Question 1

Graph the inverse circular function.
-$y=\sin ^{-1} x$

Accepted Answer

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

Question 2

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

Accepted Answer

A)  2 
B)  $\cos ^ { 2 } x \sin ^ { 2 } x$ 
C)  1 
D)  0 
A)  2 
B)  $\cos ^ { 2 } x \sin ^ { 2 } x$ 
C)  1 
D)  0

Question 3

Use a sum or difference identity to find the exact value.
-$\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $\frac { 5 } { 12 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $\frac { 5 } { 12 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 4

Verify that each equation is an identity.
-$$\tan x ( \csc x - \sin x ) = \cos x$$

Accepted Answer

\[\begin{array} { l } 
\tan x ( \csc x -

Question 5

Solve the problem.
-A coil of wire rotating in a magnetic field induces a voltage given by $e = 20 \sin \left( \frac { \pi t } { 4 } - \frac { \pi } { 2 } \right)$, where $t$ is time in seconds. Find the smallest positive time to produce a voltage of $10 \sqrt { 2 }$. Round values to the nearest hundredth.

Accepted Answer

A)  $2.8 \mathrm { sec }$ 
B)  $3 \pi \mathrm { sec }$ 
C)  $2.8 \pi \mathrm { sec }$ 
D)  $3 \mathrm { sec }$ 
A)  $2.8 \mathrm { sec }$ 
B)  $3 \pi \mathrm { sec }$ 
C)  $2.8 \pi \mathrm { sec }$ 
D)  $3 \mathrm { sec }$

Question 6

Solve the equation for exact solutions over the interval [0, 2π).
-$\sin ^ { 2 } x + \sin x = 0$

Accepted Answer

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

Question 7

Find the exact value of the real number y.
-$$y = \sin ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$$

Accepted Answer

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

Question 8

Rewrite the following as a product of trigonometric functions.
-$\sin 9 + \sin 32$

Accepted Answer

A)  $2 \sin 20.5 \sin ( - 11.5 )$ 
B)  $2 \cos 20.5 \sin ( - 11.5 )$ 
C)  $2 \cos 20.5 \cos ( - 11.5 )$ 
D)  $2 \sin 20.5 \cos ( - 11.5 )$ 
A)  $2 \sin 20.5 \sin ( - 11.5 )$ 
B)  $2 \cos 20.5 \sin ( - 11.5 )$ 
C)  $2 \cos 20.5 \cos ( - 11.5 )$ 
D)  $2 \sin 20.5 \cos ( - 11.5 )$

Question 9

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

Accepted Answer

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

Question 10

Perform the transformation.
-Write $\sin x$ in terms of $\cos x$.

Accepted Answer

A)  $\frac { \pm \sqrt { 1 - \cos ^ { 2 } x } } { \cos x }$ 
B)  $\pm \sqrt { 1 - \cos ^ { 2 } x }$ 
C)  $\frac { \pm \sqrt { 1 - \cos ^ { 2 } x } } { 1 - \cos ^ { 2 } x }$ 
D)  $\frac { \pm \cos x \sqrt { 1 - \cos ^ { 2 } x } } { 1 - \cos ^ { 2 } x }$ 
A)  $\frac { \pm \sqrt { 1 - \cos ^ { 2 } x } } { \cos x }$ 
B)  $\pm \sqrt { 1 - \cos ^ { 2 } x }$ 
C)  $\frac { \pm \sqrt { 1 - \cos ^ { 2 } x } } { 1 - \cos ^ { 2 } x }$ 
D)  $\frac { \pm \cos x \sqrt { 1 - \cos ^ { 2 } x } } { 1 - \cos ^ { 2 } x }$

Question 11

Solve the problem.
-In an electrical circuit, let $\mathrm { V }$ represent the electromagnetic force in volts at $\mathrm { t }$ seconds. Assume $\mathrm { V } = \cos 2 \pi \mathrm { t }$. Find the smallest positive value of $\mathrm { t }$ where $0 \leq \mathrm { t } \leq \frac { 1 } { 2 }$ for $\mathrm { V } = \frac { \sqrt { 2 } } { 2 }$.

