Question 1

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

Accepted Answer

A)  $\sec ^ { 2 } x + \tan ^ { 2 } x$ 
B)  2 
C)  $\sec ^ { 2 } x \left( 1 + \tan ^ { 2 } x \right)$ 
D)  1 
A)  $\sec ^ { 2 } x + \tan ^ { 2 } x$ 
B)  2 
C)  $\sec ^ { 2 } x \left( 1 + \tan ^ { 2 } x \right)$ 
D)  1

Question 2

Use a calculator to give the value to the nearest degree.
-$$\theta = \sin ^ { - 1 } ( - 0.4848 )$$

Accepted Answer

A)  $- 28 ^ { \circ }$ 
B)  $151 ^ { \circ }$ 
C)  $- 29 ^ { \circ }$ 
D)  $- 31 ^ { \circ }$ 
A)  $- 28 ^ { \circ }$ 
B)  $151 ^ { \circ }$ 
C)  $- 29 ^ { \circ }$ 
D)  $- 31 ^ { \circ }$

Question 3

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

Accepted Answer

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

Question 4

Use the cofunction identities to find an angle $$\theta$$ that makes the statement true.
-$\cot 5 \theta = \tan 4 \theta$

Accepted Answer

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

Question 5

Use a sum or difference identity to find the exact value.
-$$\frac { \tan 100 ^ { \circ } - \tan ( - 20 ) ^ { \circ } } { 1 + \tan 100 ^ { \circ } \tan ( - 20 ) ^ { \circ } }$$

Accepted Answer

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

Question 6

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

Question 7

Complete the sentence so the result is an identity. Let x be any real number.
-$$\cos x = ( \cot x ) ( \square )$$

Accepted Answer

A)  $\tan x$ 
B)  $\csc x$ 
C)  $\sec x$ 
D)  $\sin x$ 
A)  $\tan x$ 
B)  $\csc x$ 
C)  $\sec x$ 
D)  $\sin x$

Question 8

Solve the problem.

-$$\sec ^ { 2 } \frac { \mathrm { u } } { 2 } = \frac { 2 \sec \mathrm { u } } { \sec \mathrm { u } + 1 }$$

Accepted Answer

The answer of Solve the problem.

-\[\sec ^ { 2 }...

Question 9

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

Accepted Answer

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

Question 10

Use identities to find the indicated value for each angle measure.
-$\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $\frac { - 12 \sqrt { 10 } } { 49 }$ 
B)  $\frac { 12 \sqrt { 10 } } { 49 }$ 
C)  $- \frac { 31 } { 49 }$ 
D)  $\frac { 31 } { 49 }$ 
A)  $\frac { - 12 \sqrt { 10 } } { 49 }$ 
B)  $\frac { 12 \sqrt { 10 } } { 49 }$ 
C)  $- \frac { 31 } { 49 }$ 
D)  $\frac { 31 } { 49 }$

Question 11

Give the exact value of the expression.
-$$\csc \left( \csc ^ { - 1 } - 2 \right)$$

Accepted Answer

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

Question 12

Give the degree measure of $$\theta$$
-$$\theta = \sin ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)$$

Accepted Answer

A)  $120 ^ { \circ }$ 
B)  $135 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $60 ^ { \circ }$ 
A)  $120 ^ { \circ }$ 
B)  $135 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $60 ^ { \circ }$

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

Solve the equation in the interval $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$ ). Give solutions to the nearest tenth, if necessary.
-$\sin ^ { 2 } \theta - \sin \theta - 12 = 0$

Accepted Answer

A)  $\left\{ 45 ^ { \circ } , 315 ^ { \circ } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ 45 ^ { \circ } \right\}$ 
D)  $\left\{ 45 ^ { \circ } , 135 ^ { \circ } \right\}$ 
A)  $\left\{ 45 ^ { \circ } , 315 ^ { \circ } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ 45 ^ { \circ } \right\}$ 
D)  $\left\{ 45 ^ { \circ } , 135 ^ { \circ } \right\}$

Question 16

Give the degree measure of $$\theta$$
-$$\theta = \arcsin ( - 1 )$$

Accepted Answer

A)  $- 90 ^ { \circ }$ 
B)  $0 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $180 ^ { \circ }$ 
A)  $- 90 ^ { \circ }$ 
B)  $0 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $180 ^ { \circ }$

Question 17

Give the exact value of the expression.
-$\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)$

Accepted Answer

A)  $\frac { 7 } { 8 }$ 
B)  $\frac { 1 } { 8 }$ 
C)  $\frac { 5 } { 8 }$ 
D)  $\frac { 3 } { 8 }$ 
A)  $\frac { 7 } { 8 }$ 
B)  $\frac { 1 } { 8 }$ 
C)  $\frac { 5 } { 8 }$ 
D)  $\frac { 3 } { 8 }$

Question 18

Solve the equation for exact solutions.
-$$\sin ^ { - 1 } x + \cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right) = \pi$$

Accepted Answer

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

Question 19

Verify that the equation is an identity.
-$$\sin \left( x + \frac { \pi } { 2 } \right) = \cos x$$

Accepted Answer

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

Question 20

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\frac { \sin 28 ^ { \circ } } { 1 + \cos 28 ^ { \circ } }$$

Accepted Answer

A)  $\cos 14 ^ { \circ }$ 
B)  $\tan 14 ^ { \circ }$ 
C)  $\sin 14 ^ { \circ }$ 
D)  $\cot 14 ^ { \circ }$ 
A)  $\cos 14 ^ { \circ }$ 
B)  $\tan 14 ^ { \circ }$ 
C)  $\sin 14 ^ { \circ }$ 
D)  $\cot 14 ^ { \circ }$

