Question 1

Find the exact value of the expression using the provided information.
-$$( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta$$

Accepted Answer

A)  $2 \cot ^ { 2 } \theta$ 
B)  0 
C)  2 
D)  $- 2 \cot ^ { 2 } \theta$ 
A)  $2 \cot ^ { 2 } \theta$ 
B)  0 
C)  2 
D)  $- 2 \cot ^ { 2 } \theta$

Question 2

Use the fundamental identities to find the value of the trigonometric function.
-tan ϴ = cot (30°+ 5ϴ)

Accepted Answer

A) ϴ = 75° 
B) ϴ = 10° 
C) ϴ = 16° 
D) ϴ = 6° 
A) ϴ = 75° 
B) ϴ = 10° 
C) ϴ = 16° 
D) ϴ = 6°

Question 3

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-$$1 - \sin ^ { 2 } \theta$$

Accepted Answer

A)  $\cos \theta$ 
B)  $- \cos ^ { 2 } \theta$ 
C)  $\cot ^ { 2 } \theta$ 
D)  $\cos ^ { 2 } \theta$ 
A)  $\cos \theta$ 
B)  $- \cos ^ { 2 } \theta$ 
C)  $\cot ^ { 2 } \theta$ 
D)  $\cos ^ { 2 } \theta$

Question 4

Verify that each equation is an identity.
-$$\frac { \sin x + \cos x } { \sin x - \cos x } = \frac { 1 + 2 \sin x \cos x } { 2 \sin ^ { 2 } x - 1 }$$

Accepted Answer

\[\begin{array} { l } 
\frac { \sin x +

Question 5

Use a sum or difference identity to find the exact value.
-cot x cos x + sin x = csc x

Accepted Answer

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

Question 6

Determine all solutions of the equation in radians.
-$\cos \theta = \frac { 20 } { 29 } , \quad \sin \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $- \frac { 41 } { 841 }$ 
B)  $\frac { 840 } { 841 }$ 
C)  -------- 
D)  $- \frac { 840 } { 841 }$ 
A)  $- \frac { 41 } { 841 }$ 
B)  $\frac { 840 } { 841 }$ 
C)  -------- 
D)  $- \frac { 840 } { 841 }$

Question 7

Use the fundamental identities to simplify the expression.
-$$1 - \ldots = \sin ^ { 2 } x$$

Accepted Answer

A)  $\sec ^ { 2 } x$ 
B)  $\cot ^ { 2 } x$ 
C)  $\tan ^ { 2 } x$ 
D)  $\cos ^ { 2 } x$ 
A)  $\sec ^ { 2 } x$ 
B)  $\cot ^ { 2 } x$ 
C)  $\tan ^ { 2 } x$ 
D)  $\cos ^ { 2 } x$

Question 8

Write in terms of the cofunction of a complementary angle.
-$$\frac { \tan \frac { 7 \pi } { 24 } - \tan \frac { \pi } { 8 } } { 1 + \tan \frac { 7 \pi } { 24 } \tan \frac { \pi } { 8 } }$$

Accepted Answer

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

Question 9

Solve the problem.
-$$\sin x = ( - ) ( \cos x )$$

Accepted Answer

A)  $\cot x$ 
B)  $\csc x$ 
C)  $\sec x$ 
D)  $\tan x$ 
A)  $\cot x$ 
B)  $\csc x$ 
C)  $\sec x$ 
D)  $\tan x$

Question 10

Use Identities to find the exact value.
-$$( \cos \theta + \cot \theta ) ( 1 - \csc \theta ) = \cot ^ { 2 } \theta$$

Accepted Answer

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

Question 11

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-cos(x + y)= cos x + cos y

Accepted Answer

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

Question 12

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\frac { \sin x + \csc x } { \sin x - \csc x } = - 2 \tan ^ { 2 } x$$

Accepted Answer

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

Question 13

Rewrite the following as a product of trigonometric functions.
-Find $\cot \theta$ if $\sec \theta = - \frac { \sqrt { 29 } } { 5 }$ and $\theta$ is in quadrant II.

