Question 1

Find the point $( x , y )$ on the unit circle that corresponds to the real number $t = - \frac { \pi } { 6 }$ .Use your results to evaluate cos t. ​

Accepted Answer

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

Question 2

Evaluate the trigonometric function.Round your answer to four decimal places. ​
Csc(-310°)
​

Accepted Answer

A) 2.3054 
B) 3.3054 
C) 1.3054 
D) 4.3054 
E) 1.8054 
A) 2.3054 
B) 3.3054 
C) 1.3054 
D) 4.3054 
E) 1.8054 C

Question 3

Find the reference angle $\theta ^ { \prime }$ .​ $$\theta = 5.3$$ ​

Accepted Answer

A)  $\theta ^ { \prime } = \pi - 5.3$ 
B)  $$\theta ^ { \prime } = 5.3 - 2 \pi$$ 
C)  $\theta ^ { \prime } = \pi + 5.3$ 
D)  $\theta ^ { \prime } = 2 \pi - 5.3$ 
E)  $\theta ^ { \prime } = 2 \pi + 5.3$ 
A)  $\theta ^ { \prime } = \pi - 5.3$ 
B)  $$\theta ^ { \prime } = 5.3 - 2 \pi$$ 
C)  $\theta ^ { \prime } = \pi + 5.3$ 
D)  $\theta ^ { \prime } = 2 \pi - 5.3$ 
E)  $\theta ^ { \prime } = 2 \pi + 5.3$ D

Question 4

Determine the exact value of the $\text { tangent }$ of the quadrant angle π.

Accepted Answer

A) 0 
B) undefined 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $- \frac { \sqrt { 3 } } { 2 }$ 
A) 0 
B) undefined 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $- \frac { \sqrt { 3 } } { 2 }$

Question 5

Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ.​   ​ where $a = 5 \text { and } b = 12$ ​

Accepted Answer

A) sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$ 
B) sin θ = $\frac { 5 } { 13 }$ cos θ = $\frac { 12 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$ 
C) sin θ = $\frac { 12 } { 5 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 13 }$ 
D) sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 12 } { 5 }$
Tan θ = $\frac { 5 } { 13 }$ 
E) sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 13 }$ 
A) sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$ 
B) sin θ = $\frac { 5 } { 13 }$ cos θ = $\frac { 12 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$ 
C) sin θ = $\frac { 12 } { 5 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 13 }$ 
D) sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 12 } { 5 }$
Tan θ = $\frac { 5 } { 13 }$ 
E) sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 13 }$

Question 6

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​
Sec 0.21
​

Accepted Answer

A) 1.5225 
B) 1.1225 
C) 1.4225 
D) 1.0225 
E) 1.3225 
A) 1.5225 
B) 1.1225 
C) 1.4225 
D) 1.0225 
E) 1.3225

Question 7

The point $( - 7 , - 24 )$ is on the terminal side of an angle in standard position.Determine the exact value of $\cos \theta$ .

Accepted Answer

A)  $\cos \theta = \frac { 7 } { 24 }$ 
B)  $\cos \theta = - \frac { 18 } { 7 }$ 
C)  $\cos \theta = - \frac { 7 } { 25 }$ 
D)  $\cos \theta = \frac { 31 } { 7 }$ 
E)  $\cos \theta = \frac { 24 } { 7 }$ 
A)  $\cos \theta = \frac { 7 } { 24 }$ 
B)  $\cos \theta = - \frac { 18 } { 7 }$ 
C)  $\cos \theta = - \frac { 7 } { 25 }$ 
D)  $\cos \theta = \frac { 31 } { 7 }$ 
E)  $\cos \theta = \frac { 24 } { 7 }$

Question 8

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​
Sec 221°
​

Accepted Answer

A) -1.3250 
B) -1.0250 
C) -0.8250 
D) -0.9250 
E) -1.2250 
A) -1.3250 
B) -1.0250 
C) -0.8250 
D) -0.9250 
E) -1.2250

Question 9

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​
Sin 11°
​

Accepted Answer

A) 0.4908 
B) 0.1908 
C) 0.5908 
D) 0.2908 
E) 0.6908 
A) 0.4908 
B) 0.1908 
C) 0.5908 
D) 0.2908 
E) 0.6908

Question 10

The point $( 5,12 )$ is on the terminal side of an angle in standard position.Determine the exact value of $\sec \theta$ .

