Question 1

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$, given that $\cos \theta = \frac { 4 } { 9 }$ and $\theta$ is in quadrant IV.

Accepted Answer

A)  $- \frac { \sqrt { 65 } } { 4 }$ 
B)  $- \sqrt { 65 }$ 
C)  $- \frac { 9 } { 4 }$ 
D)  $- \frac { \sqrt { 65 } } { 9 }$ 
A)  $- \frac { \sqrt { 65 } } { 4 }$ 
B)  $- \sqrt { 65 }$ 
C)  $- \frac { 9 } { 4 }$ 
D)  $- \frac { \sqrt { 65 } } { 9 }$

Question 2

Decide whether the statement is possible or impossible for an angle θ.
-$\cos \theta = \frac { 5 } { 11 }$ and $\sec \theta = \frac { 11 } { 5 }$

Accepted Answer

A)  Impossible 
B)  Possible 
A)  Impossible 
B)  Possible

Question 3

Solve the problem.
-A $6.1$-ft fence is $10.738 \mathrm { ft }$ away from a plant in the direction of the sun. It is observed that the shadow of the fence extends exactly to the bottom of the plant. (See drawing) Find $\theta$, the angle of elevation of the sun at that time. Round the measure of the angle to the nearest tenth of a degree when necessary.

Accepted Answer

A)  $\theta = 29.8 ^ { \circ }$ 
B)  $\theta = 29.6 ^ { \circ }$ 
C)  $\theta = 31 ^ { \circ }$ 
D)  $\theta = 29.4 ^ { \circ }$ 
A)  $\theta = 29.8 ^ { \circ }$ 
B)  $\theta = 29.6 ^ { \circ }$ 
C)  $\theta = 31 ^ { \circ }$ 
D)  $\theta = 29.4 ^ { \circ }$

Question 4

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$, given that $\tan \theta = - \frac { 4 } { 7 }$ and $\theta$ is in quadrant II.

Accepted Answer

A)  $\frac { \sqrt { 65 } } { 4 }$ 
B)  $- \frac { 7 \sqrt { 65 } } { 65 }$ 
C)  $\frac { 7 \sqrt { 65 } } { 65 }$ 
D)  $- \frac { \sqrt { 65 } } { 7 }$ 
A)  $\frac { \sqrt { 65 } } { 4 }$ 
B)  $- \frac { 7 \sqrt { 65 } } { 65 }$ 
C)  $\frac { 7 \sqrt { 65 } } { 65 }$ 
D)  $- \frac { \sqrt { 65 } } { 7 }$

Question 5

Sketch an angle $\theta$ in standard position such that $\theta$ has the least positive measure and the given point is on the terminal
side of $\theta _ { . }$
-$( 4 , - 2 )$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 6

Determine the signs of the given trigonometric functions of an angle in standard position with the given measure.
-$\sec \left( - 1 ^ { \circ } \right)$ and $\sin \left( - 1 ^ { \circ } \right)$

Accepted Answer

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

Question 7

Decide whether the statement is possible or impossible for an angle θ.
-$\tan \theta = - 2.14$

Accepted Answer

A)  Possible 
B)  Impossible 
A)  Possible 
B)  Impossible

Question 8

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cot \theta$, given that $\cos \theta = \frac { 21 } { 29 }$ and $\theta$ is in quadrant IV.

Accepted Answer

A)  $- \frac { 21 \sqrt { 2 } } { 4 }$ 
B)  $- \frac { 21 } { 20 }$ 
C)  $- \frac { 20 } { 21 }$ 
D)  $\frac { 29 } { 21 }$ 
A)  $- \frac { 21 \sqrt { 2 } } { 4 }$ 
B)  $- \frac { 21 } { 20 }$ 
C)  $- \frac { 20 } { 21 }$ 
D)  $\frac { 29 } { 21 }$

Question 9

Find the exact value of the expression.
-$\cot 30 ^ { \circ }$

Accepted Answer

A)  1 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\sqrt { 3 }$ 
D)  $\frac { \sqrt { 3 } } { 3 }$ 
A)  1 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\sqrt { 3 }$ 
D)  $\frac { \sqrt { 3 } } { 3 }$

Question 10

Solve the problem for the given information.
-Find the equation of a line passing through the origin so that the cosine of the angle between the line in quadrant $I$ and the positive $x$-axis is $\frac { \sqrt { 3 } } { 2 }$.