Accepted Answer

A)  $\frac { 1 } { 8 } \mathrm { sec }$ 
B)  $\frac { 1 } { 2 } \mathrm { sec }$ 
C)  $0 \mathrm { sec }$ 
D)  $\frac { 1 } { 4 } \mathrm { sec }$ 
A)  $\frac { 1 } { 8 } \mathrm { sec }$ 
B)  $\frac { 1 } { 2 } \mathrm { sec }$ 
C)  $0 \mathrm { sec }$ 
D)  $\frac { 1 } { 4 } \mathrm { sec }$

Question 12

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 13

Provide an appropriate response.
-True or false? The statement $\sin ^ { - 1 } ( \sin x ) = x$ for all real numbers in the interval $0 \leq x \leq \pi$.

Accepted Answer

A)  True 
B)  False 
A)  True 
B)  False

Question 14

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

Question 15

Find the exact value of the real number y.
-$$y = \arccos \left( \frac { \sqrt { 3 } } { 2 } \right)$$

Accepted Answer

A)  $\frac { 5 \pi } { 6 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $- \frac { \pi } { 3 }$ 
A)  $\frac { 5 \pi } { 6 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $- \frac { \pi } { 3 }$

Question 16

Find the exact value of the expression using the provided information.
-Find $\cos ( s - t )$ given that $\cos s = - \frac { 4 } { 5 }$, with $s$ in quadrant II, and $\cos t = \frac { 12 } { 13 }$, with $t$ in quadrant IV.

Accepted Answer

A)  $- \frac { 33 } { 65 }$ 
B)  $- \frac { 63 } { 65 }$ 
C)  $\frac { 63 } { 65 }$ 
D)  $\frac { 33 } { 65 }$ 
A)  $- \frac { 33 } { 65 }$ 
B)  $- \frac { 63 } { 65 }$ 
C)  $\frac { 63 } { 65 }$ 
D)  $\frac { 33 } { 65 }$

Question 17

Use a calculator to give the value to the nearest degree.
-$$\theta = \tan ^ { - 1 } ( - 0.7002 )$$

Accepted Answer

A)  $- 38 ^ { \circ }$ 
B)  $- 33 ^ { \circ }$ 
C)  $- 35 ^ { \circ }$ 
D)  $145 ^ { \circ }$ 
A)  $- 38 ^ { \circ }$ 
B)  $- 33 ^ { \circ }$ 
C)  $- 35 ^ { \circ }$ 
D)  $145 ^ { \circ }$

Question 18

Use a sum or difference identity to find the exact value.
-$\sin \frac { 11 \pi } { 12 }$

Accepted Answer

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

Question 19

Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth.
-$3 \cos ^ { 2 } \theta + 2 \cos \theta = 1$

Accepted Answer

A)  $\left\{ 70.5 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 180 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 289.5 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
B)  $\left\{ 49.8 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 130.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 229.8 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 310.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
C)  $\left\{ 51.8 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 128.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
D)  $\left\{ 103.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 145.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 283.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 325.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
A)  $\left\{ 70.5 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 180 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 289.5 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
B)  $\left\{ 49.8 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 130.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 229.8 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 310.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
C)  $\left\{ 51.8 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 128.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
D)  $\left\{ 103.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 145.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 283.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 325.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$

Question 20

Use Identities to find the exact value.
-$$\cos \frac { 7 \pi } { 12 } \cos \frac { 5 \pi } { 12 } + \sin \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 }$$

Accepted Answer

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

Graph the inverse circular function. - $y=\sin ^{-1} x$ $Graph the inverse circular function. - y=\sin ^{-1} x$

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

Use a sum or difference identity to find the exact value. - $\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }$

Verify that each equation is an identity. - $\tan x ( \csc x - \sin x ) = \cos x$

Solve the equation for exact solutions over the interval [0, 2π). - $\sin ^ { 2 } x + \sin x = 0$

Find the exact value of the real number y. - $y = \sin ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$

Rewrite the following as a product of trigonometric functions. - $\sin 9 + \sin 32$

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

Perform the transformation. -Write $\sin x$ in terms of $\cos x$ .

