Question 1

Use a sum or difference identity to find the exact value.
-$\sin \left( - 15 ^ { \circ } \right)$

Accepted Answer

A)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
C)  $\frac { - \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 2 } + \sqrt { 6 } } { 4 }$ 
A)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
C)  $\frac { - \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 2 } + \sqrt { 6 } } { 4 }$

Question 2

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\sec ( \arctan 4.839 )$

Accepted Answer

A)  $0.5$ 
B)  $4.9412$ 
C)  $0.2024$ 
D)  1 
A)  $0.5$ 
B)  $4.9412$ 
C)  $0.2024$ 
D)  1

Question 3

Use the fundamental identities to simplify the expression.
-$$\frac { \tan \theta } { \sec \theta }$$

Accepted Answer

A)  $\cos ^ { 3 } \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sin \theta$ 
D)  $\sec ^ { 2 } \theta$ 
A)  $\cos ^ { 3 } \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sin \theta$ 
D)  $\sec ^ { 2 } \theta$

Question 4

Give the degree measure of $$\theta$$
-$\theta = \arctan ( - 1 )$

Accepted Answer

A)  $- 45 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $- 135 ^ { \circ }$ 
D)  $225 ^ { \circ }$ 
A)  $- 45 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $- 135 ^ { \circ }$ 
D)  $225 ^ { \circ }$

Question 5

Use the cofunction identities to find an angle $$\theta$$ that makes the statement true.
-$$\sin \left( \theta + 4 ^ { \circ } \right) = \cos \left( \theta + 6 ^ { \circ } \right)$$

Accepted Answer

A)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
B)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
C)  $\theta = 40 ^ { \circ }$ 
D)  $\theta = 10 ^ { \circ }$ 
A)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
B)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
C)  $\theta = 40 ^ { \circ }$ 
D)  $\theta = 10 ^ { \circ }$

Question 6

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta }$$

Accepted Answer

A)  1 
B)  $- \sec ^ { 2 } \theta$ 
C)  $1 - \sin \theta$ 
D)  0 
A)  1 
B)  $- \sec ^ { 2 } \theta$ 
C)  $1 - \sin \theta$ 
D)  0

Question 7

Give the degree measure of $$\theta$$
-$$\theta = \cot ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 3 } \right)$$

Accepted Answer

A)  $60 ^ { \circ }$ 
B)  $240 ^ { \circ }$ 
C)  $120 ^ { \circ }$ 
D)  $330 ^ { \circ }$ 
A)  $60 ^ { \circ }$ 
B)  $240 ^ { \circ }$ 
C)  $120 ^ { \circ }$ 
D)  $330 ^ { \circ }$

Question 8

Use a calculator to give the value to the nearest degree.
-$\theta = \sin ^ { - 1 } ( 0.2079 )$

Accepted Answer

A)  $10 ^ { \circ }$ 
B)  $12 ^ { \circ }$ 
C)  $15 ^ { \circ }$ 
D)  $14 ^ { \circ }$ 
A)  $10 ^ { \circ }$ 
B)  $12 ^ { \circ }$ 
C)  $15 ^ { \circ }$ 
D)  $14 ^ { \circ }$

Question 9

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \cos ^ { - 1 } ( 0.8910 )$$

Accepted Answer

A)  $1.1000010$ 
B)  $0.4365083$ 
C)  $0.5063492$ 
D)  $0.4712533$ 
A)  $1.1000010$ 
B)  $0.4365083$ 
C)  $0.5063492$ 
D)  $0.4712533$

Question 10

Verify that the equation is an identity.
-$$\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$$

Accepted Answer

\[\begin{array} { l } 
\tan \left( \frac

Question 11

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$
-$\cos x = \sin x$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { 3 \pi } { 4 } , \frac { 7 \pi } { 2 } \right\}$ 
C)  $\left\{ \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { 3 \pi } { 4 } , \frac { 7 \pi } { 2 } \right\}$ 
C)  $\left\{ \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$

Question 12

Use identities to write the expression as a single function of x or $$\theta$$
-$$\tan \left( x + \frac { \pi } { 6 } \right)$$

Accepted Answer

A)  $\tan x$ 
B)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
C)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$ 
D)  $- \tan x$ 
A)  $\tan x$ 
B)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
C)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$ 
D)  $- \tan x$

Question 13

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \theta$$

Accepted Answer

A)  Not an Identity 
B)  Identity 
A)  Not an Identity 
B)  Identity

Question 14

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \cot ^ { - 1 } ( 2.5181552 )$$

Accepted Answer

A)  $0.3780178$ 
B)  $6.8341176$ 
C)  $2.1658824$ 
D)  $1.1927785$ 
A)  $0.3780178$ 
B)  $6.8341176$ 
C)  $2.1658824$ 
D)  $1.1927785$

Question 15

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$$\sin x \cos x = \frac { 1 } { 2 }$$

Accepted Answer

A)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
A)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$

Question 16

Write in terms of the cofunction of a complementary angle.
-$\sin 70 ^ { \circ }$

Accepted Answer

A)  $\cos 110 ^ { \circ }$ 
B)  $\csc 110 ^ { \circ }$ 
C)  $\csc 20 ^ { \circ }$ 
D)  $\cos 20 ^ { \circ }$ 
A)  $\cos 110 ^ { \circ }$ 
B)  $\csc 110 ^ { \circ }$ 
C)  $\csc 20 ^ { \circ }$ 
D)  $\cos 20 ^ { \circ }$

