Question 1

Solve the equation for solutions in the interval [0, 2 π).
-$\tan 2 x - \tan x = 0$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
C)  $\{ 0 , \pi \}$ 
D)  $\{ 0 \}$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
C)  $\{ 0 , \pi \}$ 
D)  $\{ 0 \}$

Question 2

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\cot \theta = - 2$ and $\cos \theta < 0$.

Accepted Answer

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

Question 3

Write in terms of the cofunction of a complementary angle.
-$\tan 91 ^ { \circ } 10 ^ { \prime }$

Accepted Answer

A)  $\tan - 1 ^ { \circ } 50 ^ { \prime }$ 
B)  $\cot 1 ^ { \circ } 50 ^ { \prime }$ 
C)  $\cot 0 ^ { \circ } 50 ^ { \prime }$ 
D)  $\cot - 1 ^ { \circ } 10 ^ { \prime }$ 
A)  $\tan - 1 ^ { \circ } 50 ^ { \prime }$ 
B)  $\cot 1 ^ { \circ } 50 ^ { \prime }$ 
C)  $\cot 0 ^ { \circ } 50 ^ { \prime }$ 
D)  $\cot - 1 ^ { \circ } 10 ^ { \prime }$

Question 4

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\cos \left( \cos ^ { - 1 } ( - 0.9372 ) \right)$

Accepted Answer

A)  $- 0.4686$ 
B)  $0.9372$ 
C)  $- 0.9372$ 
D)  $0 \quad .4686$ 
A)  $- 0.4686$ 
B)  $0.9372$ 
C)  $- 0.9372$ 
D)  $0 \quad .4686$

Question 5

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

Accepted Answer

\[\begin{array} { l } 
\left( \cos \frac

Question 6

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

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  $- \frac { \sqrt { 26 } } { 26 }$ 
C)  $- 26$ 
D)  $- \frac { \sqrt { 26 } } { 5 }$ 
A)  $\frac { 1 } { 5 }$ 
B)  $- \frac { \sqrt { 26 } } { 26 }$ 
C)  $- 26$ 
D)  $- \frac { \sqrt { 26 } } { 5 }$

Question 7

Use a calculator to give the value to the nearest degree.
-$$\theta = \cos ^ { - 1 } ( - 0.9397 )$$

Accepted Answer

A)  $157 ^ { \circ }$ 
B)  $160 ^ { \circ }$ 
C)  $-20 ^ { \circ }$ 
D)  $162 ^ { \circ }$ 
A)  $157 ^ { \circ }$ 
B)  $160 ^ { \circ }$ 
C)  $-20 ^ { \circ }$ 
D)  $162 ^ { \circ }$

Question 8

Verify that each equation is an identity.
-$$\frac { 1 + \cos t } { 1 - \cos t } - \frac { 1 - \cos t } { 1 + \cos t } = 4 \cot t \csc t$$

Accepted Answer

\[\begin{array} { l } 
\frac { 1 + \cos

Question 9

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.
-$2 \sin ^ { 2 } x + \sin x = 1$

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

Rewrite the following as a product of trigonometric functions.
-$\cos 11 ^ { \circ } - \cos 24 ^ { \circ }$

Accepted Answer

A)  $- 2 \sin 17.5 ^ { \circ } \sin \left( - 6.5 ^ { \circ } \right)$ 
B)  $- 2 \sin 17.5 ^ { \circ } \cos \left( - 6.5 ^ { \circ } \right)$ 
C)  $- 2 \cos 17.5 ^ { \circ } \sin \left( - 6.5 ^ { \circ } \right)$ 
D)  $- 2 \cos 17.5 ^ { \circ } \cos \left( - 6.5 ^ { \circ } \right)$ 
A)  $- 2 \sin 17.5 ^ { \circ } \sin \left( - 6.5 ^ { \circ } \right)$ 
B)  $- 2 \sin 17.5 ^ { \circ } \cos \left( - 6.5 ^ { \circ } \right)$ 
C)  $- 2 \cos 17.5 ^ { \circ } \sin \left( - 6.5 ^ { \circ } \right)$ 
D)  $- 2 \cos 17.5 ^ { \circ } \cos \left( - 6.5 ^ { \circ } \right)$