Accepted Answer

A)  $y = \frac { \sqrt { 3 } } { 2 } x$ 
B)  $y = \frac { \sqrt { 3 } } { 3 } x$ 
C)  $y = x$ 
D)  $y = \sqrt { 3 } x$ 
A)  $y = \frac { \sqrt { 3 } } { 2 } x$ 
B)  $y = \frac { \sqrt { 3 } } { 3 } x$ 
C)  $y = x$ 
D)  $y = \sqrt { 3 } x$

Question 11

Without using a calculator, give the exact trigonometric function value with rational denominator.
-$\sec 45 ^ { \circ }$

Accepted Answer

A)  $\frac { 2 \sqrt { 3 } } { 3 }$ 
B)  1 
C)  $\frac { 1 } { 2 }$ 
D)  $\sqrt { 2 }$ 
A)  $\frac { 2 \sqrt { 3 } } { 3 }$ 
B)  1 
C)  $\frac { 1 } { 2 }$ 
D)  $\sqrt { 2 }$

Question 12

If n is an integer, n ∙ 180° represents an integer multiple of 180°, and (2n + 1) ∙ 90° represents an odd integer multiple of
90°. Decide whether the expression is equal to 0, 1, -1, or is undefined.
-$\cos \left( ( 2 n + 1 ) \cdot 90 ^ { \circ } \right)$

Accepted Answer

A)  0 
B)  $- 1$ 
C)  Undefined 
D)  1 
A)  0 
B)  $- 1$ 
C)  Undefined 
D)  1

Question 13

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.
-$\cot \theta$, given that $\tan \theta = 0.2352$

Accepted Answer

A)  $4.252$ 
B)  $4.266$ 
C)  $4.259$ 
D)  $4.245$ 
A)  $4.252$ 
B)  $4.266$ 
C)  $4.259$ 
D)  $4.245$

Question 14

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sec \theta$, given that $\tan \theta = 3.7320508$ and $\theta$ is in quadrant $I$.

Accepted Answer

A)  $- 3.8637033$ 
B)  $- 2.1753277$ 
C)  $2.1753277$ 
D)  $3.8637033$ 
A)  $- 3.8637033$ 
B)  $- 2.1753277$ 
C)  $2.1753277$ 
D)  $3.8637033$

Question 15

Find all values of θ, if θ is in the interval [0, 360°) and has the given function value.
-$\sec \theta$ is undefined

Accepted Answer

A)  $90 ^ { \circ }$ 
B)  $90 ^ { \circ }$ and $270 ^ { \circ }$ 
C)  $0 ^ { \circ }$ 
D)  $0 ^ { \circ }$ and $180 ^ { \circ }$ 
A)  $90 ^ { \circ }$ 
B)  $90 ^ { \circ }$ and $270 ^ { \circ }$ 
C)  $0 ^ { \circ }$ 
D)  $0 ^ { \circ }$ and $180 ^ { \circ }$

Question 16

Solve the problem.
-Radio direction finders are set up at points $A$ and $B , 8.68 \mathrm { mi }$ apart on an east-west line. From $A$ it is found that the bearing of a signal from a transmitter is $\mathrm { N } 54.3 ^ { \circ } \mathrm { E }$, while from $\mathrm { B }$ it is $\mathrm { N } 35.7 ^ { \circ } \mathrm { W }$. Find the distance of the transmitter from B, to the nearest hundredth of a mile.

Accepted Answer

A)  $5.07 \mathrm { mi }$ 
B)  $4.57 \mathrm { mi }$ 
C)  $7.05 \mathrm { mi }$ 
D)  $7.55 \mathrm { mi }$ 
A)  $5.07 \mathrm { mi }$ 
B)  $4.57 \mathrm { mi }$ 
C)  $7.05 \mathrm { mi }$ 
D)  $7.55 \mathrm { mi }$

Question 17

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length
using the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle.
Rationalize the denominator if applicable.
-Find $\csc \mathrm { A }$ when $\mathrm { b } = 16$ and $\mathrm { c } = 34$

Accepted Answer

A)  $\frac { 8 } { 15 }$ 
B)  $\frac { 17 } { 15 }$ 
C)  $\frac { 15 } { 17 }$ 
D)  $\frac { 17 } { 8 }$ 
A)  $\frac { 8 } { 15 }$ 
B)  $\frac { 17 } { 15 }$ 
C)  $\frac { 15 } { 17 }$ 
D)  $\frac { 17 } { 8 }$

Question 18

Evaluate the expression.
-$\cot \left( - 90 ^ { \circ } \right)$

Accepted Answer

A)  $- 1$ 
B)  $\frac { \sqrt { 2 } } { 2 }$ 
C)  Undefined 
D)  0 
A)  $- 1$ 
B)  $\frac { \sqrt { 2 } } { 2 }$ 
C)  Undefined 
D)  0

Question 19

Find the reference angle for the given angle.
-$109 ^ { \circ }$

Accepted Answer

A)  $81 ^ { \circ }$ 
B)  $29 ^ { \circ }$ 
C)  $19 ^ { \circ }$ 
D)  $71 ^ { \circ }$ 
A)  $81 ^ { \circ }$ 
B)  $29 ^ { \circ }$ 
C)  $19 ^ { \circ }$ 
D)  $71 ^ { \circ }$

Question 20

If r is a positive number and the point (x, y) is in the indicated quadrant, decide whether the given ratio is positive or
negative.
-II, $\frac { x } { r }$

Accepted Answer

A)  Negative 
B)  Positive 
A)  Negative 
B)  Positive

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ , given that $\cos \theta = \frac { 4 } { 9 }$ and $\theta$ is in quadrant IV.

