Question 1

Find the cotangent (if it exists) of the real number $t = \frac { 5 \pi } { 4 }$.

Accepted Answer

A)  $- \frac { \sqrt { 2 } } { 2 }$ 
B)  1 
C)  $- 1$ 
D)  0 
E)  $\cot \left( \frac { 5 \pi } { 4 } \right)$ does not exist 
A)  $- \frac { \sqrt { 2 } } { 2 }$ 
B)  1 
C)  $- 1$ 
D)  0 
E)  $\cot \left( \frac { 5 \pi } { 4 } \right)$ does not exist

Question 2

Determine two coterminal angles (one positive and one negative) for $\theta = \frac { 5 \pi } { 6 }$.

Accepted Answer

A)  $\frac { 11 \pi } { 6 } , - \frac { 13 \pi } { 6 }$ 
B)  $\frac { 19 \pi } { 6 } , - \frac { 17 \pi } { 6 }$ 
C)  $\frac { 17 \pi } { 6 } , - \frac { 7 \pi } { 6 }$ 
D)  $\frac { 13 \pi } { 6 } , - \frac { 7 \pi } { 12 }$ 
E)  $\frac { 5 \pi } { 3 } , - \frac { 7 \pi } { 18 }$ 
A)  $\frac { 11 \pi } { 6 } , - \frac { 13 \pi } { 6 }$ 
B)  $\frac { 19 \pi } { 6 } , - \frac { 17 \pi } { 6 }$ 
C)  $\frac { 17 \pi } { 6 } , - \frac { 7 \pi } { 6 }$ 
D)  $\frac { 13 \pi } { 6 } , - \frac { 7 \pi } { 12 }$ 
E)  $\frac { 5 \pi } { 3 } , - \frac { 7 \pi } { 18 }$

Question 3

Approximate $\tan ^ { - 1 } ( 3 )$. Round your answer to four decimal places.

Accepted Answer

A)  $1.2490$ 
B)  $0.3463$ 
C)  $- 0.1425$ 
D)  $0.8006$ 
E)  $- 7.0153$ 
A)  $1.2490$ 
B)  $0.3463$ 
C)  $- 0.1425$ 
D)  $0.8006$ 
E)  $- 7.0153$

Question 4

Convert the angle measure to decimal degree form. $427 ^ { \circ } 50 ^ { \prime } 55 ^ { \prime \prime }$

Accepted Answer

A)  $426.151 ^ { \circ }$ 
B)  $427.050 ^ { \circ }$ 
C)  $427.849 ^ { \circ }$ 
D)  $7.454 ^ { \circ }$ 
E)  $24,471.314 ^ { \circ }$ 
A)  $426.151 ^ { \circ }$ 
B)  $427.050 ^ { \circ }$ 
C)  $427.849 ^ { \circ }$ 
D)  $7.454 ^ { \circ }$ 
E)  $24,471.314 ^ { \circ }$

Question 5

Find the cotangent (if it exists) of the real number $t = - \frac { 5 \pi } { 4 }$.

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 }$ 
B)  $- 1$ 
C)  1 
D)  0 
E)  $\cot \left( - \frac { 5 \pi } { 4 } \right)$ does not exist 
A)  $\frac { \sqrt { 2 } } { 2 }$ 
B)  $- 1$ 
C)  1 
D)  0 
E)  $\cot \left( - \frac { 5 \pi } { 4 } \right)$ does not exist

Question 6

Find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$.
radius: $r = 12$ kilometers arc length: $s = 64$ kilometers

Accepted Answer

A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 3 \pi } { 16 }$ 
C)  $\frac { 16 \pi } { 3 }$ 
D)  $\frac { 8 } { 3 }$ 
E)  $\frac { 16 } { 3 }$ 
A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 3 \pi } { 16 }$ 
C)  $\frac { 16 \pi } { 3 }$ 
D)  $\frac { 8 } { 3 }$ 
E)  $\frac { 16 } { 3 }$

Question 7

Find the reference angle for the angle $6.1$ radians. Round your answer to one decimal place.