Accepted Answer

A)  $\sec \theta = \frac { 5 } { 12 }$ 
B)  $\sec \theta = \frac { 13 } { 5 }$ 
C)  $\sec \theta = \frac { 12 } { 5 }$ 
D)  $\sec \theta = - \frac { 5 } { 13 }$ 
E)  $\sec \theta = - \frac { 5 } { 12 }$ 
A)  $\sec \theta = \frac { 5 } { 12 }$ 
B)  $\sec \theta = \frac { 13 } { 5 }$ 
C)  $\sec \theta = \frac { 12 } { 5 }$ 
D)  $\sec \theta = - \frac { 5 } { 13 }$ 
E)  $\sec \theta = - \frac { 5 } { 12 }$

Question 11

The point is on the terminal side of an angle in standard position.Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ.​ $( 15,8 )$ ​

Accepted Answer

A) sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 8 } { 15 }$
Tan θ = $\frac { 15 } { 17 }$ 
B) sin θ = $\frac { 8 } { 15 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 17 }$ 
C) sin θ = $\frac { 15 } { 17 }$ cos θ = $\frac { 8 } { 17 }$
Tan θ = $\frac { 8 } { 15 }$ 
D) sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 15 }$ 
E) sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 17 }$ 
A) sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 8 } { 15 }$
Tan θ = $\frac { 15 } { 17 }$ 
B) sin θ = $\frac { 8 } { 15 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 17 }$ 
C) sin θ = $\frac { 15 } { 17 }$ cos θ = $\frac { 8 } { 17 }$
Tan θ = $\frac { 8 } { 15 }$ 
D) sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 15 }$ 
E) sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 17 }$

Question 12

The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by $y = 2 e ^ { - t } \cos 6 t$ where y is the displacement in centimeters and t is the time in seconds (see figure).Find the displacement when $t = \frac { 1 } { 5 }$ .Round your answer to two decimal places.​   ​ ​

Accepted Answer

A) 0.89 cm 
B) 0.59 cm 
C) 0.99 cm 
D) 1.09 cm 
E) 0.79 cm 
A) 0.89 cm 
B) 0.59 cm 
C) 0.99 cm 
D) 1.09 cm 
E) 0.79 cm

Question 13

The point (-7,-24)is on the terminal side of an angle in standard position.Determine the exact value of $\cot \theta$ .

Accepted Answer

A)  $\cot \theta = - \frac { 25 } { 24 }$ 
B)  $\cot \theta = \frac { 1 } { 25 }$ 
C)  $\cot \theta = \frac { 7 } { 24 }$ 
D)  $\cot \theta = \frac { 31 } { 7 }$ 
E)  $\cot \theta = - \frac { 24 } { 25 }$ 
A)  $\cot \theta = - \frac { 25 } { 24 }$ 
B)  $\cot \theta = \frac { 1 } { 25 }$ 
C)  $\cot \theta = \frac { 7 } { 24 }$ 
D)  $\cot \theta = \frac { 31 } { 7 }$ 
E)  $\cot \theta = - \frac { 24 } { 25 }$

Question 14

Determine the exact value of the $\text { tangent }$ of the quadrant angle $\frac { \pi } { 2 }$ .

Accepted Answer

A)  $- \frac { \sqrt { 2 } } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D) 0 
E) undefined 
A)  $- \frac { \sqrt { 2 } } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D) 0 
E) undefined

Question 15

Find the values of the trigonometric functions sine and sec of θ with the given constraint. ​
Function Value
Constra int
Tan θ = - $\frac { 4 } { 3 }$ sin θ > 0
​​

