Question 1

Evaluate the cosine 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 C

Question 2

Find the point $( x , y )$ on the unit circle corresponding to the real number $t = \frac { 11 \pi } { 6 }$.

Accepted Answer

A)  $\left( - \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$ 
B)  $\left( \frac { \sqrt { 3 } } { 2 } , - \frac { 1 } { 2 } \right)$ 
C)  $\left( - \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$ 
D)  $\left( \frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } \right)$ 
E)  $\left( \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right)$ 
A)  $\left( - \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$ 
B)  $\left( \frac { \sqrt { 3 } } { 2 } , - \frac { 1 } { 2 } \right)$ 
C)  $\left( - \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$ 
D)  $\left( \frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } \right)$ 
E)  $\left( \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right)$ B

Question 3

A certain satellite orbits 13,000 miles above Earth's surface (see figure). Find the angle of depression $\alpha$ from the satellite to the horizon. Assume the radius of the Earth is 4000 miles. Round your answer to the nearest hundredth of a degree.

Accepted Answer

A)  $13.61 ^ { \circ }$ 
B)  $17.92 ^ { \circ }$ 
C)  $72.08 ^ { \circ }$ 
D)  $76.39 ^ { \circ }$ 
E)  $13.24 ^ { \circ }$ 
A)  $13.61 ^ { \circ }$ 
B)  $17.92 ^ { \circ }$ 
C)  $72.08 ^ { \circ }$ 
D)  $76.39 ^ { \circ }$ 
E)  $13.24 ^ { \circ }$ D

Question 4

Determine the graph of $ y=\frac{1}{3} \cos (x) $.

Accepted Answer

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

Question 5

Find the amplitude of $y = - \frac { \pi } { 7 } \cos \left( \frac { 3 x } { 2 } \right)$.

Accepted Answer

A)  $- \frac { \pi } { 7 }$ 
B)  $\frac { 4 \pi } { 3 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 1 } { 7 }$ 
E)  $\frac { \pi } { 7 }$ 
A)  $- \frac { \pi } { 7 }$ 
B)  $\frac { 4 \pi } { 3 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 1 } { 7 }$ 
E)  $\frac { \pi } { 7 }$

Question 6

Find (if possible) the complement and supplement of the given angle. $51 ^ { \circ }$

Accepted Answer

A)  complement: $129 ^ { \circ }$; supplement: $39 ^ { \circ }$ 
B)  complement: $51 ^ { \circ }$; supplement: $129 ^ { \circ }$ 
C)  complement: $129 ^ { \circ }$; supplement: $309 ^ { \circ }$ 
D)  complement: $39 ^ { \circ }$; supplement: $309 ^ { \circ }$ 
E)  complement: $39 ^ { \circ }$; supplement: $129 ^ { \circ }$ 
A)  complement: $129 ^ { \circ }$; supplement: $39 ^ { \circ }$ 
B)  complement: $51 ^ { \circ }$; supplement: $129 ^ { \circ }$ 
C)  complement: $129 ^ { \circ }$; supplement: $309 ^ { \circ }$ 
D)  complement: $39 ^ { \circ }$; supplement: $309 ^ { \circ }$ 
E)  complement: $39 ^ { \circ }$; supplement: $129 ^ { \circ }$

Question 7

Use a calculator to evaluate the function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.)

Accepted Answer

A)  $- 0.3524$ 
B)  $0.1146$ 
C)  $0.9813$ 
D)  $0.1925$ 
E)  $5.0970$ 
A)  $- 0.3524$ 
B)  $0.1146$ 
C)  $0.9813$ 
D)  $0.1925$ 
E)  $5.0970$

Question 8

A submarine, cruising at a depth $d = 65$ meters, is on a trajectory that passes directly below a ship (see figure). If $\theta$ is the angle of depression from the ship to the submarine, find the distance $L$ from the ship to the sub when $\theta = 70 ^ { \circ }$. Round to the nearest meter.