Accepted Answer

A)  $- 3.0$ 
B)  $- 0.2$ 
C)  $3.0$ 
D)  $0.2$ 
E)  $- 12.4$ 
A)  $- 3.0$ 
B)  $- 0.2$ 
C)  $3.0$ 
D)  $0.2$ 
E)  $- 12.4$

Question 8

Evaluate the tangent of the angle without using a calculator. $- 120 ^ { \circ }$

Accepted Answer

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

Question 9

Use trigonometric identities to transform the left side of the equation into the right side. Assume all angles are positive acute angles, and show all of your work.
$$( \csc x - \cot x ) ( \csc x + \cot x ) = 1$$

Accepted Answer

\[\begin{aligned}
( \csc x - \cot x ) (

Question 10

The circular blade of a saw has a diameter of $6.5$ inches and rotates at 2000 revolutions per minute. Find the linear speed of the saw teeth in feet per second. Round your answer to two decimal places.

Accepted Answer

A)  $113.45$ feet per second 
B)  $680.68$ feet per second 
C)  $9.03$ feet per second 
D)  $56.72$ feet per second 
E)  $209.44$ feet per second 
A)  $113.45$ feet per second 
B)  $680.68$ feet per second 
C)  $9.03$ feet per second 
D)  $56.72$ feet per second 
E)  $209.44$ feet per second

Question 11

Find the exact value of $\sec \left( \arctan \frac { 8 } { 15 } \right)$.

Accepted Answer

A)  $\frac { 8 } { 15 }$ 
B)  $\frac { 32 } { 15 }$ 
C)  $\frac { 17 } { 15 }$ 
D)  $\frac { 17 } { 32 }$ 
E)  $\frac { 15 } { 8 }$ 
A)  $\frac { 8 } { 15 }$ 
B)  $\frac { 32 } { 15 }$ 
C)  $\frac { 17 } { 15 }$ 
D)  $\frac { 17 } { 32 }$ 
E)  $\frac { 15 } { 8 }$

Question 12

Find (if possible) the complement of $\frac { \pi } { 7 }$.

Accepted Answer

A)  $\frac { 3 \pi } { 7 }$ 
B)  $\frac { 2 \pi } { 7 }$ 
C)  $\frac { 6 \pi } { 7 }$ 
D)  $\frac { 5 \pi } { 14 }$ 
E)  not possible 
A)  $\frac { 3 \pi } { 7 }$ 
B)  $\frac { 2 \pi } { 7 }$ 
C)  $\frac { 6 \pi } { 7 }$ 
D)  $\frac { 5 \pi } { 14 }$ 
E)  not possible

Question 13

Determine two coterminal angles (one positive and one negative) for the given angle. Give your answer in degrees.

Accepted Answer

Answers may vary. On

Question 14

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $\tan 1.7$

Accepted Answer

A)  $0.0297$ 
B)  $0.9917$ 
C)  $- 0.1288$ 
D)  $- 7.6966$ 
E)  $1.0084$ 
A)  $0.0297$ 
B)  $0.9917$ 
C)  $- 0.1288$ 
D)  $- 7.6966$ 
E)  $1.0084$

Question 15

Use a graphing utility to graph the function below, making sure to show at least two periods.
$\sec \frac { x } { 2 }$

Accepted Answer

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

Question 16

The terminal side of $\theta$ lies on the the line $12 x + 5 y = 0$ in the second quadrant. Find the exact value of $\tan \theta$.

Accepted Answer

A)  $\tan \theta = - \frac { 5 } { 12 }$ 
B)  $\tan \theta = - \frac { 7 } { 5 }$ 
C)  $\tan \theta = \frac { 12 } { 7 }$ 
D)  $\tan \theta = - \frac { 12 } { 5 }$ 
E)  $\tan \theta = - \frac { 37 } { 14 }$ 
A)  $\tan \theta = - \frac { 5 } { 12 }$ 
B)  $\tan \theta = - \frac { 7 } { 5 }$ 
C)  $\tan \theta = \frac { 12 } { 7 }$ 
D)  $\tan \theta = - \frac { 12 } { 5 }$ 
E)  $\tan \theta = - \frac { 37 } { 14 }$

Question 17

Evaluate the sine of the angle without using a calculator.