Accepted Answer

A) sin θ = - $\frac { 4 } { 5 }$ sec θ = $\frac { 5 } { 3 }$ 
B) sin θ = $\frac { 4 } { 5 }$ sec θ = - $\frac { 5 } { 3 }$ 
C) sin θ = $\frac { 5 } { 3 }$ sec θ = - $\frac { 4 } { 5 }$ 
D) sin θ = $\frac { 4 } { 5 }$ sec θ = $\frac { 5 } { 3 }$ 
E) sin θ = - $\frac { 4 } { 5 }$ sec θ = - $\frac { 5 } { 3 }$ 
A) sin θ = - $\frac { 4 } { 5 }$ sec θ = $\frac { 5 } { 3 }$ 
B) sin θ = $\frac { 4 } { 5 }$ sec θ = - $\frac { 5 } { 3 }$ 
C) sin θ = $\frac { 5 } { 3 }$ sec θ = - $\frac { 4 } { 5 }$ 
D) sin θ = $\frac { 4 } { 5 }$ sec θ = $\frac { 5 } { 3 }$ 
E) sin θ = - $\frac { 4 } { 5 }$ sec θ = - $\frac { 5 } { 3 }$

Question 16

Determine the exact values of the three trigonometric functions csc,sec,and cot of the angle θ. ​
​​   where $a = 5 \text { and } b = 12$ ​

Accepted Answer

A) csc θ = $\frac { 13 } { 5 }$ sec θ = $\frac { 13 } { 12 }$
Cot θ = $\frac { 5 } { 12 }$ 
B) csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 13 } { 12 }$ 
C) csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 5 } { 12 }$ 
D) csc θ = $\frac { 5 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 13 } { 12 }$ 
E) csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 5 } { 12 }$
Cot θ = $\frac { 13 } { 5 }$ 
A) csc θ = $\frac { 13 } { 5 }$ sec θ = $\frac { 13 } { 12 }$
Cot θ = $\frac { 5 } { 12 }$ 
B) csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 13 } { 12 }$ 
C) csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 5 } { 12 }$ 
D) csc θ = $\frac { 5 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 13 } { 12 }$ 
E) csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 5 } { 12 }$
Cot θ = $\frac { 13 } { 5 }$

Question 17

Find the reference angle $\theta ^ { \prime }$ .​ $\theta = 11.8$ ​

Accepted Answer

A)  $$\theta ^ { \prime } = 2 \pi + 11.8$$ ​ 
B)  $\theta ^ { \prime } = 2 \pi - 11.8$ 
C)  $$\theta ^ { \prime } = 4 \pi - 11.8$$ 
D)  $\theta ^ { \prime } = 4 \pi + 11.8$ 
E)  $\theta ^ { \prime } = 11.8 - 2 \pi$ 
A)  $$\theta ^ { \prime } = 2 \pi + 11.8$$ ​ 
B)  $\theta ^ { \prime } = 2 \pi - 11.8$ 
C)  $$\theta ^ { \prime } = 4 \pi - 11.8$$ 
D)  $\theta ^ { \prime } = 4 \pi + 11.8$ 
E)  $\theta ^ { \prime } = 11.8 - 2 \pi$

Question 18

State the quadrant in which θ lies if $\cot \theta$  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 19

Find the reference angle $\theta^\prime$ ,and sketch $\theta$ and $\theta^\prime$ in standard position.​ $\theta = - 135 ^ { \circ }$ ​

Accepted Answer

A)  $\theta ^ { \prime } = - 45 ^ { \circ }$   
B)  $\theta^\prime = 45 ^ { \circ }$   
C)  $\theta ^ { \prime }$ = 35 ${}^\circ$   
D)  $\theta ^ { \prime }$ = 65 ${}^\circ$   
E)  $\theta ^ { \prime }$ = 55 ${}^\circ$   
A)  $\theta ^ { \prime } = - 45 ^ { \circ }$   
B)  $\theta^\prime = 45 ^ { \circ }$   
C)  $\theta ^ { \prime }$ = 35 ${}^\circ$   
D)  $\theta ^ { \prime }$ = 65 ${}^\circ$   
E)  $\theta ^ { \prime }$ = 55 ${}^\circ$

Question 20

An airplane,flying at an altitude of 8 miles,is on a flight path that passes directly over an observer (see figure).If $\theta$ is the angle of elevation from the observer to the plane,find the distance from the observer to the plane when $\theta = 60$ °.(Round your answers to two decimal place. )​   ​

Accepted Answer

A) 11.24 mi 
B) 10.24 mi 
C) 9.24 mi 
D) 12.24 mi 
E) 13.24 mi 
A) 11.24 mi 
B) 10.24 mi 
C) 9.24 mi 
D) 12.24 mi 
E) 13.24 mi

Exam 26: Trigonometric Functions of Any Angle

Find the point (x,y)( x , y )(x,y) on the unit circle that corresponds to the real number t=−π6t = - \frac { \pi } { 6 }t=−6π​ .Use your results to evaluate cos t. ​

Evaluate the trigonometric function.Round your answer to four decimal places. ​ Csc(-310°) ​

Find the reference angle θ′\theta ^ { \prime }θ′ .​ θ=5.3\theta = 5.3θ=5.3 ​

Determine the exact value of the tangent \text { tangent } tangent of the quadrant angle π.

Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ.​ ​ where a=5 and b=12a = 5 \text { and } b = 12a=5 and b=12 ​

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​ Sec 0.21 ​

The point (−7,−24)( - 7 , - 24 )(−7,−24) is on the terminal side of an angle in standard position.Determine the exact value of cos⁡θ\cos \thetacosθ .

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​ Sec 221° ​

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​ Sin 11° ​

The point (5,12)( 5,12 )(5,12) is on the terminal side of an angle in standard position.Determine the exact value of sec⁡θ\sec \thetasecθ .

The point is on the terminal side of an angle in standard position.Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ.​ (15,8)( 15,8 )(15,8) ​

The point (-7,-24)is on the terminal side of an angle in standard position.Determine the exact value of cot⁡θ\cot \thetacotθ .

Determine the exact value of the tangent \text { tangent } tangent of the quadrant angle π2\frac { \pi } { 2 }2π​ .

Find the values of the trigonometric functions sine and sec of θ with the given constraint. ​ Function Value Constra int Tan θ = - 43\frac { 4 } { 3 }34​ sin θ > 0 ​​

Determine the exact values of the three trigonometric functions csc,sec,and cot of the angle θ. ​ ​​ where a=5 and b=12a = 5 \text { and } b = 12a=5 and b=12 ​

Find the reference angle θ′\theta ^ { \prime }θ′ .​ θ=11.8\theta = 11.8θ=11.8 ​

State the quadrant in which θ lies if cot⁡θ\cot \thetacotθ < 0 and sec⁡θ\sec \thetasecθ > 0​.

Find the reference angle θ′\theta^\primeθ′ ,and sketch θ\thetaθ and θ′\theta^\primeθ′ in standard position.​ θ=−135∘\theta = - 135 ^ { \circ }θ=−135∘ ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Hyperbolas

Find the point $( x , y )$ on the unit circle that corresponds to the real number $t = - \frac { \pi } { 6 }$ .Use your results to evaluate cos t.

Evaluate the trigonometric function.Round your answer to four decimal places. Csc(-310°)

Find the reference angle $\theta ^ { \prime }$ . $\theta = 5.3$

Determine the exact value of the $\text { tangent }$ of the quadrant angle π.

Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ. $Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ. where a = 5 \text { and } b = 12 $ where $a = 5 \text { and } b = 12$

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) Sec 0.21

The point $( - 7 , - 24 )$ is on the terminal side of an angle in standard position.Determine the exact value of $\cos \theta$ .

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) Sec 221°

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) Sin 11°

The point $( 5,12 )$ is on the terminal side of an angle in standard position.Determine the exact value of $\sec \theta$ .

The point is on the terminal side of an angle in standard position.Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ. $( 15,8 )$

The point (-7,-24)is on the terminal side of an angle in standard position.Determine the exact value of $\cot \theta$ .

Determine the exact value of the $\text { tangent }$ of the quadrant angle $\frac { \pi } { 2 }$ .

Find the values of the trigonometric functions sine and sec of θ with the given constraint. Function Value Constra int Tan θ = - $\frac { 4 } { 3 }$ sin θ > 0

Determine the exact values of the three trigonometric functions csc,sec,and cot of the angle θ. $Determine the exact values of the three trigonometric functions csc,sec,and cot of the angle θ. where a = 5 \text { and } b = 12 $ where $a = 5 \text { and } b = 12$

Find the reference angle $\theta ^ { \prime }$ . $\theta = 11.8$

State the quadrant in which θ lies if $\cot \theta$ < 0 and $\sec \theta$ > 0.

Find the reference angle $\theta^\prime$ ,and sketch $\theta$ and $\theta^\prime$ in standard position. $\theta = - 135 ^ { \circ }$