Question 1

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$\sin 2 x = 2 \sin x \cos x$

Accepted Answer

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

Question 2

Use identities to find the indicated value for each angle measure.
-$\sin \theta = \frac { 7 } { 25 } , \quad \cos \theta > 0 \quad$ Find $\cos ( 2 \theta )$

Accepted Answer

A)  $\frac { 527 } { 625 }$ 
B)  $\frac { 528 } { 625 }$ 
C)  $- \frac { 527 } { 625 }$ 
D)  $\frac { 336 } { 625 }$ 
A)  $\frac { 527 } { 625 }$ 
B)  $\frac { 528 } { 625 }$ 
C)  $- \frac { 527 } { 625 }$ 
D)  $\frac { 336 } { 625 }$

Question 3

Solve the problem.
-Suppose that a household appliance draws a current represented by the relation $\mathrm { I } ( t ) = 7 \cos ( 120 \pi t )$, where $t$ is time measured in seconds. The power consumed by the appliance is $P = I ^ { 2 } R$, where $R$ is a constant. Take $R$ to be 16 and graph the power in [0, 0.04, 0.01] by $[ - 200,2000,200 ]$ and use an identity to write the expression for the power in the form $\mathrm { P } = \mathrm { a } \cos ( \mathrm { k } \pi \mathrm { t } ) + \mathrm { d }$, where $\mathrm { a }$, $\mathrm { k }$, and $\mathrm { d }$ are constants.

Accepted Answer

A)  $\mathrm { P } = - 392 \cos ( 240 \pi \mathrm { t } ) + 392$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
B)  $\mathrm { P } = 392 \cos ( 120 \pi \mathrm { t } ) + 392$
 $[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
C)  $\mathrm { P } = 392 \cos ( 240 \pi \mathrm { t } ) + 392$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
D)  $P = 784 \cos ( 240 \pi t ) + 784$
 $[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
A)  $\mathrm { P } = - 392 \cos ( 240 \pi \mathrm { t } ) + 392$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
B)  $\mathrm { P } = 392 \cos ( 120 \pi \mathrm { t } ) + 392$
 $[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
C)  $\mathrm { P } = 392 \cos ( 240 \pi \mathrm { t } ) + 392$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
D)  $P = 784 \cos ( 240 \pi t ) + 784$
 $[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$

Question 4

Solve the problem.
-The power dissipated in an electric circuit is given by the expression $\mathrm { P } = \mathrm { RI } ^ { 2 }$, where $\mathrm { R }$ is the resistance of the circuit and $\mathrm { I }$ is the current through the circuit. For a sinusoidal alternating current, the current might be represented by the relation $\mathrm { I } = \mathrm { A } \sin ( 2 \pi \mathrm { ft } )$, where $\mathrm { A }$ is the amplitude, $\mathrm { f }$ is the frequency, and $t$ is time. Write an expression for P involving the sine function, and use a fundamental identity to write $P$ in terms of the cosine function.

Accepted Answer

A)  $P = R A \sin ^ { 2 } ( 2 \pi f \mathrm { t } ) ; \mathrm { P } = \mathrm { RA } - \mathrm { RA } \cos ^ { 2 } ( 2 \pi \mathrm { ft } )$ 
B)  $P = R A ^ { 2 } \sin ^ { 2 } ( 2 \pi f t ) ; P = - R A ^ { 2 } \cos ^ { 2 } ( 2 \pi f t )$ 
C)  $P = R A \sin ^ { 2 } ( 2 \pi f \mathrm { ft } ) ; \mathrm { P } = \mathrm { RA } - \cos ^ { 2 } ( 2 \pi \mathrm { ft } )$ 
D)  $P = R A ^ { 2 } \sin ^ { 2 } ( 2 \pi f t ) ; P = R A ^ { 2 } - R A ^ { 2 } \cos ^ { 2 } ( 2 \pi f t )$ 
A)  $P = R A \sin ^ { 2 } ( 2 \pi f \mathrm { t } ) ; \mathrm { P } = \mathrm { RA } - \mathrm { RA } \cos ^ { 2 } ( 2 \pi \mathrm { ft } )$ 
B)  $P = R A ^ { 2 } \sin ^ { 2 } ( 2 \pi f t ) ; P = - R A ^ { 2 } \cos ^ { 2 } ( 2 \pi f t )$ 
C)  $P = R A \sin ^ { 2 } ( 2 \pi f \mathrm { ft } ) ; \mathrm { P } = \mathrm { RA } - \cos ^ { 2 } ( 2 \pi \mathrm { ft } )$ 
D)  $P = R A ^ { 2 } \sin ^ { 2 } ( 2 \pi f t ) ; P = R A ^ { 2 } - R A ^ { 2 } \cos ^ { 2 } ( 2 \pi f t )$

