Question 1

Use Identities to find the exact value.
-$\cos 255 ^ { \circ }$

Accepted Answer

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

Question 2

Solve the problem.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\theta$ is the vertex angle of the cone and $m$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { m }$ $( m > 1 )$. Find $\theta$ to the nearest degree if $m = 1.5$.

Accepted Answer

A)  $89 ^ { \circ }$ 
B)  $82 ^ { \circ }$ 
C)  $84 ^ { \circ }$ 
D)  $87 ^ { \circ }$ 
A)  $89 ^ { \circ }$ 
B)  $82 ^ { \circ }$ 
C)  $84 ^ { \circ }$ 
D)  $87 ^ { \circ }$

Question 3

Solve the equation for solutions in the interval [0, 2 π).
-$$\sqrt { 2 } \cos 2 x = 1$$

Accepted Answer

A)  $\varnothing$ 
B)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$ 
A)  $\varnothing$ 
B)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$

Question 4

Verify that each equation is an identity.
-$$\sec \beta + \tan \beta = \frac { \cos \beta } { 1 - \sin \beta }$$

Accepted Answer

\[\begin{array} { l } 
\sec \beta + \tan

Question 5

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.
-$$\cos ^ { 2 } x - \cos x = 0$$

Accepted Answer

A)  $\left\{ \frac { \pi } { 2 } + 2 n \pi \right\}$ 
B)  $\left\{ 2 \mathrm { n } \pi , \frac { \pi } { 2 } + \mathrm { n } \pi \right\}$ 
C)  $\{ \pi + 2 n \pi \}$ 
D)  $\left\{ \frac { 5 \pi } { 3 } + 2 n \pi \right\}$ 
A)  $\left\{ \frac { \pi } { 2 } + 2 n \pi \right\}$ 
B)  $\left\{ 2 \mathrm { n } \pi , \frac { \pi } { 2 } + \mathrm { n } \pi \right\}$ 
C)  $\{ \pi + 2 n \pi \}$ 
D)  $\left\{ \frac { 5 \pi } { 3 } + 2 n \pi \right\}$

Question 6

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta }$$

Accepted Answer

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

Question 7

Use the cofunction identities to find an angle θ that makes the statement true.
-$\tan \left( 2 \theta - 140 ^ { \circ } \right) = \cot \left( \theta + 5 ^ { \circ } \right)$

Accepted Answer

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

Question 8

Factor the trigonometric expression and simplify.
-$1 - \sin ^ { 3 } x$

Accepted Answer

A)  $( 1 - \sin x ) \left( 1 - 2 \sin x + \sin ^ { 2 } x \right)$ 
B)  $( 1 - \sin x ) ^ { 3 }$ 
C)  $( 1 - \sin x ) \left( \cos x + \sin ^ { 2 } x \right)$ 
D)  $( 1 - \sin x ) \left( 1 + \sin x + \sin ^ { 2 } x \right)$ 
A)  $( 1 - \sin x ) \left( 1 - 2 \sin x + \sin ^ { 2 } x \right)$ 
B)  $( 1 - \sin x ) ^ { 3 }$ 
C)  $( 1 - \sin x ) \left( \cos x + \sin ^ { 2 } x \right)$ 
D)  $( 1 - \sin x ) \left( 1 + \sin x + \sin ^ { 2 } x \right)$

Question 9

Write the following as an algebraic expression in u, u > 0.
-$\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 2 } } \right)$

Accepted Answer

A)  $u \sqrt { u ^ { 2 } + 2 }$ 
B)  $\frac { \sqrt { u ^ { 2 } + 2 } } { u ^ { 2 } + 2 }$ 
C)  $\frac { u \sqrt { \mathrm { u } ^ { 2 } + 2 } } { \mathrm { u } ^ { 2 } + 2 }$ 
D)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } - 2 } } { \mathrm { u } ^ { 2 } - 2 }$ 
A)  $u \sqrt { u ^ { 2 } + 2 }$ 
B)  $\frac { \sqrt { u ^ { 2 } + 2 } } { u ^ { 2 } + 2 }$ 
C)  $\frac { u \sqrt { \mathrm { u } ^ { 2 } + 2 } } { \mathrm { u } ^ { 2 } + 2 }$ 
D)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } - 2 } } { \mathrm { u } ^ { 2 } - 2 }$

Question 10

Graph the inverse circular function.
-$y=\tan ^{-1} x$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 11

Use a sum or difference identity to find the exact value.
-$$\frac { \tan \frac { 7 \pi } { 24 } - \tan \frac { \pi } { 8 } } { 1 + \tan \frac { 7 \pi } { 24 } \tan \frac { \pi } { 8 } }$$

