Question 1

Solve the equation for exact solutions.
-$$\tan ^ { - 1 } ( \mathrm { x } + 2 ) + \tan ^ { - 1 } \mathrm { x } = \tan ^ { - 1 } \frac { 1 } { 4 }$$

Accepted Answer

A)  $\{ 2 \}$ 
B)  $\left\{ \frac { 1 } { 2 } \right\}$ 
C)  $\left\{ - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right\}$ 
D)  $\{ - 5 + 3 \sqrt { 2 } \}$ 
A)  $\{ 2 \}$ 
B)  $\left\{ \frac { 1 } { 2 } \right\}$ 
C)  $\left\{ - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right\}$ 
D)  $\{ - 5 + 3 \sqrt { 2 } \}$

Question 2

Use the fundamental identities to simplify the expression.
-$\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta$

Accepted Answer

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

Question 3

Find the exact value by using a half-angle identity.
-$\sin 75 ^ { \circ }$

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 4

Give the degree measure of θ.
-$$\theta = \arcsin ( 0 )$$

Accepted Answer

A)  $0 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $270 ^ { \circ }$ 
A)  $0 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $270 ^ { \circ }$

Question 5

Write the product as a sum or difference of trigonometric functions.
-$8 \sin 57 ^ { \circ } \cos 110 ^ { \circ }$

Accepted Answer

A)  $4 \left[ \cos 167 ^ { \circ } + \cos \left( - 53 ^ { \circ } \right) \right]$ 
B)  $8 \left[ \cos 167 ^ { \circ } + \cos \left( - 53 ^ { \circ } \right) \right]$ 
C)  $4 \left[ \sin 167 ^ { \circ } - \sin \left( - 53 ^ { \circ } \right) \right]$ 
D)  $4 \left[ \sin 167 ^ { \circ } + \sin \left( - 53 ^ { \circ } \right) \right]$ 
A)  $4 \left[ \cos 167 ^ { \circ } + \cos \left( - 53 ^ { \circ } \right) \right]$ 
B)  $8 \left[ \cos 167 ^ { \circ } + \cos \left( - 53 ^ { \circ } \right) \right]$ 
C)  $4 \left[ \sin 167 ^ { \circ } - \sin \left( - 53 ^ { \circ } \right) \right]$ 
D)  $4 \left[ \sin 167 ^ { \circ } + \sin \left( - 53 ^ { \circ } \right) \right]$

Question 6

Solve the equation for exact solutions over the interval [0, 2π).
-$$\cos ^ { 2 } x + 2 \cos x + 1 = 0$$

Accepted Answer

A)  $\{ \pi \}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
C)  $\left\{ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right\}$ 
D)  $\{ 2 \pi \}$ 
A)  $\{ \pi \}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
C)  $\left\{ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right\}$ 
D)  $\{ 2 \pi \}$

Question 7

Use a sum or difference identity to find the exact value.
-$\tan 345 ^ { \circ }$

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 8

Solve the equation for x, where x is restricted to the given interval.
-$y = 4 \cot \frac { x } { 2 }$, for $x$ in $( 0,2 \pi )$

Accepted Answer

A)  $x = \frac { 1 } { 2 } \operatorname { arccot } 4 y$ 
B)  $x = 2 \operatorname { arccot } \frac { y } { 4 }$ 
C)  $x = 2 \operatorname { arccot } 4 y$ 
D)  $x = \frac { 1 } { 2 } \operatorname { arccot } \frac { y } { 4 }$ 
A)  $x = \frac { 1 } { 2 } \operatorname { arccot } 4 y$ 
B)  $x = 2 \operatorname { arccot } \frac { y } { 4 }$ 
C)  $x = 2 \operatorname { arccot } 4 y$ 
D)  $x = \frac { 1 } { 2 } \operatorname { arccot } \frac { y } { 4 }$

Question 9

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$$\frac { 1 + \tan ^ { 2 } x } { \sec x }$$

Accepted Answer

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

Question 10

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$( \cos \theta + \cot \theta ) ( 1 - \csc \theta ) = \cot ^ { 2 } \theta$$

Accepted Answer

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

Question 11

Verify that the equation is an identity.
-$$\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }$$

Accepted Answer

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

Question 12

Factor the trigonometric expression and simplify.
-$\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } x$

Accepted Answer

A)  $\sec ^ { 2 } x + 2$ 
B)  $3 \sec ^ { 2 } x - 2$ 
C)  $4 \sec ^ { 2 } x$ 
D)  $\tan ^ { 2 } x - 1$ 
A)  $\sec ^ { 2 } x + 2$ 
B)  $3 \sec ^ { 2 } x - 2$ 
C)  $4 \sec ^ { 2 } x$ 
D)  $\tan ^ { 2 } x - 1$

Question 13

Verify that the equation is an identity.
-$$\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$$