Accepted Answer

A)  $- \frac { \sqrt { 29 } } { 2 }$ 
B)  $\frac { \sqrt { 29 } } { 2 }$ 
C)  $- \frac { 5 } { 2 }$ 
D)  $\frac { 2 } { 5 }$ 
A)  $- \frac { \sqrt { 29 } } { 2 }$ 
B)  $\frac { \sqrt { 29 } } { 2 }$ 
C)  $- \frac { 5 } { 2 }$ 
D)  $\frac { 2 } { 5 }$

Question 14

Find the exact value of the expression using the provided information.
-Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { \sqrt { 3 } } { 3 }$, with $\mathrm { s }$ in quadrant $\mathrm { IV }$, and $\sin \mathrm { t } = - \frac { \sqrt { 5 } } { 6 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$ 
B)  $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$ 
C)  $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$ 
D)  $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$ 
A)  $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$ 
B)  $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$ 
C)  $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$ 
D)  $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$

Question 15

Use Identities 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 16

Using a sum or difference identity, write the following as an expression involving functions of x.
-$\tan \left( x + \frac { \pi } { 6 } \right)$

Accepted Answer

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

Question 17

Perform the indicated operations and simplify the result so there are no quotients.
-$\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta$

Accepted Answer

A)  $\sec \theta \csc \theta$ 
B)  $1 + \cot \theta$ 
C)  $- 2 \tan ^ { 2 } \theta$ 
D)  $\sin \theta \tan \theta$ 
A)  $\sec \theta \csc \theta$ 
B)  $1 + \cot \theta$ 
C)  $- 2 \tan ^ { 2 } \theta$ 
D)  $\sin \theta \tan \theta$

Question 18

Use the fundamental identities to simplify the expression.
-$\cos \theta - \cos \theta \sin ^ { 2 } \theta$

Accepted Answer

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

Question 19

Use a sum or difference identity to find the exact value.
-Find $\sin ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = \frac { 1 } { 3 }$, with $\mathrm { s }$ in quadrant $\mathrm { I }$, and $\sin t = - \frac { 1 } { 2 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $\frac { \sqrt { 3 } - 2 \sqrt { 2 } } { 6 }$ 
B)  $\frac { 2 \sqrt { 6 } + 1 } { 6 }$ 
C)  $\frac { \sqrt { 3 } + 2 \sqrt { 2 } } { 6 }$ 
D)  $\frac { 2 \sqrt { 6 } - 1 } { 6 }$ 
A)  $\frac { \sqrt { 3 } - 2 \sqrt { 2 } } { 6 }$ 
B)  $\frac { 2 \sqrt { 6 } + 1 } { 6 }$ 
C)  $\frac { \sqrt { 3 } + 2 \sqrt { 2 } } { 6 }$ 
D)  $\frac { 2 \sqrt { 6 } - 1 } { 6 }$

Question 20

Use identities to find the indicated value for each angle measure.
-$\sin \theta = - \frac { 4 } { 5 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\cos ( 2 \theta )$.

Accepted Answer

A)  $\frac { 24 } { 25 }$ 
B)  $- \frac { 24 } { 25 }$ 
C)  $\frac { 7 } { 25 }$ 
D)  $- \frac { 7 } { 25 }$ 
A)  $\frac { 24 } { 25 }$ 
B)  $- \frac { 24 } { 25 }$ 
C)  $\frac { 7 } { 25 }$ 
D)  $- \frac { 7 } { 25 }$

Find the exact value of the expression using the provided information. - $( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta$

Use the fundamental identities to find the value of the trigonometric function. -tan ϴ = cot (30°+ 5ϴ)

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. - $1 - \sin ^ { 2 } \theta$

Verify that each equation is an identity. - $\frac { \sin x + \cos x } { \sin x - \cos x } = \frac { 1 + 2 \sin x \cos x } { 2 \sin ^ { 2 } x - 1 }$

Use a sum or difference identity to find the exact value. -cot x cos x + sin x = csc x