Question 5

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
-$2 \cos 2 \theta + 7 \sin \theta = 5$

Accepted Answer

A)  $\left\{ 30 ^ { \circ } , 330 ^ { \circ } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ 30 ^ { \circ } , 210 ^ { \circ } \right\}$ 
D)  $\left\{ 90 ^ { \circ } , 48.6 ^ { \circ } , 131.4 ^ { \circ } \right\}$ 
A)  $\left\{ 30 ^ { \circ } , 330 ^ { \circ } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ 30 ^ { \circ } , 210 ^ { \circ } \right\}$ 
D)  $\left\{ 90 ^ { \circ } , 48.6 ^ { \circ } , 131.4 ^ { \circ } \right\}$

Question 6

Solve the problem.
-The area of a triangle is given by
$$\mathrm { A } = \frac { 1 } { 2 } \mathrm { ab } \sin \mathrm { C } \text {, }$$
where $\mathrm { a }$ and $\mathrm { b }$ are the lengths of two of the sides and $\mathrm { C }$ is the included angle. If $\mathrm { A } = 15$ in. 2 , $\mathrm { a } = 5$ in., $\mathrm { b } = 8$ in., and $\mathrm { C }$ is an acute angle, what must $\mathrm { C }$ be? Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $24.30 ^ { \circ }$ 
B)  $131.41 ^ { \circ }$ 
C)  $48.59 ^ { \circ }$ 
D)  No such triangle exists. 
A)  $24.30 ^ { \circ }$ 
B)  $131.41 ^ { \circ }$ 
C)  $48.59 ^ { \circ }$ 
D)  No such triangle exists.

Question 7

Use identities to find the indicated value for each angle measure.
-$\sin \theta = \frac { 2 \sqrt { 6 } } { 5 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $\frac { - 4 \sqrt { 6 } } { 25 }$ 
B)  $\frac { 23 } { 25 }$ 
C)  $- \frac { 23 } { 25 }$ 
D)  $\frac { 4 \sqrt { 6 } } { 25 }$ 
A)  $\frac { - 4 \sqrt { 6 } } { 25 }$ 
B)  $\frac { 23 } { 25 }$ 
C)  $- \frac { 23 } { 25 }$ 
D)  $\frac { 4 \sqrt { 6 } } { 25 }$

Question 8

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

Accepted Answer

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

Question 9

Use Identities to find the exact value.
-$\cos \left( - 165 ^ { \circ } \right)$

Accepted Answer

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

Question 10

Use the fundamental identities to simplify the expression.
-$\frac { 1 + \tan ^ { 2 } x } { \sec x }$

Accepted Answer

A)  $\csc x$ 
B)  $- \sec x$ 
C)  $\sec x$ 
D)  1 
A)  $\csc x$ 
B)  $- \sec x$ 
C)  $\sec x$ 
D)  1

Question 11

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\sin ( \arctan 2 )$

Accepted Answer

A)  Undefined 
B)  $- 0.8172$ 
C)  $0.8944$ 
D)  0 
A)  Undefined 
B)  $- 0.8172$ 
C)  $0.8944$ 
D)  0

Question 12

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

Accepted Answer

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

Question 13

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

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 14

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$$( \sin x + \cos x ) ^ { 2 } = 1$$

Accepted Answer

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

Question 15

Find the exact value by using a half-angle identity.
-$\sin \frac { 5 \pi } { 12 }$

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)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$

Question 16

Find the exact value of the expression using the provided information.
-Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 5 } { 13 }$, with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = \frac { 8 } { 17 }$, with $\mathrm { t }$ in quadrant II.

