Question 1

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

Accepted Answer

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

Question 2

Use identities to write the expression as a single function of x or θ.
-$\tan \left( x - \frac { \pi } { 3 } \right)$

Accepted Answer

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

Question 3

Solve the equation for exact solutions over the interval [0, 2π).
-$\tan x + \sec x = 1$

Accepted Answer

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

Question 4

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.
-$\tan x ( \tan x - 2 ) = 7$

Accepted Answer

A)  $\{ 1.3153 + \mathrm { n } \pi , 5.2129 + \mathrm { n } \pi \}$ 
B)  $\{ 1.2608 + \mathrm { n } \pi , 2.2991 + \mathrm { n } \pi \}$ 
C)  $\{ 1.3153 + \mathrm { n } \pi , 2.0713 + \mathrm { n } \pi \}$ 
D)  $\{ 1.2608 + \mathrm { n } \pi , 5 + \mathrm { n } \pi \}$ 
A)  $\{ 1.3153 + \mathrm { n } \pi , 5.2129 + \mathrm { n } \pi \}$ 
B)  $\{ 1.2608 + \mathrm { n } \pi , 2.2991 + \mathrm { n } \pi \}$ 
C)  $\{ 1.3153 + \mathrm { n } \pi , 2.0713 + \mathrm { n } \pi \}$ 
D)  $\{ 1.2608 + \mathrm { n } \pi , 5 + \mathrm { n } \pi \}$

Question 5

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

Accepted Answer

\[\begin{array} { l } 
\frac { \sin x +

Question 6

Perform the indicated operations and simplify the result so there are no quotients.
-$2 \tan \theta - ( 1 + \tan \theta ) ^ { 2 }$

Accepted Answer

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

Question 7

Solve the problem.
-Suppose that for an electrical appliance, voltage is given by $\mathrm { V } = 189 \sin 120 \pi \mathrm { t }$ and amperage by $\mathrm { I } = 1.07 \sin 120 \pi \mathrm { t }$, where $\mathrm { t }$ is time in seconds. The wattage is given by $\mathrm { W } = \mathrm { VI }$. Find the maximum wattage used by the appliance.

Accepted Answer

A)  $404.46$ 
B)  $202.23$ 
C)  $101.115$ 
D)  189 
A)  $404.46$ 
B)  $202.23$ 
C)  $101.115$ 
D)  189

Question 8

Solve the equation for exact solutions.
-$6 \arcsin x = \pi$

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Use the fundamental identities to simplify the expression.
-$$\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta$$

Accepted Answer

A)  $\cos ^ { 3 } \theta$ 
B)  $\sin \theta$ 
C)  $\tan ^ { 2 } \theta$ 
D)  $\sec ^ { 2 } \theta$ 
A)  $\cos ^ { 3 } \theta$ 
B)  $\sin \theta$ 
C)  $\tan ^ { 2 } \theta$ 
D)  $\sec ^ { 2 } \theta$

Question 11

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

Accepted Answer

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

Question 12

Provide an appropriate response.
-The equation $\sin 2 x - \cos 4 x = 3$ has no solution in the interval $[ 0,2 \pi )$. Explain what this tells you about the graph of $y = \sin 2 x - \cos 4 x - 3$.

Accepted Answer

The graph

Question 13

Give the exact value of the expression.
-$\csc \left( \csc ^ { - 1 } \sqrt { 3 } \right)$

Accepted Answer

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

Question 14

Rewrite the following as a product of trigonometric functions.
-$\sin 2 ^ { \circ } + \sin 32 ^ { \circ }$