Accepted Answer

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

Question 12

Verify that each equation is an identity.
-$$6 \csc ^ { 2 } \theta - 4 \cot ^ { 2 } \theta = 2 \csc ^ { 2 } \theta + 4$$

Accepted Answer

\[\begin{array} { l } 
6 \csc ^ { 2 } \t

Question 13

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

Accepted Answer

A)  $\left\{ 0 ^ { \circ } , 60 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
B)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } , 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } , 180 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$ 
A)  $\left\{ 0 ^ { \circ } , 60 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
B)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } , 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } , 180 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$

Question 14

Find the exact value of the expression using the provided information.
-Find $\cos ( s + t )$ given that $\cos s = - \frac { 1 } { 6 }$, with s in quadrant III, and $\cos t = - \frac { 3 } { 5 }$, with $t$ in quadrant III.

Accepted Answer

A)  $\frac { - 3 - 4 \sqrt { 35 } } { 30 }$ 
B)  $\frac { 3 + 4 \sqrt { 35 } } { 30 }$ 
C)  $\frac { 3 - 4 \sqrt { 35 } } { 30 }$ 
D)  $\frac { 3 - 4 \sqrt { 37 } } { 30 }$ 
A)  $\frac { - 3 - 4 \sqrt { 35 } } { 30 }$ 
B)  $\frac { 3 + 4 \sqrt { 35 } } { 30 }$ 
C)  $\frac { 3 - 4 \sqrt { 35 } } { 30 }$ 
D)  $\frac { 3 - 4 \sqrt { 37 } } { 30 }$

Question 15

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$\mathrm { y } = \cos ^ { - 1 } ( - 0.9397 )$$

Accepted Answer

A)  $2.7925484$ 
B)  $2.7412723$ 
C)  $2.8285740$ 
D)  $- 0.3492063$ 
A)  $2.7925484$ 
B)  $2.7412723$ 
C)  $2.8285740$ 
D)  $- 0.3492063$

Question 16

Solve the problem.
-It can be shown that if the angle of elevation from an observer to the top of an object is A and the angle of elevation $\mathrm { d } \mathrm { ft }$ closer is $\mathrm { B }$, then the distance from the object to the closest point of observation is given by
$$\mathrm { D } = \frac { \mathrm { d } \cot \mathrm { B } } { \cot \mathrm { A } - \cot \mathrm { B } } \mathrm { ft } \text {. }$$
Find $B$ if $D = 40 \mathrm { ft } , \mathrm { d } = 20 \mathrm { ft }$, and $A = 33 ^ { \circ }$. Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $44.25 ^ { \circ }$ 
B)  $33.00 ^ { \circ }$ 
C)  $62.83 ^ { \circ }$ 
D)  $17.99 ^ { \circ }$ 
A)  $44.25 ^ { \circ }$ 
B)  $33.00 ^ { \circ }$ 
C)  $62.83 ^ { \circ }$ 
D)  $17.99 ^ { \circ }$

Question 17

Use the fundamental identities to simplify the expression.
-$\sin \theta \cos \theta \sec \theta \csc \theta$

Accepted Answer

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

Question 18

Solve the equation for exact solutions.
-$\sin ^ { - 1 } \mathrm { x } + \tan ^ { - 1 } \mathrm { x } = 0$

Accepted Answer

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

Question 19

Verify that each equation is an identity.
-$$\cos ( 4 u ) = 2 \cos ^ { 2 } ( 2 u ) - 1$$

Accepted Answer

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

Question 20

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

Accepted Answer

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

Use Identities to find the exact value. - $\cos 255 ^ { \circ }$

Solve the equation for solutions in the interval [0, 2 π). - $\sqrt { 2 } \cos 2 x = 1$

Verify that each equation is an identity. - $\sec \beta + \tan \beta = \frac { \cos \beta } { 1 - \sin \beta }$

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. - $\cos ^ { 2 } x - \cos x = 0$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta }$

Use the cofunction identities to find an angle θ that makes the statement true. - $\tan \left( 2 \theta - 140 ^ { \circ } \right) = \cot \left( \theta + 5 ^ { \circ } \right)$

Factor the trigonometric expression and simplify. - $1 - \sin ^ { 3 } x$

Write the following as an algebraic expression in u, u > 0. - $\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 2 } } \right)$

Graph the inverse circular function. - $y=\tan ^{-1} x$ $Graph the inverse circular function. - y=\tan ^{-1} x$

Use a sum or difference identity to find the exact value. - $\frac { \tan \frac { 7 \pi } { 24 } - \tan \frac { \pi } { 8 } } { 1 + \tan \frac { 7 \pi } { 24 } \tan \frac { \pi } { 8 } }$

Verify that each equation is an identity. - $6 \csc ^ { 2 } \theta - 4 \cot ^ { 2 } \theta = 2 \csc ^ { 2 } \theta + 4$

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

Find the exact value of the expression using the provided information. -Find $\cos ( s + t )$ given that $\cos s = - \frac { 1 } { 6 }$ , with s in quadrant III, and $\cos t = - \frac { 3 } { 5 }$ , with $t$ in quadrant III.

