Question 1

Use the fundamental identities to find the value of the trigonometric function.
-$$\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \theta$$

Accepted Answer

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

Question 2

Use the cofunction identities to find an angle that makes the statement true.
-$\cot \theta \sec \theta \sin \theta$

Accepted Answer

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

Question 3

Verify that the equation is an identity.
-$$\cos \left( \frac { 3 \pi } { 2 } - x \right) = - \sin x$$

Accepted Answer

The answer of Verify that the equation is an identity.
-\[\cos...

Question 4

Use a sum or difference identity to find the exact value.
-$\csc \theta ( \sin \theta + \cos \theta )$

Accepted Answer

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

Question 5

Use a sum or difference identity to find the exact value.
-$$\cos 9 ^ { \circ } - \cos 33 ^ { \circ }$$

Accepted Answer

A)  $- 2 \sin 21 ^ { \circ } \cos \left( - 12 ^ { \circ } \right)$ 
B)  $- 2 \cos 21 ^ { \circ } \sin \left( - 12 ^ { \circ } \right)$ 
C)  $- 2 \cos 21 ^ { \circ } \cos \left( - 12 ^ { \circ } \right)$ 
D)  $- 2 \sin 21 ^ { \circ } \sin \left( - 12 ^ { \circ } \right)$ 
A)  $- 2 \sin 21 ^ { \circ } \cos \left( - 12 ^ { \circ } \right)$ 
B)  $- 2 \cos 21 ^ { \circ } \sin \left( - 12 ^ { \circ } \right)$ 
C)  $- 2 \cos 21 ^ { \circ } \cos \left( - 12 ^ { \circ } \right)$ 
D)  $- 2 \sin 21 ^ { \circ } \sin \left( - 12 ^ { \circ } \right)$

Question 6

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 12 } { 13 }$, with $\mathrm { s }$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$, with $\mathrm { t }$ in quadrant II.

Accepted Answer

A)  $- \frac { 220 } { 221 }$ 
B)  $\frac { 140 } { 221 }$ 
C)  $\frac { 220 } { 221 }$ 
D)  $- \frac { 140 } { 221 }$ 
A)  $- \frac { 220 } { 221 }$ 
B)  $\frac { 140 } { 221 }$ 
C)  $\frac { 220 } { 221 }$ 
D)  $- \frac { 140 } { 221 }$

Question 7

Tell whether the statement is true or false.
-$$\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }$$

Accepted Answer

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

Question 8

Find the exact value of the expression using the provided information.
-$$\frac { \tan \frac { \pi } { 15 } + \tan \frac { 4 \pi } { 15 } } { 1 - \tan \frac { \pi } { 15 } \tan \frac { 4 \pi } { 15 } }$$

Accepted Answer

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

Question 9

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { \csc \theta \cot \theta } { \sec \theta }$$

Accepted Answer

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

Question 10

Use a sum or difference identity to find the exact value.
-$\frac { \sin ^ { 2 } x - 1 } { \cos ( - x ) }$

Accepted Answer

A)  $- \cos x$ 
B)  $\sin x$ 
C)  $\cos x$ 
D)  $- \sin x$ 
A)  $- \cos x$ 
B)  $\sin x$ 
C)  $\cos x$ 
D)  $- \sin x$

Question 11

Solve the equation for exact solutions.
-$$\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }$$

Accepted Answer

The answer of Solve the equation for exact solutions.
-\[\cot (...

Question 12

Complete the sentence so the result is an identity. Let x be any real number.
-$2 \sin 8 x \sin 17 x$

Accepted Answer

A)  $\sin 25 x + \sin 9 x$ 
B)  $\frac { 1 } { 2 } ( \cos 25 x + \cos 9 x )$ 
C)  $\cos 25 x + \cos 9 x$ 
D)  $\cos 9 x - \cos 25 x$ 
A)  $\sin 25 x + \sin 9 x$ 
B)  $\frac { 1 } { 2 } ( \cos 25 x + \cos 9 x )$ 
C)  $\cos 25 x + \cos 9 x$ 
D)  $\cos 9 x - \cos 25 x$

Question 13

Use the fundamental identities to simplify the expression.
-$$\cos x = ( \cot x ) ( - )$$

Accepted Answer

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

Question 14

Use a sum or difference identity to find the exact value.
-$\tan 75 ^ { \circ }$

Accepted Answer

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

Question 15

Use the fundamental identities to simplify the expression.
-8 sin 57° cos 101°

Accepted Answer

A) 8[cos 158° + cos(-44°)] 
B) 4[cos 158° + cos(-44°)] 
C) 4[sin 158° + sin(-44°)] 
D) 4[sin 158° - sin(-44°)] 
A) 8[cos 158° + cos(-44°)] 
B) 4[cos 158° + cos(-44°)] 
C) 4[sin 158° + sin(-44°)] 
D) 4[sin 158° - sin(-44°)]

Question 16

Use the fundamental identities to simplify the expression.
-$\tan \frac { 11 \pi } { 12 }$