Accepted Answer

A)  $2 \sin 17 ^ { \circ } \sin \left( - 15 ^ { \circ } \right)$ 
B)  $2 \sin 17 ^ { \circ } \cos \left( - 15 ^ { \circ } \right)$ 
C)  $2 \cos 17 ^ { \circ } \cos \left( - 15 ^ { \circ } \right)$ 
D)  $2 \cos 17 ^ { \circ } \sin \left( - 15 ^ { \circ } \right)$ 
A)  $2 \sin 17 ^ { \circ } \sin \left( - 15 ^ { \circ } \right)$ 
B)  $2 \sin 17 ^ { \circ } \cos \left( - 15 ^ { \circ } \right)$ 
C)  $2 \cos 17 ^ { \circ } \cos \left( - 15 ^ { \circ } \right)$ 
D)  $2 \cos 17 ^ { \circ } \sin \left( - 15 ^ { \circ } \right)$

Question 15

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

Accepted Answer

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

Question 16

Solve the equation for solutions over the interval [0, 2π). Write solutions as exact values or to four decimal places, as
appropriate.
-$$\sin \frac { x } { 2 } + \cos \frac { x } { 2 } = \sqrt { 2 }$$

Accepted Answer

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

Question 17

Find the exact value of the expression using the provided information.
-Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos s = \frac { \sqrt { 5 } } { 5 }$, with $\mathrm { s }$ in quadrant $\mathrm { I }$, and $\sin \mathrm { t } = - \frac { \sqrt { 3 } } { 2 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $\frac { \sqrt { 5 } + 2 \sqrt { 15 } } { 10 }$ 
B)  $\frac { \sqrt { 5 } - \sqrt { 15 } } { 9 }$ 
C)  $- \frac { \sqrt { 5 } } { 10 }$ 
D)  $\frac { \sqrt { 5 } - 2 \sqrt { 15 } } { 10 }$ 
A)  $\frac { \sqrt { 5 } + 2 \sqrt { 15 } } { 10 }$ 
B)  $\frac { \sqrt { 5 } - \sqrt { 15 } } { 9 }$ 
C)  $- \frac { \sqrt { 5 } } { 10 }$ 
D)  $\frac { \sqrt { 5 } - 2 \sqrt { 15 } } { 10 }$

Question 18

Use Identities to find the exact value.
-$\cos \left( - 75 ^ { \circ } \right)$

Accepted Answer

A)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
B)  $- \sqrt { 6 } - \sqrt { 2 }$ 
C)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
D)  $\sqrt { 2 } - \sqrt { 6 }$ 
A)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
B)  $- \sqrt { 6 } - \sqrt { 2 }$ 
C)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
D)  $\sqrt { 2 } - \sqrt { 6 }$

Question 19

Use the cofunction identities to find an angle θ that makes the statement true.
-$\tan \theta = \cot \left( 30 ^ { \circ } + 5 \theta \right)$

Accepted Answer

A)  $\theta = 75 ^ { \circ }$ 
B)  $\theta = 10 ^ { \circ }$ 
C)  $\theta = 16 ^ { \circ }$ 
D)  $\theta = 6 ^ { \circ }$ 
A)  $\theta = 75 ^ { \circ }$ 
B)  $\theta = 10 ^ { \circ }$ 
C)  $\theta = 16 ^ { \circ }$ 
D)  $\theta = 6 ^ { \circ }$

Question 20

Solve the equation in the interval [0°, 360°). 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)  $\left\{ 45 ^ { \circ } , 135 ^ { \circ } \right\}$ 
D)  $\left\{ 45 ^ { \circ } \right\}$ 
A)  $\left\{ 45 ^ { \circ } , 315 ^ { \circ } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ 45 ^ { \circ } , 135 ^ { \circ } \right\}$ 
D)  $\left\{ 45 ^ { \circ } \right\}$

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

Use identities to write the expression as a single function of x or θ. - $\tan \left( x - \frac { \pi } { 3 } \right)$

Solve the equation for exact solutions over the interval [0, 2π). - $\tan x + \sec x = 1$

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. - $\tan x ( \tan x - 2 ) = 7$

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

Perform the indicated operations and simplify the result so there are no quotients. - $2 \tan \theta - ( 1 + \tan \theta ) ^ { 2 }$

Solve the equation for exact solutions. - $6 \arcsin x = \pi$

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

Use the fundamental identities to simplify the expression. - $\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta$

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

Provide an appropriate response. -The equation $\sin 2 x - \cos 4 x = 3$ has no solution in the interval $[ 0,2 \pi )$ . Explain what this tells you about the graph of $y = \sin 2 x - \cos 4 x - 3$ .

