Question 1

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

Accepted Answer

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

Question 2

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\tan \theta = - \frac { 12 } { 5 }$ and $\cos \theta > 0$.

Accepted Answer

A)  $\frac { 5 } { 13 }$ 
B)  $- \frac { 5 } { 13 }$ 
C)  $- \frac { 12 } { 13 }$ 
D)  $- \frac { 13 } { 12 }$ 
A)  $\frac { 5 } { 13 }$ 
B)  $- \frac { 5 } { 13 }$ 
C)  $- \frac { 12 } { 13 }$ 
D)  $- \frac { 13 } { 12 }$

Question 3

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \sin ^ { - 1 } ( - 0.4848 )$$

Accepted Answer

A)  $2.6365103$ 
B)  $- 0.5412698$ 
C)  $- 0.5061345$ 
D)  $- 0.4888889$ 
A)  $2.6365103$ 
B)  $- 0.5412698$ 
C)  $- 0.5061345$ 
D)  $- 0.4888889$

Question 4

Complete the sentence so the result is an identity. Let x be any real number.
-$$1 - ــــــــــــــ = \sin ^ { 2 } x$$

Accepted Answer

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

Question 5

Determine all solutions of the equation in radians.
-Find $\tan \frac { x } { 2 }$, given that $\tan x = - 3$ and $x$ terminates in $90 ^ { \circ } < x < 180 ^ { \circ }$.

Accepted Answer

A)  $\frac { \sqrt { 10 } + 1 } { - 3 }$ 
B)  $\frac { \sqrt { 10 } + 1 } { 3 }$ 
C)  $\frac { \sqrt { 10 } - 1 } { 3 }$ 
D)  $\frac { \sqrt { 10 } - 1 } { - 3 }$ 
A)  $\frac { \sqrt { 10 } + 1 } { - 3 }$ 
B)  $\frac { \sqrt { 10 } + 1 } { 3 }$ 
C)  $\frac { \sqrt { 10 } - 1 } { 3 }$ 
D)  $\frac { \sqrt { 10 } - 1 } { - 3 }$

Question 6

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sec \theta$ if $\tan \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant $\mathrm { I }$.

Accepted Answer

A)  $\frac { 3 \sqrt { 7 } } { 7 }$ 
B)  $\frac { 5 } { 4 }$ 
C)  $- \frac { \sqrt { 7 } } { 9 }$ 
D)  $- \frac { 3 } { 2 }$ 
A)  $\frac { 3 \sqrt { 7 } } { 7 }$ 
B)  $\frac { 5 } { 4 }$ 
C)  $- \frac { \sqrt { 7 } } { 9 }$ 
D)  $- \frac { 3 } { 2 }$

Question 7

Solve the equation for exact solutions.
-$4 \cos ^ { - 1 } x = \pi$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { \sqrt { 2 } } { 2 } \right\}$ 
C)  $\left\{ \frac { \sqrt { 3 } } { 2 } \right\}$ 
D)  $\left\{ \frac { 1 } { 2 } \right\}$ 
A)  $\left\{ \frac { \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { \sqrt { 2 } } { 2 } \right\}$ 
C)  $\left\{ \frac { \sqrt { 3 } } { 2 } \right\}$ 
D)  $\left\{ \frac { 1 } { 2 } \right\}$

Question 8

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

Accepted Answer

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

Question 9

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }$$

Accepted Answer

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

Question 10

Find the exact value of the expression using the provided information.
-Find $\sin ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 1 } { 6 }$, with $\mathrm { s }$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$, with $t$ in quadrant III.

