Question 1

Verify that each equation is an identity.
-$$\csc ^ { 2 } t - \cos t \sec t = \cot ^ { 2 } t$$

Accepted Answer

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

Question 2

Determine all solutions of the equation in radians.
-Find $\cos \frac { x } { 2 }$, given that $\sec x = 4$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 4 }$ 
C)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 4 }$ 
C)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$

Question 3

Verify that each equation is an identity.
-$$\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }$$

Accepted Answer

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

Question 4

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$
-$$\cos ^ { 2 } x + 2 \cos x + 1 = 0$$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
B)  $\{ \pi \}$ 
C)  $\{ 2 \pi \}$ 
D)  $\left\{ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right\}$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
B)  $\{ \pi \}$ 
C)  $\{ 2 \pi \}$ 
D)  $\left\{ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right\}$

Question 5

The statement tan-1(tan x) = x for all real numbers in the interval
-Explain why you can find sin 1.262 on your calculator, but $\sin ^ { - 1 } 1.262$ results in an erro
message.

Accepted Answer

1.262 is n

Question 6

Write the following as an algebraic expression in u, u > 0.
-$\tan \left(\cos ^{-1} \frac{\mathrm{u}}{3}\right)$

Accepted Answer

A)   $\frac{\sqrt{u^{2}-9}}{u}$ 
B)  $\frac { u \sqrt { 9 - u ^ { 2 } } } { 9 - u ^ { 2 } }$ 
C)  $u ^ { 2 } \sqrt { 9 }$ 
D)  $\frac { \sqrt { 9 - u ^ { 2 } } } { u }$ 
A)   $\frac{\sqrt{u^{2}-9}}{u}$ 
B)  $\frac { u \sqrt { 9 - u ^ { 2 } } } { 9 - u ^ { 2 } }$ 
C)  $u ^ { 2 } \sqrt { 9 }$ 
D)  $\frac { \sqrt { 9 - u ^ { 2 } } } { u }$

Question 7

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

Accepted Answer

\[\begin{array} { l } 
\frac { 1 - \cot

Question 8

Solve the problem.

-$$\left( \cos \frac { x } { 2 } - \sin \frac { x } { 2 } \right) ^ { 2 } = 1 - \sin x$$

Accepted Answer

\[\begin{array} { l } 
\left( \cos \frac

Question 9

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

Accepted Answer

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

Question 10

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). 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)  $\{ \pi \}$ 
B)  $\left\{ \frac { \pi } { 4 } \right\}$ 
C)  $\{ 0 , \pi \}$ 
D)  $\left\{ \frac { \pi } { 2 } \right\}$ 
A)  $\{ \pi \}$ 
B)  $\left\{ \frac { \pi } { 4 } \right\}$ 
C)  $\{ 0 , \pi \}$ 
D)  $\left\{ \frac { \pi } { 2 } \right\}$

Question 11

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\frac { 2 \tan 30 ^ { \circ } } { 1 - \tan ^ { 2 } 30 ^ { \circ } }$$

Accepted Answer

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

Question 12

Use identities to fill in the blank with the appropriate trigonometric function name.
-$\tan 14 ^ { \circ } = \frac { 1 } {....... 76 ^ { \circ } }$

Accepted Answer

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

Question 13

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

Accepted Answer

A)  $2 \sin 4 x$ 
B)  $\cos 8 x$ 
C)  $\cos 4 x$ 
D)  $\frac { 1 } { 2 } \sin 16 x$ 
A)  $2 \sin 4 x$ 
B)  $\cos 8 x$ 
C)  $\cos 4 x$ 
D)  $\frac { 1 } { 2 } \sin 16 x$

Question 14

Solve the equation for x, where x is restricted to the given interval.
-$y = \cos ( x + 12 )$, for $x$ in $[ - 12 , \pi - 12 ]$

Accepted Answer

A)  $x = \arccos ( y + 12 )$ 
B)  $x = \arccos y - 12$ 
C)  $x = \arccos y + 12$ 
D)  $x = \arccos ( y - 12 )$ 
A)  $x = \arccos ( y + 12 )$ 
B)  $x = \arccos y - 12$ 
C)  $x = \arccos y + 12$ 
D)  $x = \arccos ( y - 12 )$

