \quad2x+3y=0,x≤0; Find csc⁡θ\csc \thetacscθ

Convert the angle to decimal degrees and round to the nearest hundredth of a degree. - 268∘50′1′′268 ^ { \circ } 50 ^ { \prime } 1 ^ { \prime \prime }268∘50′1′′

Identify the quadrant for the angle θ\thetaθ satisfying the following conditions. - sin⁡θ>0 and cos⁡θ>0\sin \theta > 0 \text { and } \cos \theta > 0sinθ>0 and cosθ>0

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary. - cos⁡θ=0.53445102\cos \theta = 0.53445102cosθ=0.53445102