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

Provide an appropriate response. -True or false? The statement $\sin ^ { - 1 } ( \sin x ) = x$ for all real numbers in the interval $0 \leq x \leq \pi$ .

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

Find the exact value of the real number y. - $y = \arccos \left( \frac { \sqrt { 3 } } { 2 } \right)$

Find the exact value of the expression using the provided information. -Find $\cos ( s - t )$ given that $\cos s = - \frac { 4 } { 5 }$ , with $s$ in quadrant II, and $\cos t = \frac { 12 } { 13 }$ , with $t$ in quadrant IV.

Use a calculator to give the value to the nearest degree. - $\theta = \tan ^ { - 1 } ( - 0.7002 )$

Use a sum or difference identity to find the exact value. - $\sin \frac { 11 \pi } { 12 }$

Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - $3 \cos ^ { 2 } \theta + 2 \cos \theta = 1$

Use Identities to find the exact value. - $\cos \frac { 7 \pi } { 12 } \cos \frac { 5 \pi } { 12 } + \sin \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 }$

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

Review of Basic Concepts

Filters

Exam 7: Trigonometric Identities and Equations

Graph the inverse circular function. - y=sin⁡−1xy=\sin ^{-1} xy=sin−1x

Use the fundamental identities to simplify the expression. - cos⁡(−x)cos⁡x−sin⁡(−x)sin⁡x\cos ( - x ) \cos x - \sin ( - x ) \sin xcos(−x)cosx−sin(−x)sinx

Use a sum or difference identity to find the exact value. - sin⁡25∘cos⁡35∘+cos⁡25∘sin⁡35∘\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }sin25∘cos35∘+cos25∘sin35∘

Verify that each equation is an identity. - tan⁡x(csc⁡x−sin⁡x)=cos⁡x\tan x ( \csc x - \sin x ) = \cos xtanx(cscx−sinx)=cosx

Solve the equation for exact solutions over the interval [0, 2π). - sin⁡2x+sin⁡x=0\sin ^ { 2 } x + \sin x = 0sin2x+sinx=0

Find the exact value of the real number y. - y=sin⁡−1(32)y = \sin ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)y=sin−1(23​​)

Rewrite the following as a product of trigonometric functions. - sin⁡9+sin⁡32\sin 9 + \sin 32sin9+sin32

Use the cofunction identities to find an angle θ 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 transformation. -Write sin⁡x\sin xsinx in terms of cos⁡x\cos xcosx .

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)

Provide an appropriate response. -True or false? The statement sin⁡−1(sin⁡x)=x\sin ^ { - 1 } ( \sin x ) = xsin−1(sinx)=x for all real numbers in the interval 0≤x≤π0 \leq x \leq \pi0≤x≤π .

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)

Find the exact value of the real number y. - y=arccos⁡(32)y = \arccos \left( \frac { \sqrt { 3 } } { 2 } \right)y=arccos(23​​)

Find the exact value of the expression using the provided information. -Find cos⁡(s−t)\cos ( s - t )cos(s−t) given that cos⁡s=−45\cos s = - \frac { 4 } { 5 }coss=−54​ , with sss in quadrant II, and cos⁡t=1213\cos t = \frac { 12 } { 13 }cost=1312​ , with ttt in quadrant IV.

Use a calculator to give the value to the nearest degree. - θ=tan⁡−1(−0.7002)\theta = \tan ^ { - 1 } ( - 0.7002 )θ=tan−1(−0.7002)

Use a sum or difference identity to find the exact value. - sin⁡11π12\sin \frac { 11 \pi } { 12 }sin1211π​

Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - 3cos⁡2θ+2cos⁡θ=13 \cos ^ { 2 } \theta + 2 \cos \theta = 13cos2θ+2cosθ=1

Use Identities to find the exact value. - cos⁡7π12cos⁡5π12+sin⁡7π12sin⁡5π12\cos \frac { 7 \pi } { 12 } \cos \frac { 5 \pi } { 12 } + \sin \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 }cos127π​cos125π​+sin127π​sin125π​