Question 1

Give the degree measure of $$\theta$$
-$$\theta = \cot ^ { - 1 } ( - 1 )$$

Accepted Answer

A)  $180 ^ { \circ }$ 
B)  $90 ^ { \circ }$ 
C)  $135 ^ { \circ }$ 
D)  $45 ^ { \circ }$ 
A)  $180 ^ { \circ }$ 
B)  $90 ^ { \circ }$ 
C)  $135 ^ { \circ }$ 
D)  $45 ^ { \circ }$

Question 2

Use an identity to write the expression as a single trigonometric function or as a single number.
-$2 \cos ^ { 2 } 22.5 ^ { \circ } - 1$

Accepted Answer

A)  $\sqrt { 3 }$ 
B)  $\frac { \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 3 } } { 3 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
A)  $\sqrt { 3 }$ 
B)  $\frac { \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 3 } } { 3 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$

Question 3

Give the exact value of the expression.
-$$\csc \left( \sin ^ { - 1 } \frac { 18 } { 30 } \right)$$

Accepted Answer

A)  $\frac { 18 } { 24 }$ 
B)  $\frac { 30 } { 18 }$ 
C)  $\frac { 24 } { 18 }$ 
D)  $\frac { 18 } { 30 }$ 
A)  $\frac { 18 } { 24 }$ 
B)  $\frac { 30 } { 18 }$ 
C)  $\frac { 24 } { 18 }$ 
D)  $\frac { 18 } { 30 }$

Question 4

Use a sum or difference identity to find the exact value.
-$$\frac { \tan \frac { 7 \pi } { 24 } - \tan \frac { \pi } { 8 } } { 1 + \tan \frac { 7 \pi } { 24 } \tan \frac { \pi } { 8 } }$$

Accepted Answer

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

Question 5

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$y = \operatorname { arcsec } ( 2.8842912 )$

Accepted Answer

A)  $6.9714051$ 
B)  $1.5139822$ 
C)  $1.2167397$ 
D)  $0.3540566$ 
A)  $6.9714051$ 
B)  $1.5139822$ 
C)  $1.2167397$ 
D)  $0.3540566$

Question 6

Verify that each equation is an identity.
-$$\sin ( 4 x ) = ( 4 \sin x \cos x ) \left( 2 \cos ^ { 2 } x - 1 \right)$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\sin...

Question 7

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\cos \left( \cos ^ { - 1 } ( - 0.9372 ) \right)$

Accepted Answer

A)  $0 \quad .4686$ 
B)  $- 0.4686$ 
C)  $0.9372$ 
D)  $- 0.9372$ 
A)  $0 \quad .4686$ 
B)  $- 0.4686$ 
C)  $0.9372$ 
D)  $- 0.9372$

Question 8

Use identities to write the expression as a single function of x or $$\theta$$
-$\tan \left( x - 360 ^ { \circ } \right)$

Accepted Answer

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

Question 9

Write in terms of the cofunction of a complementary angle.
-$\tan 49 ^ { \circ }$

Accepted Answer

A)  $\cot 131 ^ { \circ }$ 
B)  $\cot 41 ^ { \circ }$ 
C)  $\cot 139 ^ { \circ }$ 
D)  $\tan 41 ^ { \circ }$ 
A)  $\cot 131 ^ { \circ }$ 
B)  $\cot 41 ^ { \circ }$ 
C)  $\cot 139 ^ { \circ }$ 
D)  $\tan 41 ^ { \circ }$

Question 10

Write in terms of the cofunction of a complementary angle.
-$$\sin \frac { \pi } { 15 }$$

Accepted Answer

A)  $\cos \frac { 13 \pi } { 30 }$ 
B)  $\csc \frac { 13 \pi } { 30 }$ 
C)  $\csc \frac { 14 \pi } { 15 }$ 
D)  $\cos \frac { 14 \pi } { 15 }$ 
A)  $\cos \frac { 13 \pi } { 30 }$ 
B)  $\csc \frac { 13 \pi } { 30 }$ 
C)  $\csc \frac { 14 \pi } { 15 }$ 
D)  $\cos \frac { 14 \pi } { 15 }$

Question 11

Use identities to find the indicated value for each angle measure.
-$\cos \theta = - \frac { 5 } { 13 } , \quad \frac { \pi } { 2 } < \theta < \pi \quad$ Find $\cos ( 2 \theta )$

Accepted Answer

A)  $- \frac { 119 } { 169 }$ 
B)  $\frac { 120 } { 169 }$ 
C)  $- \frac { 120 } { 169 }$ 
D)  $\frac { 119 } { 169 }$ 
A)  $- \frac { 119 } { 169 }$ 
B)  $\frac { 120 } { 169 }$ 
C)  $- \frac { 120 } { 169 }$ 
D)  $\frac { 119 } { 169 }$