Accepted Answer

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

Question 17

Solve the problem.
-$\tan ^ { 4 } x - \sec ^ { 4 } x$

Accepted Answer

A)  $\sec ^ { 2 } x + \tan ^ { 2 } x$ 
B)  $\tan ^ { 2 } - \sec ^ { 2 } x$ 
C)  $\sec ^ { 2 } x$ 
D)  $- 2 \tan ^ { 2 } x - 1$ 
A)  $\sec ^ { 2 } x + \tan ^ { 2 } x$ 
B)  $\tan ^ { 2 } - \sec ^ { 2 } x$ 
C)  $\sec ^ { 2 } x$ 
D)  $- 2 \tan ^ { 2 } x - 1$

Question 18

Use Identities to find the exact value.
-$\tan 75 ^ { \circ }$

Accepted Answer

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

Question 19

Use a sum or difference identity to find the exact value.
-$$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta }$$

Accepted Answer

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

Question 20

Write in terms of the cofunction of a complementary angle.
-sin 75°

Accepted Answer

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

Use the fundamental identities to find the value of the trigonometric function. - $\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \theta$

Use the cofunction identities to find an angle that makes the statement true. - $\cot \theta \sec \theta \sin \theta$

Verify that the equation is an identity. - $\cos \left( \frac { 3 \pi } { 2 } - x \right) = - \sin x$

Use a sum or difference identity to find the exact value. - $\csc \theta ( \sin \theta + \cos \theta )$

Use a sum or difference identity to find the exact value. - $\cos 9 ^ { \circ } - \cos 33 ^ { \circ }$

Tell whether the statement is true or false. - $\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }$

Find the exact value of the expression using the provided information. - $\frac { \tan \frac { \pi } { 15 } + \tan \frac { 4 \pi } { 15 } } { 1 - \tan \frac { \pi } { 15 } \tan \frac { 4 \pi } { 15 } }$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { \csc \theta \cot \theta } { \sec \theta }$

Use a sum or difference identity to find the exact value. - $\frac { \sin ^ { 2 } x - 1 } { \cos ( - x ) }$

Solve the equation for exact solutions. - $\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }$

Complete the sentence so the result is an identity. Let x be any real number. - $2 \sin 8 x \sin 17 x$

Use the fundamental identities to simplify the expression. - $\cos x = ( \cot x ) ( - )$

Use a sum or difference identity to find the exact value. - $\tan 75 ^ { \circ }$

Use the fundamental identities to simplify the expression. -8 sin 57° cos 101°

Use the fundamental identities to simplify the expression. - $\tan \frac { 11 \pi } { 12 }$

Solve the problem. - $\tan ^ { 4 } x - \sec ^ { 4 } x$

Use Identities to find the exact value. - $\tan 75 ^ { \circ }$

Use a sum or difference identity to find the exact value. - $\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta }$

Write in terms of the cofunction of a complementary angle. -sin 75°

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

Use the fundamental identities to find the value of the trigonometric function. - sin⁡θ+1cos⁡θ+cot⁡θ=tan⁡θ\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \thetacosθ+cotθsinθ+1​=tanθ

Use the cofunction identities to find an angle that makes the statement true. - cot⁡θsec⁡θsin⁡θ\cot \theta \sec \theta \sin \thetacotθsecθsinθ

Verify that the equation is an identity. - cos⁡(3π2−x)=−sin⁡x\cos \left( \frac { 3 \pi } { 2 } - x \right) = - \sin xcos(23π​−x)=−sinx

Use a sum or difference identity to find the exact value. - csc⁡θ(sin⁡θ+cos⁡θ)\csc \theta ( \sin \theta + \cos \theta )cscθ(sinθ+cosθ)

Use a sum or difference identity to find the exact value. - cos⁡9∘−cos⁡33∘\cos 9 ^ { \circ } - \cos 33 ^ { \circ }cos9∘−cos33∘

Tell whether the statement is true or false. - 1cos⁡2θ−1cot⁡2θ\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }cos2θ1​−cot2θ1​

Find the exact value of the expression using the provided information. - tan⁡π15+tan⁡4π151−tan⁡π15tan⁡4π15\frac { \tan \frac { \pi } { 15 } + \tan \frac { 4 \pi } { 15 } } { 1 - \tan \frac { \pi } { 15 } \tan \frac { 4 \pi } { 15 } }1−tan15π​tan154π​tan15π​+tan154π​​

Perform the indicated operations and simplify the result so there are no quotients. - csc⁡θcot⁡θsec⁡θ\frac { \csc \theta \cot \theta } { \sec \theta }secθcscθcotθ​

Use a sum or difference identity to find the exact value. - sin⁡2x−1cos⁡(−x)\frac { \sin ^ { 2 } x - 1 } { \cos ( - x ) }cos(−x)sin2x−1​

Solve the equation for exact solutions. - cot⁡(2θ)=csc⁡2θ−22cot⁡θ\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }cot(2θ)=2cotθcsc2θ−2​

Complete the sentence so the result is an identity. Let x be any real number. - 2sin⁡8xsin⁡17x2 \sin 8 x \sin 17 x2sin8xsin17x

Use the fundamental identities to simplify the expression. - cos⁡x=(cot⁡x)(−)\cos x = ( \cot x ) ( - )cosx=(cotx)(−)

Use a sum or difference identity to find the exact value. - tan⁡75∘\tan 75 ^ { \circ }tan75∘