Question 12

Solve the equation for exact solutions.
-$\arcsin x + 2 \arctan x = \pi$

Accepted Answer

A)  $\{ 1 \}$ 
B)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$ 
C)  $\{ 0 \}$ 
D)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
A)  $\{ 1 \}$ 
B)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$ 
C)  $\{ 0 \}$ 
D)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$

Question 13

Verify that each equation is an identity.
-$$\sec ( 2 \theta ) = \frac { \csc ^ { 2 } \theta } { \csc ^ { 2 } \theta - 2 }$$

Accepted Answer

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

Question 14

Verify that the equation is an identity.
-$$\csc ( s + t ) = \frac { \operatorname { cscs } \csc t } { \cot t + \cot s }$$

Accepted Answer

\[\begin{array} { l } 
\csc ( s + t ) =

Question 15

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\cos \theta = \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

Accepted Answer

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

Question 16

Perform the transformation.
-Write $\cot x$ in terms of sec $x$.

Accepted Answer

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

Question 17

Verify that each equation is an identity.
-$$\left( 1 + \tan ^ { 2 } s \right) \left( 1 - \sin ^ { 2 } s \right) = 1$$

Accepted Answer

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

Question 18

Use a sum or difference identity to find the exact value.
-$\tan 105 ^ { \circ }$

Accepted Answer

A)  $\frac { 2 + \sqrt { 3 } } { 4 }$ 
B)  $2 + \sqrt { 3 }$ 
C)  $\frac { 2 - \sqrt { 3 } } { 4 }$ 
D)  $- 2 - \sqrt { 3 }$ 
A)  $\frac { 2 + \sqrt { 3 } } { 4 }$ 
B)  $2 + \sqrt { 3 }$ 
C)  $\frac { 2 - \sqrt { 3 } } { 4 }$ 
D)  $- 2 - \sqrt { 3 }$

Question 19

Express the function as a trigonometric function of x.
-$\sin 4 x$

Accepted Answer

A)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
B)  $1 - 8 \sin ^ { 2 } x \cos ^ { 2 } x$ 
C)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
D)  $3 \sin x - 4 \sin ^ { 3 } x$ 
A)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
B)  $1 - 8 \sin ^ { 2 } x \cos ^ { 2 } x$ 
C)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
D)  $3 \sin x - 4 \sin ^ { 3 } x$

Question 20

Write the word or phrase that best completes each statement or answers the question.
-On the given graph of $y = \sec x$ sketch the graph of $y = \sec ^ { - 1 } x$ as defined in the text. Give the domain and the range.

Accepted Answer

The answer of Write the word or phrase that best...

Solve the equation for solutions in the interval [0, 2 π). - $\tan 2 x - \tan x = 0$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\cot \theta = - 2$ and $\cos \theta < 0$ .

Write in terms of the cofunction of a complementary angle. - $\tan 91 ^ { \circ } 10 ^ { \prime }$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\cos \left( \cos ^ { - 1 } ( - 0.9372 ) \right)$

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

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

Use a calculator to give the value to the nearest degree. - $\theta = \cos ^ { - 1 } ( - 0.9397 )$

Verify that each equation is an identity. - $\frac { 1 + \cos t } { 1 - \cos t } - \frac { 1 - \cos t } { 1 + \cos t } = 4 \cot t \csc t$

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. - $2 \sin ^ { 2 } x + \sin x = 1$

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

Rewrite the following as a product of trigonometric functions. - $\cos 11 ^ { \circ } - \cos 24 ^ { \circ }$

Solve the equation for exact solutions. - $\arcsin x + 2 \arctan x = \pi$

Verify that each equation is an identity. - $\sec ( 2 \theta ) = \frac { \csc ^ { 2 } \theta } { \csc ^ { 2 } \theta - 2 }$

Verify that the equation is an identity. - $\csc ( s + t ) = \frac { \operatorname { cscs } \csc t } { \cot t + \cot s }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\cos \theta = \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

Perform the transformation. -Write $\cot x$ in terms of sec $x$ .