Accepted Answer

A)  $\frac { 3 + 4 \sqrt { 35 } } { 30 }$ 
B)  $\frac { 3 \sqrt { 35 } - 4 } { 30 }$ 
C)  $\frac { 3 - 4 \sqrt { 35 } } { 30 }$ 
D)  $\frac { 3 \sqrt { 35 } + 4 } { 30 }$ 
A)  $\frac { 3 + 4 \sqrt { 35 } } { 30 }$ 
B)  $\frac { 3 \sqrt { 35 } - 4 } { 30 }$ 
C)  $\frac { 3 - 4 \sqrt { 35 } } { 30 }$ 
D)  $\frac { 3 \sqrt { 35 } + 4 } { 30 }$

Question 11

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.
-$4 \csc ^ { 2 } \theta = 5 \csc \theta$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 53.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 126.9 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 180 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
B)  $\left\{ 0 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 53.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 126.9 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 233.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
C)  
$\left\{ 53.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 126.9 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
D)  
$\left\{ 0 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 53.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 126.9 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
A)  $\left\{ 0 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 53.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 126.9 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 180 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
B)  $\left\{ 0 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 53.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 126.9 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 233.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
C)  
$\left\{ 53.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 126.9 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$ 
D)  
$\left\{ 0 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 53.1 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 126.9 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$

Question 12

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

Question 13

Determine all solutions of the equation in radians.
-Find $\cos \frac { x } { 2 }$, given that $\sec x = 4$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
A)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$

Question 14

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$\mathrm { y } = \cot ^ { - 1 } ( 2.5181552 )$$

Accepted Answer

A)  $2.1658824$ 
B)  $6.8341176$ 
C)  $1.1927785$ 
D)  $0.3780178$ 
A)  $2.1658824$ 
B)  $6.8341176$ 
C)  $1.1927785$ 
D)  $0.3780178$

Question 15

Verify that each equation is an identity.
-$$\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )$$

Accepted Answer

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

Question 16

Solve the problem.
-It can be shown that if the angle of elevation from an observer to the top of an object is A and the angle of elevation $\mathrm { d } \mathrm { ft }$ closer is $\mathrm { B }$, then the height of the object is given by $h = \frac { d } { \cot A - \cot B } f t$.
Find $\mathrm { A }$ if $\mathrm { h } = 80 \mathrm { ft } , \mathrm { d } = 50 \mathrm { ft }$, and $\mathrm { B } = 28 ^ { \circ }$. Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $21.76 ^ { \circ }$ 
B)  $19.25 ^ { \circ }$ 
C)  $26.51 ^ { \circ }$ 
D)  $24.18 ^ { \circ }$ 
A)  $21.76 ^ { \circ }$ 
B)  $19.25 ^ { \circ }$ 
C)  $26.51 ^ { \circ }$ 
D)  $24.18 ^ { \circ }$

Question 17

Verify that each equation is an identity.
-$$\tan^ 2 u ( 1 + \cos ( 2 u ) ) = 1 - \cos ( 2 u )$$

Accepted Answer

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

Question 18

Use an identity to write the expression as a single trigonometric function or as a single number.
-$4 \sin 2 x \cos 2 x$

Accepted Answer

A)  $\cos 4 x$ 
B)  $\cos 8 x$ 
C)  $2 \sin 4 x$ 
D)  $\frac { 1 } { 2 } \sin 16 x$ 
A)  $\cos 4 x$ 
B)  $\cos 8 x$ 
C)  $2 \sin 4 x$ 
D)  $\frac { 1 } { 2 } \sin 16 x$

Question 19

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

Accepted Answer

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

Question 20

Use identities to write the expression as a single function of x or θ.
-$\cos \left( \theta + 270 ^ { \circ } \right)$

Accepted Answer

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

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

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\tan \theta = - \frac { 12 } { 5 }$ and $\cos \theta > 0$ .

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \sin ^ { - 1 } ( - 0.4848 )$

Complete the sentence so the result is an identity. Let x be any real number. - $1 - ــــــــــــــ = \sin ^ { 2 } x$

Determine all solutions of the equation in radians. -Find $\tan \frac { x } { 2 }$ , given that $\tan x = - 3$ and $x$ terminates in $90 ^ { \circ } < x < 180 ^ { \circ }$ .

Use the fundamental identities to find the value of the trigonometric function. -Find $\sec \theta$ if $\tan \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant $\mathrm { I }$ .

Solve the equation for exact solutions. - $4 \cos ^ { - 1 } x = \pi$

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

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }$

Find the exact value of the expression using the provided information. -Find $\sin ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 1 } { 6 }$ , with $\mathrm { s }$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$ , with $t$ in quadrant III.

