Question 1

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$\mathrm { y } = \sin ^ { - 1 } ( 0.2079 )$$

Accepted Answer

A)  $0.2444444$ 
B)  $0.1746032$ 
C)  $0.2619048$ 
D)  $0.2094276$ 
A)  $0.2444444$ 
B)  $0.1746032$ 
C)  $0.2619048$ 
D)  $0.2094276$

Question 2

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\tan ( \arcsin 2 )$

Accepted Answer

A)  $- 0.8172$ 
B)  Undefined 
C)  0 
D)  $0.8944$ 
A)  $- 0.8172$ 
B)  Undefined 
C)  0 
D)  $0.8944$

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\cos \theta = \frac { 1 } { 2 }$ and $\theta$ is in quadrant IV.

Accepted Answer

A)  $- \sqrt { 5 }$ 
B)  2 
C)  $- \frac { \sqrt { 3 } } { 3 }$ 
D)  $- \sqrt { 3 }$ 
A)  $- \sqrt { 5 }$ 
B)  2 
C)  $- \frac { \sqrt { 3 } } { 3 }$ 
D)  $- \sqrt { 3 }$

Question 6

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$\mathrm { y } = \tan ^ { - 1 } ( - 0.7002 )$$

Accepted Answer

A)  $- 0.6634921$ 
B)  $- 0.6108602$ 
C)  $- 0.5761905$ 
D)  $2.5317483$ 
A)  $- 0.6634921$ 
B)  $- 0.6108602$ 
C)  $- 0.5761905$ 
D)  $2.5317483$

Question 7

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 5 } } { 2 }$ and $\sin \theta < 0$.

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $- \sqrt { 5 }$ 
C)  - 2 
D)  2 
A)  $- \frac { 1 } { 2 }$ 
B)  $- \sqrt { 5 }$ 
C)  - 2 
D)  2

Question 8

Write in terms of the cofunction of a complementary angle.
-$$\cos \frac { \pi } { 20 }$$

Accepted Answer

A)  $\sin \frac { 19 \pi } { 20 }$ 
B)  $\sec \frac { 19 \pi } { 20 }$ 
C)  $\sin \frac { 9 \pi } { 20 }$ 
D)  $\sec \frac { 9 \pi } { 20 }$ 
A)  $\sin \frac { 19 \pi } { 20 }$ 
B)  $\sec \frac { 19 \pi } { 20 }$ 
C)  $\sin \frac { 9 \pi } { 20 }$ 
D)  $\sec \frac { 9 \pi } { 20 }$

Question 9

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

Accepted Answer

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

Question 10

Solve the problem.
-The position of a weight on a spring relative to the point of equilibrium is given by $y = 2 \cos 3 t - 7 \sin 3 t$,
where the arguments are in radians and $t$ is in seconds. Find the smallest value of $t$ for which the weight is at the point of equilibrium $( \mathrm { y } = 0 )$. Give your answer to the nearest hundredth.

Accepted Answer

A)  $0.43$ 
B)  $0.17$ 
C)  $0.48$ 
D)  $0.09$ 
A)  $0.43$ 
B)  $0.17$ 
C)  $0.48$ 
D)  $0.09$

Question 11

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$\cot x \cos x + \sin x = \csc x$

Accepted Answer

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

Question 12

Verify that the equation is an identity.
-$$\sin ( x + y ) - \sin ( x - y ) = 2 \cos x \sin y$$

Accepted Answer

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

Question 13

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }$$

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

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 16

Use the cofunction identities to find an angle θ that makes the statement true.
-$\sin \theta = \cos \left( 4 \theta + 60 ^ { \circ } \right)$

Accepted Answer

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

Question 17

Use a calculator to give the value to the nearest degree.
-$$\theta = \tan ^ { - 1 } ( 2.2460 )$$

Accepted Answer

A)  $62 ^ { \circ }$ 
B)  $114 ^ { \circ }$ 
C)  $66 ^ { \circ }$ 
D)  $69 ^ { \circ }$ 
A)  $62 ^ { \circ }$ 
B)  $114 ^ { \circ }$ 
C)  $66 ^ { \circ }$ 
D)  $69 ^ { \circ }$

Question 18

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\sqrt { \frac { 1 - \cos 20 ^ { \circ } } { 2 } }$$

Accepted Answer

A)  $\cos 10 ^ { \circ }$ 
B)  $\cot 10 ^ { \circ }$ 
C)  $\tan 10 ^ { \circ }$ 
D)  $\sin 10 ^ { \circ }$ 
A)  $\cos 10 ^ { \circ }$ 
B)  $\cot 10 ^ { \circ }$ 
C)  $\tan 10 ^ { \circ }$ 
D)  $\sin 10 ^ { \circ }$

Question 19

Provide an appropriate response.
-Graph the expression $\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Accepted Answer

The answer of Provide an appropriate response.
-Graph the expression \(\frac...

