Question 1

Use a sum or difference identity to find the exact value.
-$\tan \frac { 11 \pi } { 12 }$

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

Solve the equation for x, where x is restricted to the given interval.
-$3 y = \tan 6 x$, for $x$ in $\left( - \frac { \pi } { 12 } , \frac { \pi } { 12 } \right)$

Accepted Answer

A)  $x = \frac { 1 } { 2 } \arctan y$ 
B)  $x = 2 \arctan y$ 
C)  $x = 6 \arctan \frac { y } { 3 }$ 
D)  $x = \frac { 1 } { 6 } \arctan 3 y$ 
A)  $x = \frac { 1 } { 2 } \arctan y$ 
B)  $x = 2 \arctan y$ 
C)  $x = 6 \arctan \frac { y } { 3 }$ 
D)  $x = \frac { 1 } { 6 } \arctan 3 y$

Question 3

Use Identities to find the exact value.
-$\cos \left( - \frac { 7 \pi } { 12 } \right)$

Accepted Answer

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

Question 4

Use identities to find the indicated value for each angle measure.
-$\cos \theta = - \frac { 5 } { 13 } , \frac { \pi } { 2 } < \theta < \pi \quad$ Find $\cos ( 2 \theta )$

Accepted Answer

A)  $- \frac { 120 } { 169 }$ 
B)  $- \frac { 119 } { 169 }$ 
C)  $\frac { 120 } { 169 }$ 
D)  $\frac { 119 } { 169 }$ 
A)  $- \frac { 120 } { 169 }$ 
B)  $- \frac { 119 } { 169 }$ 
C)  $\frac { 120 } { 169 }$ 
D)  $\frac { 119 } { 169 }$

Question 5

Solve the problem.
-The optimal angle of elevation $\theta$ that a shot-putter should aim for in order to throw the greatest distance depends on the velocity $\mathrm { v }$ of the throw and the initial height $\mathrm { h }$ of the shot. One model for $\theta$ that achieves this maximum distance is $\theta = \sin ^ { - 1 } \left( \sqrt { \frac { \mathrm { v } ^ { 2 } } { 2 \mathrm { v } ^ { 2 } + 64 \mathrm {~h} } } \right)$. Suppose a shot-putter can consistently release the steel ball with velocity of 34 feet per second from an initial height $h$ of $6.1$ feet. What angle, to the nearest degree, will maximize the distance?

Accepted Answer

A)  $64 ^ { \circ }$ 
B)  $41 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $60 ^ { \circ }$ 
A)  $64 ^ { \circ }$ 
B)  $41 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $60 ^ { \circ }$

Question 6

Use identities to find the indicated value for each angle measure.
-$\cos 2 \theta = \frac { 4 } { 5 }$ and $\theta$ terminates in quadrant I Find $\sin \theta$.

Accepted Answer

A)  $\sin \theta = - \frac { \sqrt { 10 } } { 10 }$ 
B)  $\sin \theta = 0$ 
C)  $\sin \theta = \frac { \sqrt { 10 } } { 10 }$ 
D)  $\sin \theta = \sqrt { 10 }$ 
A)  $\sin \theta = - \frac { \sqrt { 10 } } { 10 }$ 
B)  $\sin \theta = 0$ 
C)  $\sin \theta = \frac { \sqrt { 10 } } { 10 }$ 
D)  $\sin \theta = \sqrt { 10 }$

Question 7

Use a sum or difference identity to find the exact value.
-$\sin 215 ^ { \circ } \cos 95 ^ { \circ } - \cos 215 ^ { \circ } \sin 95 ^ { \circ }$

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 43 } { 12 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $- \frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 43 } { 12 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 8

Use identities to write the expression as a single function of x or θ.
-$\sin \left( 360 ^ { \circ } + x \right)$

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $\sin x$ 
C)  $- \cos x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
A)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $\sin x$ 
C)  $- \cos x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$

Question 9

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$\sin 2 \theta = - \frac { 1 } { 2 }$

Accepted Answer

A)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$ 
B)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 90 ^ { \circ } , 150 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$ 
A)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$ 
B)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 90 ^ { \circ } , 150 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$

Question 10

Find the exact value by using a half-angle identity.
-$\cos \frac { 5 \pi } { 12 }$