Accepted Answer

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

Question 18

Find the exact value of the given trigonometric function of the angle $\theta$ shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.)
Find: $\sin \theta$

Accepted Answer

A)  $8 \sqrt { 2 }$ 
B)  $\sqrt { 2 }$ 
C)  $\frac { \sqrt { 2 } } { 2 }$ 
D)  1 
E)  $\frac { \sqrt { 2 } } { 8 }$ 
A)  $8 \sqrt { 2 }$ 
B)  $\sqrt { 2 }$ 
C)  $\frac { \sqrt { 2 } } { 2 }$ 
D)  1 
E)  $\frac { \sqrt { 2 } } { 8 }$

Question 19

Find the frequency of the simple harmonic motion described by the function below. $$d = - 3 \cos ( 18 \pi t )$$

Accepted Answer

A)  9 
B)  $- \frac { 2 \pi } { 3 }$ 
C)  $18 \pi$ 
D)  $- \frac { 2 } { 3 }$ 
E)  $- 3$ 
A)  9 
B)  $- \frac { 2 \pi } { 3 }$ 
C)  $18 \pi$ 
D)  $- \frac { 2 } { 3 }$ 
E)  $- 3$

Question 20

Determine the period and amplitude of $y = 5 \cos \left( \frac { x } { 13 } + \frac { \pi } { 5 } \right)$.

Accepted Answer

A)  period: $\frac { 2 \pi } { 13 }$; amplitude: 5 
B)  period: $26 \pi$; amplitude: 5 
C)  period: $13 \pi$; amplitude: 10 
D)  period: $\frac { \pi } { 13 }$; amplitude: 5 
E)  period: $- \frac { 2 \pi } { 13 } ;$ amplitude: $- 5$ 
A)  period: $\frac { 2 \pi } { 13 }$; amplitude: 5 
B)  period: $26 \pi$; amplitude: 5 
C)  period: $13 \pi$; amplitude: 10 
D)  period: $\frac { \pi } { 13 }$; amplitude: 5 
E)  period: $- \frac { 2 \pi } { 13 } ;$ amplitude: $- 5$

Find the cotangent (if it exists) of the real number $t = \frac { 5 \pi } { 4 }$ .

Determine two coterminal angles (one positive and one negative) for $\theta = \frac { 5 \pi } { 6 }$ .

Approximate $\tan ^ { - 1 } ( 3 )$ . Round your answer to four decimal places.

Convert the angle measure to decimal degree form. $427 ^ { \circ } 50 ^ { \prime } 55 ^ { \prime \prime }$

Find the cotangent (if it exists) of the real number $t = - \frac { 5 \pi } { 4 }$ .

Find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$ . radius: $r = 12$ kilometers arc length: $s = 64$ kilometers

Find the reference angle for the angle $6.1$ radians. Round your answer to one decimal place.

Evaluate the tangent of the angle without using a calculator. $- 120 ^ { \circ }$

Use trigonometric identities to transform the left side of the equation into the right side. Assume all angles are positive acute angles, and show all of your work. $( \csc x - \cot x ) ( \csc x + \cot x ) = 1$

The circular blade of a saw has a diameter of $6.5$ inches and rotates at 2000 revolutions per minute. Find the linear speed of the saw teeth in feet per second. Round your answer to two decimal places.

Find the exact value of $\sec \left( \arctan \frac { 8 } { 15 } \right)$ .

Find (if possible) the complement of $\frac { \pi } { 7 }$ .

Determine two coterminal angles (one positive and one negative) for the given angle. Give your answer in degrees.

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $\tan 1.7$

Use a graphing utility to graph the function below, making sure to show at least two periods. $\sec \frac { x } { 2 }$

The terminal side of $\theta$ lies on the the line $12 x + 5 y = 0$ in the second quadrant. Find the exact value of $\tan \theta$ .

Evaluate the sine of the angle without using a calculator.

Find the frequency of the simple harmonic motion described by the function below. $d = - 3 \cos ( 18 \pi t )$

Determine the period and amplitude of $y = 5 \cos \left( \frac { x } { 13 } + \frac { \pi } { 5 } \right)$ .

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Calculus Practice Problems

Filters

Exam 4: Trigonometric Functions

Find the cotangent (if it exists) of the real number t=5π4t = \frac { 5 \pi } { 4 }t=45π​ .

Determine two coterminal angles (one positive and one negative) for θ=5π6\theta = \frac { 5 \pi } { 6 }θ=65π​ .

Approximate tan⁡−1(3)\tan ^ { - 1 } ( 3 )tan−1(3) . Round your answer to four decimal places.