Accepted Answer

A)  $L = 0$ meters 
B)  $L = 84$ meters 
C)  $L = 69$ meters 
D)  $L = 190$ meters 
E)  $L = 24$ meters 
A)  $L = 0$ meters 
B)  $L = 84$ meters 
C)  $L = 69$ meters 
D)  $L = 190$ meters 
E)  $L = 24$ meters

Question 9

Use a calculator to evaluate the function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.)
$$\csc 57 ^ { \circ } 27 ^ { \prime }$$

Accepted Answer

A)  $0.8429$ 
B)  $1.2753$ 
C)  $1.1863$ 
D)  $1.8586$ 
E)  $- 1.4725$ 
A)  $0.8429$ 
B)  $1.2753$ 
C)  $1.1863$ 
D)  $1.8586$ 
E)  $- 1.4725$

Question 10

Determine the exact value of $\arctan ( - 1 )$.

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $- \frac { \pi } { 2 }$ 
C)  $- \frac { \pi } { 4 }$ 
D)  0 
E)  $\frac { \pi } { 2 }$ 
A)  $\frac { \pi } { 4 }$ 
B)  $- \frac { \pi } { 2 }$ 
C)  $- \frac { \pi } { 4 }$ 
D)  0 
E)  $\frac { \pi } { 2 }$

Question 11

Rewrite the angle $\frac { \pi } { 6 }$ radians in degree measure.

Accepted Answer

A)  $60 ^ { \circ }$ 
B)  $20 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $15 ^ { \circ }$ 
E)  $330 ^ { \circ }$ 
A)  $60 ^ { \circ }$ 
B)  $20 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $15 ^ { \circ }$ 
E)  $330 ^ { \circ }$

Question 12

Determine the period and amplitude of the following function.
$y=4 \cos \left(\frac{3 x}{4}+\frac{\pi}{4}\right)$

Accepted Answer

A)  period: $2 \pi$, amplitude : 2 
B)  period: $10 \pi$, amplitude : 2 
C)  period: $5 \pi$, amplitude : 4 
D)  period : $\frac { 8 \pi } { 3 } ;$ amplitude $: 4$ 
E)  period: $3 \pi$, amplitude : 4 
A)  period: $2 \pi$, amplitude : 2 
B)  period: $10 \pi$, amplitude : 2 
C)  period: $5 \pi$, amplitude : 4 
D)  period : $\frac { 8 \pi } { 3 } ;$ amplitude $: 4$ 
E)  period: $3 \pi$, amplitude : 4

Question 13

Rewrite the given angle in radian measure as a multiple of $\pi$. (Do not use a calculator.) $72 ^ { \circ }$

Accepted Answer

A)  $\frac { 7 \pi } { 5 }$ 
B)  $\pi$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 2 \pi } { 5 }$ 
E)  $\frac { 3 \pi } { 5 }$ 
A)  $\frac { 7 \pi } { 5 }$ 
B)  $\pi$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 2 \pi } { 5 }$ 
E)  $\frac { 3 \pi } { 5 }$

Question 14

State the quadrant in which $\theta$ lies. $\cot ( \theta )  0$

Accepted Answer

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

Question 15

Determine the value of $x$.

Accepted Answer

A)  $\frac { 34 \sqrt { 3 } } { 3 }$ 
B)  $\frac { 1 } { 34 }$ 
C)  34 
D)  $\frac { \sqrt { 3 } } { 34 }$ 
E)  $\frac { 17 } { 60 }$ 
A)  $\frac { 34 \sqrt { 3 } } { 3 }$ 
B)  $\frac { 1 } { 34 }$ 
C)  34 
D)  $\frac { \sqrt { 3 } } { 34 }$ 
E)  $\frac { 17 } { 60 }$

Question 16

Determine the exact value of $\csc \theta$ when $\cos \theta = \frac { 5 } { 13 }$ and $\cot \theta < 0$.

Accepted Answer

A) 
$\csc \theta = - \frac { 1 } { 12 }$ 
B)  $\csc \theta = - \frac { 13 } { 12 }$ 
C)  $\csc \theta = - \frac { 12 } { 13 }$ 
D)  $\csc \theta = - \frac { 13 } { 6 }$ 
E)  $\csc \theta = - \frac { 14 } { 11 }$ 
A) 
$\csc \theta = - \frac { 1 } { 12 }$ 
B)  $\csc \theta = - \frac { 13 } { 12 }$ 
C)  $\csc \theta = - \frac { 12 } { 13 }$ 
D)  $\csc \theta = - \frac { 13 } { 6 }$ 
E)  $\csc \theta = - \frac { 14 } { 11 }$

Question 17

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 18

When an airplane leaves the runway, its angle of climb is $15 ^ { \circ }$ and its speed is 275 feet per second. Find the plane's altitude relative to the runway in feet after 1 minute. Round your answer to the nearest foot.

