Question 1

Rewrite the following as a product of trigonometric functions.
-$\sin 3 ^ { \circ } + \sin 37 ^ { \circ }$

Accepted Answer

A)  $2 \cos 20 ^ { \circ } \sin \left( - 17 ^ { \circ } \right)$ 
B)  $2 \sin 20 ^ { \circ } \sin \left( - 17 ^ { \circ } \right)$ 
C)  $2 \cos 20 ^ { \circ } \cos \left( - 17 ^ { \circ } \right)$ 
D)  $2 \sin 20 ^ { \circ } \cos \left( - 17 ^ { \circ } \right)$ 
A)  $2 \cos 20 ^ { \circ } \sin \left( - 17 ^ { \circ } \right)$ 
B)  $2 \sin 20 ^ { \circ } \sin \left( - 17 ^ { \circ } \right)$ 
C)  $2 \cos 20 ^ { \circ } \cos \left( - 17 ^ { \circ } \right)$ 
D)  $2 \sin 20 ^ { \circ } \cos \left( - 17 ^ { \circ } \right)$

Question 2

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

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 3

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

Question 4

Provide an appropriate response.
-$\tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1 = 0$ The equation has no solution in the interval $[ 0,2 \pi )$ Explain what this
tells you about the graph of $y = \tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1$

Accepted Answer

The graph does not c

Question 5

Solve the equation (x in radians and $$\boldsymbol { \theta }$$ 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.
-$$\cos ^ { 2 } x + 2 \cos x = - 1$$

Accepted Answer

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

Question 6

Complete the sentence so the result is an identity. Let x be any real number.
-$$\frac { \sin x } { \tan x } =$$

Accepted Answer

A)  $\sec x$ 
B)  $\cos x$ 
C)  $\cot x$ 
D)  $\csc x$ 
A)  $\sec x$ 
B)  $\cos x$ 
C)  $\cot x$ 
D)  $\csc x$

Question 7

The statement tan-1(tan x) = x for all real numbers in the interval
-On the given graph of y $$y = \sec x$$ sec x sketch the graph of y $$y = \sec ^ { - 1 } x$$ x as defined in the text. Give 
the domain and the range.

Accepted Answer

The answer of The statement tan-1(tan x) = x for...

Question 8

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

Accepted Answer

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

Question 9

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-Consider the formula $\mathrm { T } = 591 - 76 \cos 2 \theta$, where $\theta$ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of $\theta$ for which the value of $T$ will be 640 ?

Accepted Answer

A)  $65.07 ^ { \circ }$ 
B)  $62.07 ^ { \circ }$ 
C)  $133.14 ^ { \circ }$ 
D)  $130.14 ^ { \circ }$ 
A)  $65.07 ^ { \circ }$ 
B)  $62.07 ^ { \circ }$ 
C)  $133.14 ^ { \circ }$ 
D)  $130.14 ^ { \circ }$

Question 10

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

Accepted Answer

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

Question 11

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-The position of a weight on a spring relative to the point of equilibrium is given by $y = 4 \cos 3 t - 8 \sin 3 t$
where the arguments are in radians and $\mathrm { t }$ is in seconds. Find the smallest value of $\mathrm { t }$ for which the we at the point of equilibrium $( y = 0 )$. Give your answer to the nearest hundredth.

Accepted Answer

A)  $0.15$ 
B)  $0.44$ 
C)  $0.37$ 
D)  $0.26$ 
A)  $0.15$ 
B)  $0.44$ 
C)  $0.37$ 
D)  $0.26$

Question 12

Decide whether the expression is or is not an identity.
-$$\sqrt { 1 - \cos ^ { 2 } x } = 1 - \cos x$$

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\frac { 2 \tan 15 ^ { \circ } } { 1 - \tan ^ { 2 } 15 ^ { \circ } }$$

Accepted Answer

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

Question 15

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$ if $\tan \theta = 3$ and $\sin \theta < 0$

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

Solve the equation in the interval $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$ ). Give solutions to the nearest tenth, if necessary.
-$3 \sin ^ { 2 } \theta - \sin \theta - 4 = 0$

Accepted Answer

A)  $\left\{ 90 ^ { \circ } \right\}$ 
B)  $\left\{ 180 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } \right\}$ 
D)  $\left\{ 270 ^ { \circ } \right\}$ 
A)  $\left\{ 90 ^ { \circ } \right\}$ 
B)  $\left\{ 180 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } \right\}$ 
D)  $\left\{ 270 ^ { \circ } \right\}$

Question 18

Find the exact value of the real number y.
-$$y = \csc ^ { - 1 } ( - 1 )$$

Accepted Answer

A)  $\pi$ 
B)  $2 \pi$ 
C)  $- \frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 4 }$ 
A)  $\pi$ 
B)  $2 \pi$ 
C)  $- \frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 4 }$

Question 19

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-A rotating beacon is located a distance $\mathrm { d }$ from a long wall. The distance $\mathrm { d }$ is given by $\mathrm { d } = 2 \tan 2 \pi \mathrm { t }$, where $t$ is the time measured in seconds since the beacon started rotating. Solve the equation for $t$.