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. - $4 \csc ^ { 2 } \theta = 5 \csc \theta$

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

Determine all solutions of the equation in radians. -Find $\cos \frac { x } { 2 }$ , given that $\sec x = 4$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$ .

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \cot ^ { - 1 } ( 2.5181552 )$

Verify that each equation is an identity. - $\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )$

Verify that each equation is an identity. - $\tan^ 2 u ( 1 + \cos ( 2 u ) ) = 1 - \cos ( 2 u )$

Use an identity to write the expression as a single trigonometric function or as a single number. - $4 \sin 2 x \cos 2 x$

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

Use identities to write the expression as a single function of x or θ. - $\cos \left( \theta + 270 ^ { \circ } \right)$

Equations and Inequalities

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Polynomial and Rational Functions

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Exam 7: Trigonometric Identities and Equations

Use the fundamental identities to simplify the expression. - sec⁡2xtan⁡x−tan⁡x\frac { \sec ^ { 2 } x } { \tan x } - \tan xtanxsec2x​−tanx

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ if tan⁡θ=−125\tan \theta = - \frac { 12 } { 5 }tanθ=−512​ and cos⁡θ>0\cos \theta > 0cosθ>0 .

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=sin⁡−1(−0.4848)y = \sin ^ { - 1 } ( - 0.4848 )y=sin−1(−0.4848)

Complete the sentence so the result is an identity. Let x be any real number. - 1−ــــــــــــــ=sin⁡2x1 - ــــــــــــــ = \sin ^ { 2 } x1−ــــــــــــــ=sin2x

Determine all solutions of the equation in radians. -Find tan⁡x2\tan \frac { x } { 2 }tan2x​ , given that tan⁡x=−3\tan x = - 3tanx=−3 and xxx terminates in 90∘<x<180∘90 ^ { \circ } < x < 180 ^ { \circ }90∘<x<180∘ .

Use the fundamental identities to find the value of the trigonometric function. -Find sec⁡θ\sec \thetasecθ if tan⁡θ=34\tan \theta = \frac { 3 } { 4 }tanθ=43​ and θ\thetaθ is in quadrant I\mathrm { I }I .

Solve the equation for exact solutions. - 4cos⁡−1x=π4 \cos ^ { - 1 } x = \pi4cos−1x=π

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

Perform the indicated operations and simplify the result so there are no quotients. - 1cos⁡2θ−1cot⁡2θ\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }cos2θ1​−cot2θ1​

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. - 4csc⁡2θ=5csc⁡θ4 \csc ^ { 2 } \theta = 5 \csc \theta4csc2θ=5cscθ

Perform the indicated operations and simplify the result so there are no quotients. - tan⁡2θ−3sin⁡θtan⁡θsec⁡θ\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \thetatan2θ−3sinθtanθsecθ

Determine all solutions of the equation in radians. -Find cos⁡x2\cos \frac { x } { 2 }cos2x​ , given that sec⁡x=4\sec x = 4secx=4 and xxx terminates in 0<x<π20 < x < \frac { \pi } { 2 }0<x<2π​ .

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=cot⁡−1(2.5181552)\mathrm { y } = \cot ^ { - 1 } ( 2.5181552 )y=cot−1(2.5181552)

Verify that each equation is an identity. - cot⁡2x=(csc⁡x−1)(csc⁡x+1)\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )cot2x=(cscx−1)(cscx+1)

Verify that each equation is an identity. - tan⁡2u(1+cos⁡(2u))=1−cos⁡(2u)\tan^ 2 u ( 1 + \cos ( 2 u ) ) = 1 - \cos ( 2 u )tan2u(1+cos(2u))=1−cos(2u)

Use an identity to write the expression as a single trigonometric function or as a single number. - 4sin⁡2xcos⁡2x4 \sin 2 x \cos 2 x4sin2xcos2x

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

Use identities to write the expression as a single function of x or θ. - cos⁡(θ+270∘)\cos \left( \theta + 270 ^ { \circ } \right)cos(θ+270∘)