Question 12

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$
-$$\sec ^ { 2 } x - 2 = \tan ^ { 2 } x$$

Accepted Answer

A)  $\left\{ \frac { \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { \pi } { 3 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { \pi } { 4 } \right\}$ 
A)  $\left\{ \frac { \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { \pi } { 3 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { \pi } { 4 } \right\}$

Question 13

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$\sin 4 x = \frac { \sqrt { 3 } } { 2 }$

Accepted Answer

A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$ 
A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$

Question 14

Verify that each equation is an identity.
-$$5 \csc ^ { 2 } \theta - 2 \cot ^ { 2 } \theta = 3 \csc ^ { 2 } \theta + 2$$

Accepted Answer

\[\begin{array} { l } 
5 \csc ^ { 2 } \t

Question 15

Give the exact value of the expression.
-$\sin ( \arctan 2 )$

Accepted Answer

A)  $\frac { 5 \sqrt { 2 } } { 2 }$ 
B)  $5 \sqrt { 2 }$ 
C)  $2 \sqrt { 5 }$ 
D)  $\frac { 2 \sqrt { 5 } } { 5 }$ 
A)  $\frac { 5 \sqrt { 2 } } { 2 }$ 
B)  $5 \sqrt { 2 }$ 
C)  $2 \sqrt { 5 }$ 
D)  $\frac { 2 \sqrt { 5 } } { 5 }$

Question 16

Provide an appropriate response.
-Explain what is wrong with the following solution for the equation $\cos 2 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$.
$$\begin{array} { l } 
\cos 2 \theta = \sqrt { 3 } \
\cos \theta = \frac { \sqrt { 3 } } { 2 } \
\theta = \frac { \pi } { 6 } \text { or } \theta = \frac { 11 \pi } { 6 }
\end{array}$$

Accepted Answer

Answers will vary. @#LAT-DLM&

Question 17

Perform the indicated operations and simplify the result so there are no quotients.
-$\tan \theta ( \cot \theta - \cos \theta )$

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

A)  $\frac { \sqrt { 3 } + 4 } { 2 }$ 
B)  $\sqrt { 3 } - 2$ 
C)  $\frac { \sqrt { 3 } - 4 } { 2 }$ 
D)  $- \sqrt { 3 } + 2$ 
A)  $\frac { \sqrt { 3 } + 4 } { 2 }$ 
B)  $\sqrt { 3 } - 2$ 
C)  $\frac { \sqrt { 3 } - 4 } { 2 }$ 
D)  $- \sqrt { 3 } + 2$

Question 20

Solve the equation in the interval $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$ ). Give solutions to the nearest tenth, if necessary.
-$\csc \theta = 1 + \cot \theta$

Accepted Answer

A)  $\left\{ 90 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 90 ^ { \circ } \right\}$ 
C)  $\left\{ 270 ^ { \circ } \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ 90 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 90 ^ { \circ } \right\}$ 
C)  $\left\{ 270 ^ { \circ } \right\}$ 
D)  $\varnothing$

Give the degree measure of $\theta$ - $\theta = \cot ^ { - 1 } ( - 1 )$

Use an identity to write the expression as a single trigonometric function or as a single number. - $2 \cos ^ { 2 } 22.5 ^ { \circ } - 1$

Give the exact value of the expression. - $\csc \left( \sin ^ { - 1 } \frac { 18 } { 30 } \right)$

Use a sum or difference identity to find the exact value. - $\frac { \tan \frac { 7 \pi } { 24 } - \tan \frac { \pi } { 8 } } { 1 + \tan \frac { 7 \pi } { 24 } \tan \frac { \pi } { 8 } }$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \operatorname { arcsec } ( 2.8842912 )$

Verify that each equation is an identity. - $\sin ( 4 x ) = ( 4 \sin x \cos x ) \left( 2 \cos ^ { 2 } x - 1 \right)$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\cos \left( \cos ^ { - 1 } ( - 0.9372 ) \right)$

Use identities to write the expression as a single function of x or $\theta$ - $\tan \left( x - 360 ^ { \circ } \right)$

Write in terms of the cofunction of a complementary angle. - $\tan 49 ^ { \circ }$

Write in terms of the cofunction of a complementary angle. - $\sin \frac { \pi } { 15 }$

Use identities to find the indicated value for each angle measure. - $\cos \theta = - \frac { 5 } { 13 } , \quad \frac { \pi } { 2 } < \theta < \pi \quad$ Find $\cos ( 2 \theta )$

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\sec ^ { 2 } x - 2 = \tan ^ { 2 } x$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\sin 4 x = \frac { \sqrt { 3 } } { 2 }$