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \cos ^ { - 1 } ( - 0.9397 )$

Use the fundamental identities to simplify the expression. - $\sin \theta \cos \theta \sec \theta \csc \theta$

Solve the equation for exact solutions. - $\sin ^ { - 1 } \mathrm { x } + \tan ^ { - 1 } \mathrm { x } = 0$

Verify that each equation is an identity. - $\cos ( 4 u ) = 2 \cos ^ { 2 } ( 2 u ) - 1$

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

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

Filters

Exam 7: Trigonometric Identities and Equations

Use Identities to find the exact value. - cos⁡255∘\cos 255 ^ { \circ }cos255∘

Solve the equation for solutions in the interval [0, 2 π). - 2cos⁡2x=1\sqrt { 2 } \cos 2 x = 12​cos2x=1

Verify that each equation is an identity. - sec⁡β+tan⁡β=cos⁡β1−sin⁡β\sec \beta + \tan \beta = \frac { \cos \beta } { 1 - \sin \beta }secβ+tanβ=1−sinβcosβ​

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. - cos⁡2x−cos⁡x=0\cos ^ { 2 } x - \cos x = 0cos2x−cosx=0

Perform the indicated operations and simplify the result so there are no quotients. - sin⁡θcos⁡θ+cos⁡θsin⁡θ\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta }cosθsinθ​+sinθcosθ​

Use the cofunction identities to find an angle θ that makes the statement true. - tan⁡(2θ−140∘)=cot⁡(θ+5∘)\tan \left( 2 \theta - 140 ^ { \circ } \right) = \cot \left( \theta + 5 ^ { \circ } \right)tan(2θ−140∘)=cot(θ+5∘)

Factor the trigonometric expression and simplify. - 1−sin⁡3x1 - \sin ^ { 3 } x1−sin3x

Write the following as an algebraic expression in u, u > 0. - sin⁡(arctan⁡u2)\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 2 } } \right)sin(arctan2​u​)

Graph the inverse circular function. - y=tan⁡−1xy=\tan ^{-1} xy=tan−1x

Use a sum or difference identity to find the exact value. - tan⁡7π24−tan⁡π81+tan⁡7π24tan⁡π8\frac { \tan \frac { 7 \pi } { 24 } - \tan \frac { \pi } { 8 } } { 1 + \tan \frac { 7 \pi } { 24 } \tan \frac { \pi } { 8 } }1+tan247π​tan8π​tan247π​−tan8π​​

Verify that each equation is an identity. - 6csc⁡2θ−4cot⁡2θ=2csc⁡2θ+46 \csc ^ { 2 } \theta - 4 \cot ^ { 2 } \theta = 2 \csc ^ { 2 } \theta + 46csc2θ−4cot2θ=2csc2θ+4

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - sin⁡2θ=−sin⁡θ\sin 2 \theta = - \sin \thetasin2θ=−sinθ

Find the exact value of the expression using the provided information. -Find cos⁡(s+t)\cos ( s + t )cos(s+t) given that cos⁡s=−16\cos s = - \frac { 1 } { 6 }coss=−61​ , with s in quadrant III, and cos⁡t=−35\cos t = - \frac { 3 } { 5 }cost=−53​ , with ttt in quadrant III.

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=cos⁡−1(−0.9397)\mathrm { y } = \cos ^ { - 1 } ( - 0.9397 )y=cos−1(−0.9397)

Use the fundamental identities to simplify the expression. - sin⁡θcos⁡θsec⁡θcsc⁡θ\sin \theta \cos \theta \sec \theta \csc \thetasinθcosθsecθcscθ

Solve the equation for exact solutions. - sin⁡−1x+tan⁡−1x=0\sin ^ { - 1 } \mathrm { x } + \tan ^ { - 1 } \mathrm { x } = 0sin−1x+tan−1x=0

Verify that each equation is an identity. - cos⁡(4u)=2cos⁡2(2u)−1\cos ( 4 u ) = 2 \cos ^ { 2 } ( 2 u ) - 1cos(4u)=2cos2(2u)−1

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