Convert the angle measure to decimal degree form. 427∘50′55′′427 ^ { \circ } 50 ^ { \prime } 55 ^ { \prime \prime }427∘50′55′′

Find the cotangent (if it exists) of the real number t=−5π4t = - \frac { 5 \pi } { 4 }t=−45π​ .

Find the radian measure of the central angle of a circle of radius rrr that intercepts an arc of length sss . radius: r=12r = 12r=12 kilometers arc length: s=64s = 64s=64 kilometers

Find the reference angle for the angle 6.16.16.1 radians. Round your answer to one decimal place.

Evaluate the tangent of the angle without using a calculator. −120∘- 120 ^ { \circ }−120∘

Use trigonometric identities to transform the left side of the equation into the right side. Assume all angles are positive acute angles, and show all of your work. (csc⁡x−cot⁡x)(csc⁡x+cot⁡x)=1( \csc x - \cot x ) ( \csc x + \cot x ) = 1(cscx−cotx)(cscx+cotx)=1

The circular blade of a saw has a diameter of 6.56.56.5 inches and rotates at 2000 revolutions per minute. Find the linear speed of the saw teeth in feet per second. Round your answer to two decimal places.

Find the exact value of sec⁡(arctan⁡815)\sec \left( \arctan \frac { 8 } { 15 } \right)sec(arctan158​) .

Find (if possible) the complement of π7\frac { \pi } { 7 }7π​ .

Determine two coterminal angles (one positive and one negative) for the given angle. Give your answer in degrees.

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) tan⁡1.7\tan 1.7tan1.7

Use a graphing utility to graph the function below, making sure to show at least two periods. sec⁡x2\sec \frac { x } { 2 }sec2x​

The terminal side of θ\thetaθ lies on the the line 12x+5y=012 x + 5 y = 012x+5y=0 in the second quadrant. Find the exact value of tan⁡θ\tan \thetatanθ .

Evaluate the sine of the angle without using a calculator.

Find the exact value of the given trigonometric function of the angle θ\thetaθ shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) Find: sin⁡θ\sin \thetasinθ

Find the frequency of the simple harmonic motion described by the function below. d=−3cos⁡(18πt)d = - 3 \cos ( 18 \pi t )d=−3cos(18πt)

Determine the period and amplitude of y=5cos⁡(x13+π5)y = 5 \cos \left( \frac { x } { 13 } + \frac { \pi } { 5 } \right)y=5cos(13x​+5π​) .

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Calculus Practice Problems

Filters

Find the cotangent (if it exists) of the real number $t = \frac { 5 \pi } { 4 }$ .

Determine two coterminal angles (one positive and one negative) for $\theta = \frac { 5 \pi } { 6 }$ .

Approximate $\tan ^ { - 1 } ( 3 )$ . Round your answer to four decimal places.

Convert the angle measure to decimal degree form. $427 ^ { \circ } 50 ^ { \prime } 55 ^ { \prime \prime }$

Find the cotangent (if it exists) of the real number $t = - \frac { 5 \pi } { 4 }$ .

Find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$ . radius: $r = 12$ kilometers arc length: $s = 64$ kilometers

Find the reference angle for the angle $6.1$ radians. Round your answer to one decimal place.

Evaluate the tangent of the angle without using a calculator. $- 120 ^ { \circ }$

Use trigonometric identities to transform the left side of the equation into the right side. Assume all angles are positive acute angles, and show all of your work. $( \csc x - \cot x ) ( \csc x + \cot x ) = 1$

The circular blade of a saw has a diameter of $6.5$ inches and rotates at 2000 revolutions per minute. Find the linear speed of the saw teeth in feet per second. Round your answer to two decimal places.

Find the exact value of $\sec \left( \arctan \frac { 8 } { 15 } \right)$ .

Find (if possible) the complement of $\frac { \pi } { 7 }$ .

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $\tan 1.7$

Use a graphing utility to graph the function below, making sure to show at least two periods. $\sec \frac { x } { 2 }$

The terminal side of $\theta$ lies on the the line $12 x + 5 y = 0$ in the second quadrant. Find the exact value of $\tan \theta$ .

Find the frequency of the simple harmonic motion described by the function below. $d = - 3 \cos ( 18 \pi t )$

Determine the period and amplitude of $y = 5 \cos \left( \frac { x } { 13 } + \frac { \pi } { 5 } \right)$ .