Accepted Answer

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

Question 14

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\sqrt { \frac { 1 + \cos 22 ^ { \circ } } { 2 } }$$

Accepted Answer

A)  $\cos 11 ^ { \circ }$ 
B)  $\tan 11 ^ { \circ }$ 
C)  $\sin 11 ^ { \circ }$ 
D)  $\cot 11 ^ { \circ }$ 
A)  $\cos 11 ^ { \circ }$ 
B)  $\tan 11 ^ { \circ }$ 
C)  $\sin 11 ^ { \circ }$ 
D)  $\cot 11 ^ { \circ }$

Question 15

Use identities to find the indicated value for each angle measure.
-$\cos 2 \theta = \frac { 2 } { 3 }$ and $\theta$ terminates in quadrant III Find $\cos \theta$.

Accepted Answer

A)  $\cos \theta = - \frac { \sqrt { 30 } } { 5 }$ 
B)  $\cos \theta = - \frac { \sqrt { 30 } } { 6 }$ 
C)  $\cos \theta = 0$ 
D)  $\cos \theta = \frac { \sqrt { 30 } } { 6 }$ 
A)  $\cos \theta = - \frac { \sqrt { 30 } } { 5 }$ 
B)  $\cos \theta = - \frac { \sqrt { 30 } } { 6 }$ 
C)  $\cos \theta = 0$ 
D)  $\cos \theta = \frac { \sqrt { 30 } } { 6 }$

Question 16

Rewrite the following as a product of trigonometric functions.
-$\sin 16 ^ { \circ } - \sin 29 ^ { \circ }$

Accepted Answer

A)  $2 \cos 22.5 ^ { \circ } \sin \left( - 6.5 ^ { \circ } \right)$ 
B)  $2 \sin 22.5 ^ { \circ } \sin \left( - 6.5 ^ { \circ } \right)$ 
C)  $2 \cos 22.5 ^ { \circ } \cos \left( - 6.5 ^ { \circ } \right)$ 
D)  $2 \sin 22.5 ^ { \circ } \cos \left( - 6.5 ^ { \circ } \right)$ 
A)  $2 \cos 22.5 ^ { \circ } \sin \left( - 6.5 ^ { \circ } \right)$ 
B)  $2 \sin 22.5 ^ { \circ } \sin \left( - 6.5 ^ { \circ } \right)$ 
C)  $2 \cos 22.5 ^ { \circ } \cos \left( - 6.5 ^ { \circ } \right)$ 
D)  $2 \sin 22.5 ^ { \circ } \cos \left( - 6.5 ^ { \circ } \right)$

Question 17

Use identities to write the expression as a single function of x or θ.
-$\cos \left( \theta + \frac { \pi } { 2 } \right)$

Accepted Answer

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

Question 18

Solve the equation for exact solutions over the interval [0, 2π).
-$\csc ^ { 5 } x - 4 \csc x = 0$

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\cos \theta = - \frac { \sqrt { 11 } } { 6 }$ and $\theta$ is in quadrant III.

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 12 }$ 
B)  $\frac { 5 } { 6 }$ 
C)  $- \frac { 25 } { 36 }$ 
D)  $- \frac { 5 } { 6 }$ 
A)  $- \frac { \sqrt { 3 } } { 12 }$ 
B)  $\frac { 5 } { 6 }$ 
C)  $- \frac { 25 } { 36 }$ 
D)  $- \frac { 5 } { 6 }$

Solve the equation for exact solutions. - $\tan ^ { - 1 } ( \mathrm { x } + 2 ) + \tan ^ { - 1 } \mathrm { x } = \tan ^ { - 1 } \frac { 1 } { 4 }$

Use the fundamental identities to simplify the expression. - $\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta$

Find the exact value by using a half-angle identity. - $\sin 75 ^ { \circ }$

Give the degree measure of θ. - $\theta = \arcsin ( 0 )$

Write the product as a sum or difference of trigonometric functions. - $8 \sin 57 ^ { \circ } \cos 110 ^ { \circ }$

Solve the equation for exact solutions over the interval [0, 2π). - $\cos ^ { 2 } x + 2 \cos x + 1 = 0$

Use a sum or difference identity to find the exact value. - $\tan 345 ^ { \circ }$

Solve the equation for x, where x is restricted to the given interval. - $y = 4 \cot \frac { x } { 2 }$ , for $x$ in $( 0,2 \pi )$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\frac { 1 + \tan ^ { 2 } x } { \sec x }$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $( \cos \theta + \cot \theta ) ( 1 - \csc \theta ) = \cot ^ { 2 } \theta$

Verify that the equation is an identity. - $\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }$

Factor the trigonometric expression and simplify. - $\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } x$

Verify that the equation is an identity. - $\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sqrt { \frac { 1 + \cos 22 ^ { \circ } } { 2 } }$

Use identities to find the indicated value for each angle measure. - $\cos 2 \theta = \frac { 2 } { 3 }$ and $\theta$ terminates in quadrant III Find $\cos \theta$ .