Accepted Answer

A)  3417 feet 
B)  2563 feet 
C)  $5979 \mathrm { feet }$ 
D)  4271 feet 
E)  5125 feet 
A)  3417 feet 
B)  2563 feet 
C)  $5979 \mathrm { feet }$ 
D)  4271 feet 
E)  5125 feet

Question 19

Determine a pair of angles (one positive and one negative) in degree measure coterminal to the angle $117 ^ { \circ }$.

Accepted Answer

A)  $297 ^ { \circ } , - 63 ^ { \circ }$ 
B)  $477 ^ { \circ } , - 243 ^ { \circ }$ 
C)  $417 ^ { \circ } , - 183 ^ { \circ }$ 
D)  $234 ^ { \circ } , - 117 ^ { \circ }$ 
E)  $297 ^ { \circ } , - 243 ^ { \circ }$ 
A)  $297 ^ { \circ } , - 63 ^ { \circ }$ 
B)  $477 ^ { \circ } , - 243 ^ { \circ }$ 
C)  $417 ^ { \circ } , - 183 ^ { \circ }$ 
D)  $234 ^ { \circ } , - 117 ^ { \circ }$ 
E)  $297 ^ { \circ } , - 243 ^ { \circ }$

Question 20

Write an algebraic expression that is equivalent to $\sin \left( \arctan \frac { x } { 9 } \right)$.

Accepted Answer

A)  $\frac { 9 } { \sqrt { x ^ { 2 } + 81 } }$ 
B)  $\frac { 9 } { x }$ 
C)  $\frac { \sqrt { x ^ { 2 } + 81 } } { x }$ 
D)  $\frac { \sqrt { x ^ { 2 } + 81 } } { 9 }$ 
E)  $\frac { x } { \sqrt { x ^ { 2 } + 81 } }$ 
A)  $\frac { 9 } { \sqrt { x ^ { 2 } + 81 } }$ 
B)  $\frac { 9 } { x }$ 
C)  $\frac { \sqrt { x ^ { 2 } + 81 } } { x }$ 
D)  $\frac { \sqrt { x ^ { 2 } + 81 } } { 9 }$ 
E)  $\frac { x } { \sqrt { x ^ { 2 } + 81 } }$

Evaluate the cosine of the angle without using a calculator.

Find the point $( x , y )$ on the unit circle corresponding to the real number $t = \frac { 11 \pi } { 6 }$ .

Determine the graph of $y=\frac{1}{3} \cos (x)$ .

Find the amplitude of $y = - \frac { \pi } { 7 } \cos \left( \frac { 3 x } { 2 } \right)$ .

Find (if possible) the complement and supplement of the given angle. $51 ^ { \circ }$

Use a calculator to evaluate the function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.)

Use a calculator to evaluate the function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) $\csc 57 ^ { \circ } 27 ^ { \prime }$

Determine the exact value of $\arctan ( - 1 )$ .

Rewrite the angle $\frac { \pi } { 6 }$ radians in degree measure.

Determine the period and amplitude of the following function. $y=4 \cos \left(\frac{3 x}{4}+\frac{\pi}{4}\right)$ $Determine the period and amplitude of the following function. y=4 \cos \left(\frac{3 x}{4}+\frac{\pi}{4}\right)$

Rewrite the given angle in radian measure as a multiple of $\pi$ . (Do not use a calculator.) $72 ^ { \circ }$

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

Determine the value of $x$ .

Determine the exact value of $\csc \theta$ when $\cos \theta = \frac { 5 } { 13 }$ and $\cot \theta < 0$ .

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

When an airplane leaves the runway, its angle of climb is $15 ^ { \circ }$ and its speed is 275 feet per second. Find the plane's altitude relative to the runway in feet after 1 minute. Round your answer to the nearest foot.

Determine a pair of angles (one positive and one negative) in degree measure coterminal to the angle $117 ^ { \circ }$ .

Write an algebraic expression that is equivalent to $\sin \left( \arctan \frac { x } { 9 } \right)$ .