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 11

Solve the equation for x, where x is restricted to the given interval.
-$y = 3 \tan 2 x - 1$, for $x$ in $\left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$

Accepted Answer

A)  $x = \frac { 1 } { 2 } \arctan \frac { y + 1 } { 3 }$ 
B)  $x = 2 \arctan \frac { y + 1 } { 3 }$ 
C)  $x = \frac { 1 } { 2 } \arctan ( 3 y + 3 )$ 
D)  $x = 2 \arctan ( 3 y + 3 )$ 
A)  $x = \frac { 1 } { 2 } \arctan \frac { y + 1 } { 3 }$ 
B)  $x = 2 \arctan \frac { y + 1 } { 3 }$ 
C)  $x = \frac { 1 } { 2 } \arctan ( 3 y + 3 )$ 
D)  $x = 2 \arctan ( 3 y + 3 )$

Question 12

Use a calculator to give the value to the nearest degree.
-$\theta = \sin ^ { - 1 } ( 0.6561 )$

Accepted Answer

A)  $41 ^ { \circ }$ 
B)  $44 ^ { \circ }$ 
C)  $- 41 ^ { \circ }$ 
D)  $39 ^ { \circ }$ 
A)  $41 ^ { \circ }$ 
B)  $44 ^ { \circ }$ 
C)  $- 41 ^ { \circ }$ 
D)  $39 ^ { \circ }$

Question 13

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

Accepted Answer

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

Question 14

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-

Accepted Answer

A)  $f ( - x ) = - f ( x )$ 
B)  $f ( - x ) = f ( x )$ 
A)  $f ( - x ) = - f ( x )$ 
B)  $f ( - x ) = f ( x )$

Question 15

Use identities to write the expression as a single function of x or θ.
-$\sin ( x + 2 \pi )$

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $- \cos x$ 
C)  $\sin x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
A)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $- \cos x$ 
C)  $\sin x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$

Question 16

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

Accepted Answer

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

Question 17

Solve the problem.
-The weekly sales in thousands of items of a product has a seasonal sales record approximated by $\mathrm { n } = 71.69 + 18.3 \sin \frac { \pi \mathrm { t } } { 24 }$ ( $\mathrm { t }$ is time in weeks with $\mathrm { t } = 1$ referring to the first week in the year). During which week(s) will the sales equal 80,840 items?

Accepted Answer

A)  week 4 , week 20 , and week 52 
B)  week 21 and week 30 
C)  week 30 and week 47 
D)  week 4 and week 47 
A)  week 4 , week 20 , and week 52 
B)  week 21 and week 30 
C)  week 30 and week 47 
D)  week 4 and week 47

Question 18

Use a sum or difference identity to find the exact value.
-$\sin 260 ^ { \circ } \cos 20 ^ { \circ } - \cos 260 ^ { \circ } \sin 20 ^ { \circ }$

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $- \frac { 13 } { 3 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $- \frac { 1 } { 2 }$ 
B)  $- \frac { 13 } { 3 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 19

Use a sum or difference identity to find the exact value.
-$$\sin \frac { \pi } { 15 } \cos \frac { 4 \pi } { 15 } + \cos \frac { \pi } { 15 } \sin \frac { 4 \pi } { 15 }$$

Accepted Answer

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

Question 20

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$\mathrm { y } = \tan ^ { - 1 } ( 0.5774 )$$

Accepted Answer

A)  $0.5236361$ 
B)  $1.0476200$ 
C)  $0.5587302$ 
D)  $0.4714286$ 
A)  $0.5236361$ 
B)  $1.0476200$ 
C)  $0.5587302$ 
D)  $0.4714286$

Use a sum or difference identity to find the exact value. - $\tan \frac { 11 \pi } { 12 }$

Solve the equation for x, where x is restricted to the given interval. - $3 y = \tan 6 x$ , for $x$ in $\left( - \frac { \pi } { 12 } , \frac { \pi } { 12 } \right)$

Use Identities to find the exact value. - $\cos \left( - \frac { 7 \pi } { 12 } \right)$

Use identities to find the indicated value for each angle measure. - $\cos \theta = - \frac { 5 } { 13 } , \frac { \pi } { 2 } < \theta < \pi \quad$ Find $\cos ( 2 \theta )$

Use identities to find the indicated value for each angle measure. - $\cos 2 \theta = \frac { 4 } { 5 }$ and $\theta$ terminates in quadrant I Find $\sin \theta$ .