Accepted Answer

A)  $t = \frac { 1 } { 2 \pi } \arctan \frac { \mathrm { d } } { 2 }$ 
B)  $t = \frac { 1 } { 2 \pi } \tan \frac { d } { 2 }$ 
C)  $t = 2 \pi \arctan \frac { d } { 2 }$ 
D)  $t = 2 \pi \tan \frac { \mathrm { d } } { 2 }$ 
A)  $t = \frac { 1 } { 2 \pi } \arctan \frac { \mathrm { d } } { 2 }$ 
B)  $t = \frac { 1 } { 2 \pi } \tan \frac { d } { 2 }$ 
C)  $t = 2 \pi \arctan \frac { d } { 2 }$ 
D)  $t = 2 \pi \tan \frac { \mathrm { d } } { 2 }$

Question 20

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

Accepted Answer

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

Rewrite the following as a product of trigonometric functions. - $\sin 3 ^ { \circ } + \sin 37 ^ { \circ }$

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

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

Provide an appropriate response. - $\tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1 = 0$ The equation has no solution in the interval $[ 0,2 \pi )$ Explain what this tells you about the graph of $y = \tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1$

Solve the equation (x in radians and $\boldsymbol { \theta }$ 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. - $\cos ^ { 2 } x + 2 \cos x = - 1$

Complete the sentence so the result is an identity. Let x be any real number. - $\frac { \sin x } { \tan x } =$

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

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

Decide whether the expression is or is not an identity. - $\sqrt { 1 - \cos ^ { 2 } x } = 1 - \cos x$

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

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { 2 \tan 15 ^ { \circ } } { 1 - \tan ^ { 2 } 15 ^ { \circ } }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\tan \theta = 3$ and $\sin \theta < 0$

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

Solve the equation in the interval $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Give solutions to the nearest tenth, if necessary. - $3 \sin ^ { 2 } \theta - \sin \theta - 4 = 0$

Find the exact value of the real number y. - $y = \csc ^ { - 1 } ( - 1 )$

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

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

Rewrite the following as a product of trigonometric functions. - sin⁡3∘+sin⁡37∘\sin 3 ^ { \circ } + \sin 37 ^ { \circ }sin3∘+sin37∘

Use an identity to write the expression as a single trigonometric function or as a single number. - cos⁡24x−sin⁡24x\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 xcos24x−sin24x

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 .

Complete the sentence so the result is an identity. Let x be any real number. - sin⁡xtan⁡x=\frac { \sin x } { \tan x } =tanxsinx​=

The statement tan-1(tan x) = x for all real numbers in the interval -On the given graph of y y=sec⁡xy = \sec xy=secx sec x sketch the graph of y y=sec⁡−1xy = \sec ^ { - 1 } xy=sec−1x x as defined in the text. Give the domain and the range.

Perform the transformation. -Write cos⁡x\cos xcosx in terms of csc⁡x\csc xcscx .

Complete the sentence so the result is an identity. Let x be any real number. - sin⁡x=()(cos⁡x)\sin x = ( \quad ) ( \cos x )sinx=()(cosx)

Decide whether the expression is or is not an identity. - 1−cos⁡2x=1−cos⁡x\sqrt { 1 - \cos ^ { 2 } x } = 1 - \cos x1−cos2x​=1−cosx

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

Use an identity to write the expression as a single trigonometric function or as a single number. - 2tan⁡15∘1−tan⁡215∘\frac { 2 \tan 15 ^ { \circ } } { 1 - \tan ^ { 2 } 15 ^ { \circ } }1−tan215∘2tan15∘​

Use the fundamental identities to find the value of the trigonometric function. -Find cos⁡θ\cos \thetacosθ if tan⁡θ=3\tan \theta = 3tanθ=3 and sin⁡θ<0\sin \theta < 0sinθ<0

Perform the transformation. -Write sec⁡x\sec xsecx in terms of sin⁡x\sin xsinx

Solve the equation in the interval [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) ). Give solutions to the nearest tenth, if necessary. - 3sin⁡2θ−sin⁡θ−4=03 \sin ^ { 2 } \theta - \sin \theta - 4 = 03sin2θ−sinθ−4=0

Find the exact value of the real number y. - y=csc⁡−1(−1)y = \csc ^ { - 1 } ( - 1 )y=csc−1(−1)

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