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

Use a calculator to give the value to the nearest degree. - $\theta = \sin ^ { - 1 } ( - 0.4848 )$

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

Use the cofunction identities to find an angle $\theta$ that makes the statement true. - $\cot 5 \theta = \tan 4 \theta$

Use a sum or difference identity to find the exact value. - $\frac { \tan 100 ^ { \circ } - \tan ( - 20 ) ^ { \circ } } { 1 + \tan 100 ^ { \circ } \tan ( - 20 ) ^ { \circ } }$

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

Complete the sentence so the result is an identity. Let x be any real number. - $\cos x = ( \cot x ) ( \square )$

Solve the problem. - $\sec ^ { 2 } \frac { \mathrm { u } } { 2 } = \frac { 2 \sec \mathrm { u } } { \sec \mathrm { u } + 1 }$

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

Use identities to find the indicated value for each angle measure. - $\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Give the exact value of the expression. - $\csc \left( \csc ^ { - 1 } - 2 \right)$

Give the degree measure of $\theta$ - $\theta = \sin ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)$

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

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

Solve the equation in the interval $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Give solutions to the nearest tenth, if necessary. - $\sin ^ { 2 } \theta - \sin \theta - 12 = 0$

Give the degree measure of $\theta$ - $\theta = \arcsin ( - 1 )$

Give the exact value of the expression. - $\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)$

Solve the equation for exact solutions. - $\sin ^ { - 1 } x + \cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right) = \pi$

Verify that the equation is an identity. - $\sin \left( x + \frac { \pi } { 2 } \right) = \cos x$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { \sin 28 ^ { \circ } } { 1 + \cos 28 ^ { \circ } }$

Review of Basic Concepts

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

Filters

Exam 8: Trigonometric Identities and Equations

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

Use a calculator to give the value to the nearest degree. - θ=sin⁡−1(−0.4848)\theta = \sin ^ { - 1 } ( - 0.4848 )θ=sin−1(−0.4848)

Perform the transformation. -Write csc⁡x\csc xcscx in terms of sec⁡x\sec xsecx .

Use the cofunction identities to find an angle θ\thetaθ that makes the statement true. - cot⁡5θ=tan⁡4θ\cot 5 \theta = \tan 4 \thetacot5θ=tan4θ

Use a sum or difference identity to find the exact value. - tan⁡100∘−tan⁡(−20)∘1+tan⁡100∘tan⁡(−20)∘\frac { \tan 100 ^ { \circ } - \tan ( - 20 ) ^ { \circ } } { 1 + \tan 100 ^ { \circ } \tan ( - 20 ) ^ { \circ } }1+tan100∘tan(−20)∘tan100∘−tan(−20)∘​

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - sin⁡x2cos⁡x2=sin⁡x2\sin \frac { x } { 2 } \cos \frac { x } { 2 } = \frac { \sin x } { 2 }sin2x​cos2x​=2sinx​

Complete the sentence so the result is an identity. Let x be any real number. - cos⁡x=(cot⁡x)(□)\cos x = ( \cot x ) ( \square )cosx=(cotx)(□)

Solve the problem. - sec⁡2u2=2sec⁡usec⁡u+1\sec ^ { 2 } \frac { \mathrm { u } } { 2 } = \frac { 2 \sec \mathrm { u } } { \sec \mathrm { u } + 1 }sec22u​=secu+12secu​

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

Use identities to find the indicated value for each angle measure. - sin⁡θ=2107,tan⁡θ<0\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quadsinθ=7210​​,tanθ<0 Find sin⁡(2θ)\sin ( 2 \theta )sin(2θ)

Give the exact value of the expression. - csc⁡(csc⁡−1−2)\csc \left( \csc ^ { - 1 } - 2 \right)csc(csc−1−2)

Give the degree measure of θ\thetaθ - θ=sin⁡−1(22)\theta = \sin ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)θ=sin−1(22​​)

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

Verify that each equation is an identity. - (1−cos⁡x)(1+cos⁡x)=sin⁡2x( 1 - \cos x ) ( 1 + \cos x ) = \sin ^ { 2 } x(1−cosx)(1+cosx)=sin2x

Solve the equation in the interval [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) ). Give solutions to the nearest tenth, if necessary. - sin⁡2θ−sin⁡θ−12=0\sin ^ { 2 } \theta - \sin \theta - 12 = 0sin2θ−sinθ−12=0

Give the degree measure of θ\thetaθ - θ=arcsin⁡(−1)\theta = \arcsin ( - 1 )θ=arcsin(−1)

Give the exact value of the expression. - cos⁡(2arcsin⁡14)\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)cos(2arcsin41​)

Solve the equation for exact solutions. - sin⁡−1x+cos⁡−1(−12)=π\sin ^ { - 1 } x + \cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right) = \pisin−1x+cos−1(−21​)=π

Verify that the equation is an identity. - sin⁡(x+π2)=cos⁡x\sin \left( x + \frac { \pi } { 2 } \right) = \cos xsin(x+2π​)=cosx

Use an identity to write the expression as a single trigonometric function or as a single number. - sin⁡28∘1+cos⁡28∘\frac { \sin 28 ^ { \circ } } { 1 + \cos 28 ^ { \circ } }1+cos28∘sin28∘​