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

Evaluate the cosine of the angle without using a calculator.

Find the point (x,y)( x , y )(x,y) on the unit circle corresponding to the real number t=11π6t = \frac { 11 \pi } { 6 }t=611π​ .

A certain satellite orbits 13,000 miles above Earth's surface (see figure). Find the angle of depression α\alphaα from the satellite to the horizon. Assume the radius of the Earth is 4000 miles. Round your answer to the nearest hundredth of a degree.

Determine the graph of y=13cos⁡(x) y=\frac{1}{3} \cos (x) y=31​cos(x) .

Find the amplitude of y=−π7cos⁡(3x2)y = - \frac { \pi } { 7 } \cos \left( \frac { 3 x } { 2 } \right)y=−7π​cos(23x​) .

Find (if possible) the complement and supplement of the given angle. 51∘51 ^ { \circ }51∘

Use a calculator to evaluate the function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.)

Use a calculator to evaluate the function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) csc⁡57∘27′\csc 57 ^ { \circ } 27 ^ { \prime }csc57∘27′

Determine the exact value of arctan⁡(−1)\arctan ( - 1 )arctan(−1) .

Rewrite the angle π6\frac { \pi } { 6 }6π​ radians in degree measure.

Determine the period and amplitude of the following function. y=4cos⁡(3x4+π4)y=4 \cos \left(\frac{3 x}{4}+\frac{\pi}{4}\right)y=4cos(43x​+4π​)

Rewrite the given angle in radian measure as a multiple of π\piπ . (Do not use a calculator.) 72∘72 ^ { \circ }72∘

State the quadrant in which θ\thetaθ lies. cot⁡(θ)<0\cot ( \theta ) < 0cot(θ)<0 and sec⁡(θ)>0\sec ( \theta ) > 0sec(θ)>0

Determine the value of xxx .

Determine the exact value of csc⁡θ\csc \thetacscθ when cos⁡θ=513\cos \theta = \frac { 5 } { 13 }cosθ=135​ and cot⁡θ<0\cot \theta < 0cotθ<0 .

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

When an airplane leaves the runway, its angle of climb is 15∘15 ^ { \circ }15∘ and its speed is 275 feet per second. Find the plane's altitude relative to the runway in feet after 1 minute. Round your answer to the nearest foot.

Determine a pair of angles (one positive and one negative) in degree measure coterminal to the angle 117∘117 ^ { \circ }117∘ .

Write an algebraic expression that is equivalent to sin⁡(arctan⁡x9)\sin \left( \arctan \frac { x } { 9 } \right)sin(arctan9x​) .

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 point $( x , y )$ on the unit circle corresponding to the real number $t = \frac { 11 \pi } { 6 }$ .

Determine the graph of $y=\frac{1}{3} \cos (x)$ .

Find the amplitude of $y = - \frac { \pi } { 7 } \cos \left( \frac { 3 x } { 2 } \right)$ .

Find (if possible) the complement and supplement of the given angle. $51 ^ { \circ }$

Use a calculator to evaluate the function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) $\csc 57 ^ { \circ } 27 ^ { \prime }$

Determine the exact value of $\arctan ( - 1 )$ .

Rewrite the angle $\frac { \pi } { 6 }$ radians in degree measure.

Determine the period and amplitude of the following function. $y=4 \cos \left(\frac{3 x}{4}+\frac{\pi}{4}\right)$ $Determine the period and amplitude of the following function. y=4 \cos \left(\frac{3 x}{4}+\frac{\pi}{4}\right)$

Rewrite the given angle in radian measure as a multiple of $\pi$ . (Do not use a calculator.) $72 ^ { \circ }$

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

Determine the value of $x$ .

Determine the exact value of $\csc \theta$ when $\cos \theta = \frac { 5 } { 13 }$ and $\cot \theta < 0$ .

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

When an airplane leaves the runway, its angle of climb is $15 ^ { \circ }$ and its speed is 275 feet per second. Find the plane's altitude relative to the runway in feet after 1 minute. Round your answer to the nearest foot.

Determine a pair of angles (one positive and one negative) in degree measure coterminal to the angle $117 ^ { \circ }$ .

Write an algebraic expression that is equivalent to $\sin \left( \arctan \frac { x } { 9 } \right)$ .