Accepted Answer

A)  $- \frac { 171 } { 221 }$ 
B)  $- \frac { 21 } { 221 }$ 
C)  $\frac { 21 } { 221 }$ 
D)  $\frac { 171 } { 221 }$ 
A)  $- \frac { 171 } { 221 }$ 
B)  $- \frac { 21 } { 221 }$ 
C)  $\frac { 21 } { 221 }$ 
D)  $\frac { 171 } { 221 }$

Question 17

Use identities to fill in the blank with the appropriate trigonometric function name.
-

Accepted Answer

A)  $\sin$ 
B)  $\sec$ 
C)  $\cos$ 
D)  $\csc$ 
A)  $\sin$ 
B)  $\sec$ 
C)  $\cos$ 
D)  $\csc$

Question 18

Graph the inverse circular function.
-$$y = \frac { 1 } { 2 } \operatorname { arccot } x$$

Accepted Answer

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

Question 19

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$$\sin \frac { x } { 2 } \cos \frac { x } { 2 } = \frac { \sin x } { 2 }$$

Accepted Answer

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

Question 20

Verify that each equation is an identity.
-$$\sin ( 4 u ) = 2 \sin ( 2 u ) \cos ( 2 u )$$

Accepted Answer

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

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\sin 2 x = 2 \sin x \cos x$

Use identities to find the indicated value for each angle measure. - $\sin \theta = \frac { 7 } { 25 } , \quad \cos \theta > 0 \quad$ Find $\cos ( 2 \theta )$

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $2 \cos 2 \theta + 7 \sin \theta = 5$

Use identities to find the indicated value for each angle measure. - $\sin \theta = \frac { 2 \sqrt { 6 } } { 5 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

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

Use Identities to find the exact value. - $\cos \left( - 165 ^ { \circ } \right)$

Use the fundamental identities to simplify the expression. - $\frac { 1 + \tan ^ { 2 } x } { \sec x }$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\sin ( \arctan 2 )$

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

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

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $( \sin x + \cos x ) ^ { 2 } = 1$

Find the exact value by using a half-angle identity. - $\sin \frac { 5 \pi } { 12 }$

Find the exact value of the expression using the provided information. -Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 5 } { 13 }$ , with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = \frac { 8 } { 17 }$ , with $\mathrm { t }$ in quadrant II.

Use identities to fill in the blank with the appropriate trigonometric function name. -

Graph the inverse circular function. - $y = \frac { 1 } { 2 } \operatorname { arccot } x$ $Graph the inverse circular function. - y = \frac { 1 } { 2 } \operatorname { arccot } x$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\sin \frac { x } { 2 } \cos \frac { x } { 2 } = \frac { \sin x } { 2 }$

Verify that each equation is an identity. - $\sin ( 4 u ) = 2 \sin ( 2 u ) \cos ( 2 u )$

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

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - sin⁡2x=2sin⁡xcos⁡x\sin 2 x = 2 \sin x \cos xsin2x=2sinxcosx

Use identities to find the indicated value for each angle measure. - sin⁡θ=725,cos⁡θ>0\sin \theta = \frac { 7 } { 25 } , \quad \cos \theta > 0 \quadsinθ=257​,cosθ>0 Find cos⁡(2θ)\cos ( 2 \theta )cos(2θ)

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - 2cos⁡2θ+7sin⁡θ=52 \cos 2 \theta + 7 \sin \theta = 52cos2θ+7sinθ=5

Use identities to find the indicated value for each angle measure. - sin⁡θ=265,tan⁡θ<0\sin \theta = \frac { 2 \sqrt { 6 } } { 5 } , \tan \theta < 0 \quadsinθ=526​​,tanθ<0 Find sin⁡(2θ)\sin ( 2 \theta )sin(2θ)

Solve the equation for exact solutions over the interval [0, 2π). - 2sin⁡2x=sin⁡x2 \sin ^ { 2 } x = \sin x2sin2x=sinx

Use Identities to find the exact value. - cos⁡(−165∘)\cos \left( - 165 ^ { \circ } \right)cos(−165∘)

Use the fundamental identities to simplify the expression. - 1+tan⁡2xsec⁡x\frac { 1 + \tan ^ { 2 } x } { \sec x }secx1+tan2x​

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - sin⁡(arctan⁡2)\sin ( \arctan 2 )sin(arctan2)

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

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

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - (sin⁡x+cos⁡x)2=1( \sin x + \cos x ) ^ { 2 } = 1(sinx+cosx)2=1

Find the exact value by using a half-angle identity. - sin⁡5π12\sin \frac { 5 \pi } { 12 }sin125π​