Question 17

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-

Accepted Answer

A)  f(-x) = -f(x) 
B)  f(-x) = f(x) 
A)  f(-x) = -f(x) 
B)  f(-x) = f(x)

Question 18

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$\sqrt { 2 } \cos 2 x = 1$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$ 
C)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$ 
C)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
D)  $\varnothing$

Question 19

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$1 - 2 \sin ^ { 2 } 2 x$$

Accepted Answer

A)  $2 \sin 4 x$ 
B)  $\cos 4 x$ 
C)  $\frac { 1 } { 2 } \sin 16 x$ 
D)  $\cos 8 x$ 
A)  $2 \sin 4 x$ 
B)  $\cos 4 x$ 
C)  $\frac { 1 } { 2 } \sin 16 x$ 
D)  $\cos 8 x$

Question 20

Use identities to write the expression as a single function of x or $$\theta$$
-$$\sin \left( x - \frac { \pi } { 2 } \right)$$

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $\sin x$ 
C)  $- \cos x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
A)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $\sin x$ 
C)  $- \cos x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$

Use a sum or difference identity to find the exact value. - $\sin \left( - 15 ^ { \circ } \right)$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\sec ( \arctan 4.839 )$

Use the fundamental identities to simplify the expression. - $\frac { \tan \theta } { \sec \theta }$

Give the degree measure of $\theta$ - $\theta = \arctan ( - 1 )$

Use the cofunction identities to find an angle $\theta$ that makes the statement true. - $\sin \left( \theta + 4 ^ { \circ } \right) = \cos \left( \theta + 6 ^ { \circ } \right)$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta }$

Give the degree measure of $\theta$ - $\theta = \cot ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 3 } \right)$

Use a calculator to give the value to the nearest degree. - $\theta = \sin ^ { - 1 } ( 0.2079 )$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \cos ^ { - 1 } ( 0.8910 )$

Verify that the equation is an identity. - $\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\cos x = \sin x$

Use identities to write the expression as a single function of x or $\theta$ - $\tan \left( x + \frac { \pi } { 6 } \right)$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \theta$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \cot ^ { - 1 } ( 2.5181552 )$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\sin x \cos x = \frac { 1 } { 2 }$

Write in terms of the cofunction of a complementary angle. - $\sin 70 ^ { \circ }$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\sqrt { 2 } \cos 2 x = 1$

Use an identity to write the expression as a single trigonometric function or as a single number. - $1 - 2 \sin ^ { 2 } 2 x$

Use identities to write the expression as a single function of x or $\theta$ - $\sin \left( x - \frac { \pi } { 2 } \right)$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 8: Trigonometric Identities and Equations

Use a sum or difference identity to find the exact value. - sin⁡(−15∘)\sin \left( - 15 ^ { \circ } \right)sin(−15∘)

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - sec⁡(arctan⁡4.839)\sec ( \arctan 4.839 )sec(arctan4.839)

Use the fundamental identities to simplify the expression. - tan⁡θsec⁡θ\frac { \tan \theta } { \sec \theta }secθtanθ​

Give the degree measure of θ\thetaθ - θ=arctan⁡(−1)\theta = \arctan ( - 1 )θ=arctan(−1)

Use the cofunction identities to find an angle θ\thetaθ that makes the statement true. - sin⁡(θ+4∘)=cos⁡(θ+6∘)\sin \left( \theta + 4 ^ { \circ } \right) = \cos \left( \theta + 6 ^ { \circ } \right)sin(θ+4∘)=cos(θ+6∘)

Perform the indicated operations and simplify the result so there are no quotients. - (sin⁡θ+cos⁡θ)21+2sin⁡θcos⁡θ\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta }1+2sinθcosθ(sinθ+cosθ)2​

Give the degree measure of θ\thetaθ - θ=cot⁡−1(−33)\theta = \cot ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 3 } \right)θ=cot−1(−33​​)

Use a calculator to give the value to the nearest degree. - θ=sin⁡−1(0.2079)\theta = \sin ^ { - 1 } ( 0.2079 )θ=sin−1(0.2079)

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=cos⁡−1(0.8910)y = \cos ^ { - 1 } ( 0.8910 )y=cos−1(0.8910)

Verify that the equation is an identity. - tan⁡(π2+x)=−cot⁡x\tan \left( \frac { \pi } { 2 } + x \right) = - \cot xtan(2π​+x)=−cotx

Solve the equation for exact solutions over the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - cos⁡x=sin⁡x\cos x = \sin xcosx=sinx

Use identities to write the expression as a single function of x or θ\thetaθ - tan⁡(x+π6)\tan \left( x + \frac { \pi } { 6 } \right)tan(x+6π​)

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - sin⁡θ+1cos⁡θ+cot⁡θ=tan⁡θ\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \thetacosθ+cotθsinθ+1​=tanθ

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=cot⁡−1(2.5181552)y = \cot ^ { - 1 } ( 2.5181552 )y=cot−1(2.5181552)

Solve the equation for solutions in the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - sin⁡xcos⁡x=12\sin x \cos x = \frac { 1 } { 2 }sinxcosx=21​

Write in terms of the cofunction of a complementary angle. - sin⁡70∘\sin 70 ^ { \circ }sin70∘