Rewrite the following as a product of trigonometric functions. - $\sin 16 ^ { \circ } - \sin 29 ^ { \circ }$

Use identities to write the expression as a single function of x or θ. - $\cos \left( \theta + \frac { \pi } { 2 } \right)$

Solve the equation for exact solutions over the interval [0, 2π). - $\csc ^ { 5 } x - 4 \csc x = 0$

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

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\cos \theta = - \frac { \sqrt { 11 } } { 6 }$ and $\theta$ is in quadrant III.

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

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Exam 7: Trigonometric Identities and Equations

Solve the equation for exact solutions. - tan⁡−1(x+2)+tan⁡−1x=tan⁡−114\tan ^ { - 1 } ( \mathrm { x } + 2 ) + \tan ^ { - 1 } \mathrm { x } = \tan ^ { - 1 } \frac { 1 } { 4 }tan−1(x+2)+tan−1x=tan−141​

Use the fundamental identities to simplify the expression. - 1cot⁡2θ+sec⁡θcos⁡θ\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \thetacot2θ1​+secθcosθ

Find the exact value by using a half-angle identity. - sin⁡75∘\sin 75 ^ { \circ }sin75∘

Give the degree measure of θ. - θ=arcsin⁡(0)\theta = \arcsin ( 0 )θ=arcsin(0)

Write the product as a sum or difference of trigonometric functions. - 8sin⁡57∘cos⁡110∘8 \sin 57 ^ { \circ } \cos 110 ^ { \circ }8sin57∘cos110∘

Solve the equation for exact solutions over the interval [0, 2π). - cos⁡2x+2cos⁡x+1=0\cos ^ { 2 } x + 2 \cos x + 1 = 0cos2x+2cosx+1=0

Use a sum or difference identity to find the exact value. - tan⁡345∘\tan 345 ^ { \circ }tan345∘

Solve the equation for x, where x is restricted to the given interval. - y=4cot⁡x2y = 4 \cot \frac { x } { 2 }y=4cot2x​ , for xxx in (0,2π)( 0,2 \pi )(0,2π)

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - 1+tan⁡2xsec⁡x\frac { 1 + \tan ^ { 2 } x } { \sec x }secx1+tan2x​

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - (cos⁡θ+cot⁡θ)(1−csc⁡θ)=cot⁡2θ( \cos \theta + \cot \theta ) ( 1 - \csc \theta ) = \cot ^ { 2 } \theta(cosθ+cotθ)(1−cscθ)=cot2θ

Verify that the equation is an identity. - cot⁡(x+y)cot⁡(x−y)=1−tan⁡2xtan⁡2ytan⁡2x−tan⁡2y\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }cot(x+y)cot(x−y)=tan2x−tan2y1−tan2xtan2y​

Factor the trigonometric expression and simplify. - sec⁡4x+sec⁡2xtan⁡2x−2tan⁡4x\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } xsec4x+sec2xtan2x−2tan4x

Verify that the equation is an identity. - tan⁡(x−π4)=tan⁡x−11+tan⁡x\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }tan(x−4π​)=1+tanxtanx−1​

Use an identity to write the expression as a single trigonometric function or as a single number. - 1+cos⁡22∘2\sqrt { \frac { 1 + \cos 22 ^ { \circ } } { 2 } }21+cos22∘​​

Use identities to find the indicated value for each angle measure. - cos⁡2θ=23\cos 2 \theta = \frac { 2 } { 3 }cos2θ=32​ and θ\thetaθ terminates in quadrant III Find cos⁡θ\cos \thetacosθ .

Rewrite the following as a product of trigonometric functions. - sin⁡16∘−sin⁡29∘\sin 16 ^ { \circ } - \sin 29 ^ { \circ }sin16∘−sin29∘

Use identities to write the expression as a single function of x or θ. - cos⁡(θ+π2)\cos \left( \theta + \frac { \pi } { 2 } \right)cos(θ+2π​)

Solve the equation for exact solutions over the interval [0, 2π). - csc⁡5x−4csc⁡x=0\csc ^ { 5 } x - 4 \csc x = 0csc5x−4cscx=0

Provide an appropriate response. -True or false? The statement cos⁡−1(cos⁡x)=x\cos ^ { - 1 } ( \cos x ) = xcos−1(cosx)=x for all real numbers in the interval −1≤x≤1- 1 \leq x \leq 1−1≤x≤1 .

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ if cos⁡θ=−116\cos \theta = - \frac { \sqrt { 11 } } { 6 }cosθ=−611​​ and θ\thetaθ is in quadrant III.