Give the exact value of the expression. - $\csc \left( \csc ^ { - 1 } \sqrt { 3 } \right)$

Rewrite the following as a product of trigonometric functions. - $\sin 2 ^ { \circ } + \sin 32 ^ { \circ }$

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

Solve the equation for solutions over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate. - $\sin \frac { x } { 2 } + \cos \frac { x } { 2 } = \sqrt { 2 }$

Use Identities to find the exact value. - $\cos \left( - 75 ^ { \circ } \right)$

Use the cofunction identities to find an angle θ that makes the statement true. - $\tan \theta = \cot \left( 30 ^ { \circ } + 5 \theta \right)$

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

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

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Exam 7: Trigonometric Identities and Equations

Find the exact value by using a half-angle identity. - tan⁡75∘\tan 75 ^ { \circ }tan75∘

Use identities to write the expression as a single function of x or θ. - tan⁡(x−π3)\tan \left( x - \frac { \pi } { 3 } \right)tan(x−3π​)

Solve the equation for exact solutions over the interval [0, 2π). - tan⁡x+sec⁡x=1\tan x + \sec x = 1tanx+secx=1

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. - tan⁡x(tan⁡x−2)=7\tan x ( \tan x - 2 ) = 7tanx(tanx−2)=7

Verify that each equation is an identity. - sin⁡x+cos⁡xsin⁡x−cos⁡x=1+2sin⁡xcos⁡x2sin⁡2x−1\frac { \sin x + \cos x } { \sin x - \cos x } = \frac { 1 + 2 \sin x \cos x } { 2 \sin ^ { 2 } x - 1 }sinx−cosxsinx+cosx​=2sin2x−11+2sinxcosx​

Perform the indicated operations and simplify the result so there are no quotients. - 2tan⁡θ−(1+tan⁡θ)22 \tan \theta - ( 1 + \tan \theta ) ^ { 2 }2tanθ−(1+tanθ)2

Solve the equation for exact solutions. - 6arcsin⁡x=π6 \arcsin x = \pi6arcsinx=π

Find the exact value of the real number y. - y=arctan⁡(1)\mathrm { y } = \arctan ( 1 )y=arctan(1)

Use the fundamental identities to simplify the expression. - sin⁡2θ+tan⁡2θ+cos⁡2θ\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \thetasin2θ+tan2θ+cos2θ

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)

Provide an appropriate response. -The equation sin⁡2x−cos⁡4x=3\sin 2 x - \cos 4 x = 3sin2x−cos4x=3 has no solution in the interval [0,2π)[ 0,2 \pi )[0,2π) . Explain what this tells you about the graph of y=sin⁡2x−cos⁡4x−3y = \sin 2 x - \cos 4 x - 3y=sin2x−cos4x−3 .

Give the exact value of the expression. - csc⁡(csc⁡−13)\csc \left( \csc ^ { - 1 } \sqrt { 3 } \right)csc(csc−13​)

Rewrite the following as a product of trigonometric functions. - sin⁡2∘+sin⁡32∘\sin 2 ^ { \circ } + \sin 32 ^ { \circ }sin2∘+sin32∘

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

Solve the equation for solutions over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate. - sin⁡x2+cos⁡x2=2\sin \frac { x } { 2 } + \cos \frac { x } { 2 } = \sqrt { 2 }sin2x​+cos2x​=2​

Use Identities to find the exact value. - cos⁡(−75∘)\cos \left( - 75 ^ { \circ } \right)cos(−75∘)

Use the cofunction identities to find an angle θ that makes the statement true. - tan⁡θ=cot⁡(30∘+5θ)\tan \theta = \cot \left( 30 ^ { \circ } + 5 \theta \right)tanθ=cot(30∘+5θ)

Solve the equation in the interval [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