Question 20

Provide an appropriate response.
-True or false? The statement $\tan \left( \tan ^ { - 1 } x \right) = x$ for all real numbers in the interval $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$.

Accepted Answer

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

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \sin ^ { - 1 } ( 0.2079 )$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\tan ( \arcsin 2 )$

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

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

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\cos \theta = \frac { 1 } { 2 }$ and $\theta$ is in quadrant IV.

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \tan ^ { - 1 } ( - 0.7002 )$

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 5 } } { 2 }$ and $\sin \theta < 0$ .

Write in terms of the cofunction of a complementary angle. - $\cos \frac { \pi } { 20 }$

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

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\cot x \cos x + \sin x = \csc x$

Verify that the equation is an identity. - $\sin ( x + y ) - \sin ( x - y ) = 2 \cos x \sin y$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }$

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

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

Use the cofunction identities to find an angle θ that makes the statement true. - $\sin \theta = \cos \left( 4 \theta + 60 ^ { \circ } \right)$

Use a calculator to give the value to the nearest degree. - $\theta = \tan ^ { - 1 } ( 2.2460 )$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sqrt { \frac { 1 - \cos 20 ^ { \circ } } { 2 } }$

Provide an appropriate response. -Graph the expression $\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Provide an appropriate response. -True or false? The statement $\tan \left( \tan ^ { - 1 } x \right) = x$ for all real numbers in the interval $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ .

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 a calculator to give the real number value. Round the answer to 7 decimal places. - y=sin⁡−1(0.2079)\mathrm { y } = \sin ^ { - 1 } ( 0.2079 )y=sin−1(0.2079)

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - tan⁡(arcsin⁡2)\tan ( \arcsin 2 )tan(arcsin2)

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

Solve the equation for solutions in the interval [0, 2 π). - sin⁡22x=1\sin ^ { 2 } 2 x = 1sin22x=1

Use the fundamental identities to find the value of the trigonometric function. -Find tan⁡θ\tan \thetatanθ if cos⁡θ=12\cos \theta = \frac { 1 } { 2 }cosθ=21​ and θ\thetaθ is in quadrant IV.

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=tan⁡−1(−0.7002)\mathrm { y } = \tan ^ { - 1 } ( - 0.7002 )y=tan−1(−0.7002)

Use the fundamental identities to find the value of the trigonometric function. -Find tan⁡θ\tan \thetatanθ if sec⁡θ=52\sec \theta = \frac { \sqrt { 5 } } { 2 }secθ=25​​ and sin⁡θ<0\sin \theta < 0sinθ<0 .

Write in terms of the cofunction of a complementary angle. - cos⁡π20\cos \frac { \pi } { 20 }cos20π​

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

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - cot⁡xcos⁡x+sin⁡x=csc⁡x\cot x \cos x + \sin x = \csc xcotxcosx+sinx=cscx

Verify that the equation is an identity. - sin⁡(x+y)−sin⁡(x−y)=2cos⁡xsin⁡y\sin ( x + y ) - \sin ( x - y ) = 2 \cos x \sin ysin(x+y)−sin(x−y)=2cosxsiny

Perform the indicated operations and simplify the result so there are no quotients. - sin⁡θ1+sin⁡θ−sin⁡θ1−sin⁡θ\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }1+sinθsinθ​−1−sinθsinθ​

Decide whether the expression is or is not an identity. - sin⁡2x+cos⁡2x=1\sqrt { \sin ^ { 2 } x + \cos ^ { 2 } x } = 1sin2x+cos2x​=1

Use a sum or difference identity to find the exact value. - tan⁡π12\tan \frac { \pi } { 12 }tan12π​

Use the cofunction identities to find an angle θ that makes the statement true. - sin⁡θ=cos⁡(4θ+60∘)\sin \theta = \cos \left( 4 \theta + 60 ^ { \circ } \right)sinθ=cos(4θ+60∘)

Use a calculator to give the value to the nearest degree. - θ=tan⁡−1(2.2460)\theta = \tan ^ { - 1 } ( 2.2460 )θ=tan−1(2.2460)

Use an identity to write the expression as a single trigonometric function or as a single number. - 1−cos⁡20∘2\sqrt { \frac { 1 - \cos 20 ^ { \circ } } { 2 } }21−cos20∘​​

Provide an appropriate response. -Graph the expression 1−sin⁡2xsin⁡x−csc⁡x\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }sinx−cscx1−sin2x​ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Provide an appropriate response. -True or false? The statement tan⁡(tan⁡−1x)=x\tan \left( \tan ^ { - 1 } x \right) = xtan(tan−1x)=x for all real numbers in the interval −π2<x<π2- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }−2π​<x<2π​ .