Determine all solutions of the equation in radians. - $\cos \theta = \frac { 20 } { 29 } , \quad \sin \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Use the fundamental identities to simplify the expression. - $1 - \ldots = \sin ^ { 2 } x$

Write in terms of the cofunction of a complementary angle. - $\frac { \tan \frac { 7 \pi } { 24 } - \tan \frac { \pi } { 8 } } { 1 + \tan \frac { 7 \pi } { 24 } \tan \frac { \pi } { 8 } }$

Solve the problem. - $\sin x = ( - ) ( \cos x )$

Use Identities to find the exact value. - $( \cos \theta + \cot \theta ) ( 1 - \csc \theta ) = \cot ^ { 2 } \theta$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. -cos(x + y)= cos x + cos y

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { \sin x + \csc x } { \sin x - \csc x } = - 2 \tan ^ { 2 } x$

Rewrite the following as a product of trigonometric functions. -Find $\cot \theta$ if $\sec \theta = - \frac { \sqrt { 29 } } { 5 }$ and $\theta$ is in quadrant II.

Use Identities to find the exact value. - $\tan \left( - 15 ^ { \circ } \right)$

Using a sum or difference identity, write the following as an expression involving functions of x. - $\tan \left( x + \frac { \pi } { 6 } \right)$

Perform the indicated operations and simplify the result so there are no quotients. - $\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta$

Use the fundamental identities to simplify the expression. - $\cos \theta - \cos \theta \sin ^ { 2 } \theta$

Use a sum or difference identity to find the exact value. -Find $\sin ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = \frac { 1 } { 3 }$ , with $\mathrm { s }$ in quadrant $\mathrm { I }$ , and $\sin t = - \frac { 1 } { 2 }$ , with $\mathrm { t }$ in quadrant IV.

Use identities to find the indicated value for each angle measure. - $\sin \theta = - \frac { 4 } { 5 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\cos ( 2 \theta )$ .

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Inverse Circular Functions and Trigonometric Equations

Applications of Trigonometry and Vectors

Complex Numbers, Polar Equations, and Parametric Equations

Filters

Exam 5: Trigonometric Identities

Find the exact value of the expression using the provided information. - (1+cot⁡θ)(1−cot⁡θ)−csc⁡2θ( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta(1+cotθ)(1−cotθ)−csc2θ

Use the fundamental identities to find the value of the trigonometric function. -tan ϴ = cot (30°+ 5ϴ)

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. - 1−sin⁡2θ1 - \sin ^ { 2 } \theta1−sin2θ

Verify that each equation is an identity. - sin⁡x+cos⁡xsin⁡x−cos⁡x=1+2sin⁡xcos⁡x2sin⁡2x−1\frac { \sin x + \cos x } { \sin x - \cos x } = \frac { 1 + 2 \sin x \cos x } { 2 \sin ^ { 2 } x - 1 }sinx−cosxsinx+cosx​=2sin2x−11+2sinxcosx​

Use a sum or difference identity to find the exact value. -cot x cos x + sin x = csc x

Determine all solutions of the equation in radians. - cos⁡θ=2029,sin⁡θ<0\cos \theta = \frac { 20 } { 29 } , \quad \sin \theta < 0 \quadcosθ=2920​,sinθ<0 Find sin⁡(2θ)\sin ( 2 \theta )sin(2θ)

Use the fundamental identities to simplify the expression. - 1−…=sin⁡2x1 - \ldots = \sin ^ { 2 } x1−…=sin2x

Write in terms of the cofunction of a complementary angle. - 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​​

Solve the problem. - sin⁡x=(−)(cos⁡x)\sin x = ( - ) ( \cos x )sinx=(−)(cosx)

Use Identities to find the exact value. - (cos⁡θ+cot⁡θ)(1−csc⁡θ)=cot⁡2θ( \cos \theta + \cot \theta ) ( 1 - \csc \theta ) = \cot ^ { 2 } \theta(cosθ+cotθ)(1−cscθ)=cot2θ