Verify that each equation is an identity. - $\left( 1 + \tan ^ { 2 } s \right) \left( 1 - \sin ^ { 2 } s \right) = 1$

Use a sum or difference identity to find the exact value. - $\tan 105 ^ { \circ }$

Express the function as a trigonometric function of x. - $\sin 4 x$

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

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Exam 7: Trigonometric Identities and Equations

Solve the equation for solutions in the interval [0, 2 π). - tan⁡2x−tan⁡x=0\tan 2 x - \tan x = 0tan2x−tanx=0

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ if cot⁡θ=−2\cot \theta = - 2cotθ=−2 and cos⁡θ<0\cos \theta < 0cosθ<0 .

Write in terms of the cofunction of a complementary angle. - tan⁡91∘10′\tan 91 ^ { \circ } 10 ^ { \prime }tan91∘10′

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - cos⁡(cos⁡−1(−0.9372))\cos \left( \cos ^ { - 1 } ( - 0.9372 ) \right)cos(cos−1(−0.9372))

Verify that each equation is an identity. - (cos⁡x2−sin⁡x2)2=1−sin⁡x\left( \cos \frac { x } { 2 } - \sin \frac { x } { 2 } \right) ^ { 2 } = 1 - \sin x(cos2x​−sin2x​)2=1−sinx

Use the fundamental identities to find the value of the trigonometric function. -Find cos⁡θ\cos \thetacosθ if tan⁡θ=5\tan \theta = 5tanθ=5 and sin⁡θ<0\sin \theta < 0sinθ<0 .

Use a calculator to give the value to the nearest degree. - θ=cos⁡−1(−0.9397)\theta = \cos ^ { - 1 } ( - 0.9397 )θ=cos−1(−0.9397)

Verify that each equation is an identity. - 1+cos⁡t1−cos⁡t−1−cos⁡t1+cos⁡t=4cot⁡tcsc⁡t\frac { 1 + \cos t } { 1 - \cos t } - \frac { 1 - \cos t } { 1 + \cos t } = 4 \cot t \csc t1−cost1+cost​−1+cost1−cost​=4cottcsct

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. - 2sin⁡2x+sin⁡x=12 \sin ^ { 2 } x + \sin x = 12sin2x+sinx=1

Verify that each equation is an identity. - tan⁡x+cot⁡xtan⁡x−cot⁡x=1sin⁡2x−cos⁡2x\frac { \tan x + \cot x } { \tan x - \cot x } = \frac { 1 } { \sin ^ { 2 } x - \cos ^ { 2 } x }tanx−cotxtanx+cotx​=sin2x−cos2x1​

Rewrite the following as a product of trigonometric functions. - cos⁡11∘−cos⁡24∘\cos 11 ^ { \circ } - \cos 24 ^ { \circ }cos11∘−cos24∘

Solve the equation for exact solutions. - arcsin⁡x+2arctan⁡x=π\arcsin x + 2 \arctan x = \piarcsinx+2arctanx=π

Verify that each equation is an identity. - sec⁡(2θ)=csc⁡2θcsc⁡2θ−2\sec ( 2 \theta ) = \frac { \csc ^ { 2 } \theta } { \csc ^ { 2 } \theta - 2 }sec(2θ)=csc2θ−2csc2θ​

Verify that the equation is an identity. - csc⁡(s+t)=cscs⁡csc⁡tcot⁡t+cot⁡s\csc ( s + t ) = \frac { \operatorname { cscs } \csc t } { \cot t + \cot s }csc(s+t)=cott+cotscscscsct​

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ if cos⁡θ=23\cos \theta = \frac { 2 } { 3 }cosθ=32​ and θ\thetaθ is in quadrant IV.

Perform the transformation. -Write cot⁡x\cot xcotx in terms of sec xxx .

Verify that each equation is an identity. - (1+tan⁡2s)(1−sin⁡2s)=1\left( 1 + \tan ^ { 2 } s \right) \left( 1 - \sin ^ { 2 } s \right) = 1(1+tan2s)(1−sin2s)=1

Use a sum or difference identity to find the exact value. - tan⁡105∘\tan 105 ^ { \circ }tan105∘

Express the function as a trigonometric function of x. - sin⁡4x\sin 4 xsin4x

Write the word or phrase that best completes each statement or answers the question. -On the given graph of y=sec⁡xy = \sec xy=secx sketch the graph of y=sec⁡−1xy = \sec ^ { - 1 } xy=sec−1x as defined in the text. Give the domain and the range.