Decide whether the statement is possible or impossible for an angle θ. - $\cos \theta = \frac { 5 } { 11 }$ and $\sec \theta = \frac { 11 } { 5 }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ , given that $\tan \theta = - \frac { 4 } { 7 }$ and $\theta$ is in quadrant II.

Determine the signs of the given trigonometric functions of an angle in standard position with the given measure. - $\sec \left( - 1 ^ { \circ } \right)$ and $\sin \left( - 1 ^ { \circ } \right)$

Decide whether the statement is possible or impossible for an angle θ. - $\tan \theta = - 2.14$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cot \theta$ , given that $\cos \theta = \frac { 21 } { 29 }$ and $\theta$ is in quadrant IV.

Find the exact value of the expression. - $\cot 30 ^ { \circ }$

Solve the problem for the given information. -Find the equation of a line passing through the origin so that the cosine of the angle between the line in quadrant $I$ and the positive $x$ -axis is $\frac { \sqrt { 3 } } { 2 }$ .

Without using a calculator, give the exact trigonometric function value with rational denominator. - $\sec 45 ^ { \circ }$

If n is an integer, n ∙ 180° represents an integer multiple of 180°, and (2n + 1) ∙ 90° represents an odd integer multiple of 90°. Decide whether the expression is equal to 0, 1, -1, or is undefined. - $\cos \left( ( 2 n + 1 ) \cdot 90 ^ { \circ } \right)$

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. - $\cot \theta$ , given that $\tan \theta = 0.2352$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sec \theta$ , given that $\tan \theta = 3.7320508$ and $\theta$ is in quadrant $I$ .

Find all values of θ, if θ is in the interval [0, 360°) and has the given function value. - $\sec \theta$ is undefined

Evaluate the expression. - $\cot \left( - 90 ^ { \circ } \right)$

Find the reference angle for the given angle. - $109 ^ { \circ }$

If r is a positive number and the point (x, y) is in the indicated quadrant, decide whether the given ratio is positive or negative. -II, $\frac { x } { r }$

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

Review of Basic Concepts

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Exam 5: Trigonometric Functions

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ , given that cos⁡θ=49\cos \theta = \frac { 4 } { 9 }cosθ=94​ and θ\thetaθ is in quadrant IV.

Decide whether the statement is possible or impossible for an angle θ. - cos⁡θ=511\cos \theta = \frac { 5 } { 11 }cosθ=115​ and sec⁡θ=115\sec \theta = \frac { 11 } { 5 }secθ=511​

Use the fundamental identities to find the value of the trigonometric function. -Find cos⁡θ\cos \thetacosθ , given that tan⁡θ=−47\tan \theta = - \frac { 4 } { 7 }tanθ=−74​ and θ\thetaθ is in quadrant II.

Sketch an angle θ\thetaθ in standard position such that θ\thetaθ has the least positive measure and the given point is on the terminal side of θ.\theta _ { . }θ.​ - (4,−2)( 4 , - 2 )(4,−2)

Determine the signs of the given trigonometric functions of an angle in standard position with the given measure. - sec⁡(−1∘)\sec \left( - 1 ^ { \circ } \right)sec(−1∘) and sin⁡(−1∘)\sin \left( - 1 ^ { \circ } \right)sin(−1∘)

Decide whether the statement is possible or impossible for an angle θ. - tan⁡θ=−2.14\tan \theta = - 2.14tanθ=−2.14

Use the fundamental identities to find the value of the trigonometric function. -Find cot⁡θ\cot \thetacotθ , given that cos⁡θ=2129\cos \theta = \frac { 21 } { 29 }cosθ=2921​ and θ\thetaθ is in quadrant IV.

Find the exact value of the expression. - cot⁡30∘\cot 30 ^ { \circ }cot30∘

Solve the problem for the given information. -Find the equation of a line passing through the origin so that the cosine of the angle between the line in quadrant III and the positive xxx -axis is 32\frac { \sqrt { 3 } } { 2 }23​​ .

Without using a calculator, give the exact trigonometric function value with rational denominator. - sec⁡45∘\sec 45 ^ { \circ }sec45∘

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. - cot⁡θ\cot \thetacotθ , given that tan⁡θ=0.2352\tan \theta = 0.2352tanθ=0.2352

Use the fundamental identities to find the value of the trigonometric function. -Find sec⁡θ\sec \thetasecθ , given that tan⁡θ=3.7320508\tan \theta = 3.7320508tanθ=3.7320508 and θ\thetaθ is in quadrant III .

Find all values of θ, if θ is in the interval [0, 360°) and has the given function value. - sec⁡θ\sec \thetasecθ is undefined

Evaluate the expression. - cot⁡(−90∘)\cot \left( - 90 ^ { \circ } \right)cot(−90∘)

Find the reference angle for the given angle. - 109∘109 ^ { \circ }109∘

If r is a positive number and the point (x, y) is in the indicated quadrant, decide whether the given ratio is positive or negative. -II, xr\frac { x } { r }rx​