Use a sum or difference identity to find the exact value. - $\sin 215 ^ { \circ } \cos 95 ^ { \circ } - \cos 215 ^ { \circ } \sin 95 ^ { \circ }$

Use identities to write the expression as a single function of x or θ. - $\sin \left( 360 ^ { \circ } + x \right)$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\sin 2 \theta = - \frac { 1 } { 2 }$

Find the exact value by using a half-angle identity. - $\cos \frac { 5 \pi } { 12 }$

Solve the equation for x, where x is restricted to the given interval. - $y = 3 \tan 2 x - 1$ , for $x$ in $\left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$

Use a calculator to give the value to the nearest degree. - $\theta = \sin ^ { - 1 } ( 0.6561 )$

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

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Use identities to write the expression as a single function of x or θ. - $\sin ( x + 2 \pi )$

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

Use a sum or difference identity to find the exact value. - $\sin 260 ^ { \circ } \cos 20 ^ { \circ } - \cos 260 ^ { \circ } \sin 20 ^ { \circ }$

Use a sum or difference identity to find the exact value. - $\sin \frac { \pi } { 15 } \cos \frac { 4 \pi } { 15 } + \cos \frac { \pi } { 15 } \sin \frac { 4 \pi } { 15 }$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \tan ^ { - 1 } ( 0.5774 )$

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

Review of Basic Concepts

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Exam 7: Trigonometric Identities and Equations

Use a sum or difference identity to find the exact value. - tan⁡11π12\tan \frac { 11 \pi } { 12 }tan1211π​

Solve the equation for x, where x is restricted to the given interval. - 3y=tan⁡6x3 y = \tan 6 x3y=tan6x , for xxx in (−π12,π12)\left( - \frac { \pi } { 12 } , \frac { \pi } { 12 } \right)(−12π​,12π​)

Use Identities to find the exact value. - cos⁡(−7π12)\cos \left( - \frac { 7 \pi } { 12 } \right)cos(−127π​)

Use identities to find the indicated value for each angle measure. - cos⁡θ=−513,π2<θ<π\cos \theta = - \frac { 5 } { 13 } , \frac { \pi } { 2 } < \theta < \pi \quadcosθ=−135​,2π​<θ<π Find cos⁡(2θ)\cos ( 2 \theta )cos(2θ)

Use identities to find the indicated value for each angle measure. - cos⁡2θ=45\cos 2 \theta = \frac { 4 } { 5 }cos2θ=54​ and θ\thetaθ terminates in quadrant I Find sin⁡θ\sin \thetasinθ .

Use a sum or difference identity to find the exact value. - sin⁡215∘cos⁡95∘−cos⁡215∘sin⁡95∘\sin 215 ^ { \circ } \cos 95 ^ { \circ } - \cos 215 ^ { \circ } \sin 95 ^ { \circ }sin215∘cos95∘−cos215∘sin95∘

Use identities to write the expression as a single function of x or θ. - sin⁡(360∘+x)\sin \left( 360 ^ { \circ } + x \right)sin(360∘+x)

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - sin⁡2θ=−12\sin 2 \theta = - \frac { 1 } { 2 }sin2θ=−21​

Find the exact value by using a half-angle identity. - cos⁡5π12\cos \frac { 5 \pi } { 12 }cos125π​

Solve the equation for x, where x is restricted to the given interval. - y=3tan⁡2x−1y = 3 \tan 2 x - 1y=3tan2x−1 , for xxx in (−π4,π4)\left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)(−4π​,4π​)

Use a calculator to give the value to the nearest degree. - θ=sin⁡−1(0.6561)\theta = \sin ^ { - 1 } ( 0.6561 )θ=sin−1(0.6561)

Use the fundamental identities to simplify the expression. - cos⁡2θsin⁡2θ+csc⁡θsin⁡θ\frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta } + \csc \theta \sin \thetasin2θcos2θ​+cscθsinθ