Verify that each equation is an identity. - $5 \csc ^ { 2 } \theta - 2 \cot ^ { 2 } \theta = 3 \csc ^ { 2 } \theta + 2$

Give the exact value of the expression. - $\sin ( \arctan 2 )$

Provide an appropriate response. -Explain what is wrong with the following solution for the equation $\cos 2 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$ . 2\theta= \theta= \theta= or \theta=

Perform the indicated operations and simplify the result so there are no quotients. - $\tan \theta ( \cot \theta - \cos \theta )$

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

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

Solve the equation in the interval $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Give solutions to the nearest tenth, if necessary. - $\csc \theta = 1 + \cot \theta$

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

Give the degree measure of θ\thetaθ - θ=cot⁡−1(−1)\theta = \cot ^ { - 1 } ( - 1 )θ=cot−1(−1)

Use an identity to write the expression as a single trigonometric function or as a single number. - 2cos⁡222.5∘−12 \cos ^ { 2 } 22.5 ^ { \circ } - 12cos222.5∘−1

Give the exact value of the expression. - csc⁡(sin⁡−11830)\csc \left( \sin ^ { - 1 } \frac { 18 } { 30 } \right)csc(sin−13018​)

Use a sum or difference identity to find the exact value. - tan⁡7π24−tan⁡π81+tan⁡7π24tan⁡π8\frac { \tan \frac { 7 \pi } { 24 } - \tan \frac { \pi } { 8 } } { 1 + \tan \frac { 7 \pi } { 24 } \tan \frac { \pi } { 8 } }1+tan247π​tan8π​tan247π​−tan8π​​

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=arcsec⁡(2.8842912)y = \operatorname { arcsec } ( 2.8842912 )y=arcsec(2.8842912)

Verify that each equation is an identity. - sin⁡(4x)=(4sin⁡xcos⁡x)(2cos⁡2x−1)\sin ( 4 x ) = ( 4 \sin x \cos x ) \left( 2 \cos ^ { 2 } x - 1 \right)sin(4x)=(4sinxcosx)(2cos2x−1)

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - cos⁡(cos⁡−1(−0.9372))\cos \left( \cos ^ { - 1 } ( - 0.9372 ) \right)cos(cos−1(−0.9372))

Use identities to write the expression as a single function of x or θ\thetaθ - tan⁡(x−360∘)\tan \left( x - 360 ^ { \circ } \right)tan(x−360∘)

Write in terms of the cofunction of a complementary angle. - tan⁡49∘\tan 49 ^ { \circ }tan49∘

Write in terms of the cofunction of a complementary angle. - sin⁡π15\sin \frac { \pi } { 15 }sin15π​

Use identities to find the indicated value for each angle measure. - cos⁡θ=−513,π2<θ<π\cos \theta = - \frac { 5 } { 13 } , \quad \frac { \pi } { 2 } < \theta < \pi \quadcosθ=−135​,2π​<θ<π Find cos⁡(2θ)\cos ( 2 \theta )cos(2θ)

Solve the equation for exact solutions over the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - sec⁡2x−2=tan⁡2x\sec ^ { 2 } x - 2 = \tan ^ { 2 } xsec2x−2=tan2x

Solve the equation for solutions in the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - sin⁡4x=32\sin 4 x = \frac { \sqrt { 3 } } { 2 }sin4x=23​​

Verify that each equation is an identity. - 5csc⁡2θ−2cot⁡2θ=3csc⁡2θ+25 \csc ^ { 2 } \theta - 2 \cot ^ { 2 } \theta = 3 \csc ^ { 2 } \theta + 25csc2θ−2cot2θ=3csc2θ+2

Give the exact value of the expression. - sin⁡(arctan⁡2)\sin ( \arctan 2 )sin(arctan2)

Provide an appropriate response. -Explain what is wrong with the following solution for the equation cos⁡2θ=3\cos 2 \theta = \sqrt { 3 }cos2θ=3​ in the interval [0,2π)[ 0,2 \pi )[0,2π) . 2\theta= \theta= \theta= or \theta=

Perform the indicated operations and simplify the result so there are no quotients. - tan⁡θ(cot⁡θ−cos⁡θ)\tan \theta ( \cot \theta - \cos \theta )tanθ(cotθ−cosθ)

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - (csc⁡θ+cot⁡θ)(1−cos⁡θ)=sin⁡θ( \csc \theta + \cot \theta ) ( 1 - \cos \theta ) = \sin \theta(cscθ+cotθ)(1−cosθ)=sinθ

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

Solve the equation in the interval [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) ). Give solutions to the nearest tenth, if necessary. - csc⁡θ=1+cot⁡θ\csc \theta = 1 + \cot \thetacscθ=1+cotθ