Question 15

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \sin ^ { - 1 } ( 0.6561 )$$

Accepted Answer

A)  $0.7156393$ 
B)  $0.5806515$ 
C)  $0.6809524$ 
D)  $- 0.7158730$ 
A)  $0.7156393$ 
B)  $0.5806515$ 
C)  $0.6809524$ 
D)  $- 0.7158730$

Question 16

Solve the equation for exact solutions.
-$$\tan ^ { - 1 } ( x + 2 ) + \tan ^ { - 1 } x = \tan ^ { - 1 } \frac { 1 } { 4 }$$

Accepted Answer

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

Question 17

Give the degree measure of $$\theta$$
-$$\theta = \sin ^ { - 1 } ( 0.5 )$$

Accepted Answer

A)  $420 ^ { \circ }$ 
B)  $30 ^ { \circ }$ 
C)  $60 ^ { \circ }$ 
D)  $- 30 ^ { \circ }$ 
A)  $420 ^ { \circ }$ 
B)  $30 ^ { \circ }$ 
C)  $60 ^ { \circ }$ 
D)  $- 30 ^ { \circ }$

Question 18

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$ if $\sin \theta = - \frac { 5 } { 13 }$ and $\theta$ is in quadrant III.

Accepted Answer

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

Question 19

Write the following as an algebraic expression in u, u > 0.
-$\cos ( \arcsin u )$

Accepted Answer

A)  $\sqrt { u ^ { 2 } + 1 }$ 
B)  $\frac { \sqrt { u ^ { 2 } + 1 } } { u }$ 
C)  $\sqrt { u ^ { 2 } - 1 }$ 
D)  $\sqrt { 1 - \mathrm { u } ^ { 2 } }$ 
A)  $\sqrt { u ^ { 2 } + 1 }$ 
B)  $\frac { \sqrt { u ^ { 2 } + 1 } } { u }$ 
C)  $\sqrt { u ^ { 2 } - 1 }$ 
D)  $\sqrt { 1 - \mathrm { u } ^ { 2 } }$

Question 20

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

Accepted Answer

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

Verify that each equation is an identity. - $\csc ^ { 2 } t - \cos t \sec t = \cot ^ { 2 } t$

Determine all solutions of the equation in radians. -Find $\cos \frac { x } { 2 }$ , given that $\sec x = 4$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$ .

Verify that each equation is an identity. - $\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }$

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\cos ^ { 2 } x + 2 \cos x + 1 = 0$

The statement tan-1(tan x) = x for all real numbers in the interval -Explain why you can find sin 1.262 on your calculator, but $\sin ^ { - 1 } 1.262$ results in an erro message.

Write the following as an algebraic expression in u, u > 0. - $\tan \left(\cos ^{-1} \frac{\mathrm{u}}{3}\right)$

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

Solve the problem. - $\left( \cos \frac { x } { 2 } - \sin \frac { x } { 2 } \right) ^ { 2 } = 1 - \sin x$

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

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

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { 2 \tan 30 ^ { \circ } } { 1 - \tan ^ { 2 } 30 ^ { \circ } }$

Use identities to fill in the blank with the appropriate trigonometric function name. - $\tan 14 ^ { \circ } = \frac { 1 } {....... 76 ^ { \circ } }$

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

Solve the equation for x, where x is restricted to the given interval. - $y = \cos ( x + 12 )$ , for $x$ in $[ - 12 , \pi - 12 ]$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \sin ^ { - 1 } ( 0.6561 )$

Solve the equation for exact solutions. - $\tan ^ { - 1 } ( x + 2 ) + \tan ^ { - 1 } x = \tan ^ { - 1 } \frac { 1 } { 4 }$

Give the degree measure of $\theta$ - $\theta = \sin ^ { - 1 } ( 0.5 )$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\sin \theta = - \frac { 5 } { 13 }$ and $\theta$ is in quadrant III.

Write the following as an algebraic expression in u, u > 0. - $\cos ( \arcsin u )$

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

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

Verify that each equation is an identity. - csc⁡2t−cos⁡tsec⁡t=cot⁡2t\csc ^ { 2 } t - \cos t \sec t = \cot ^ { 2 } tcsc2t−costsect=cot2t

Determine all solutions of the equation in radians. -Find cos⁡x2\cos \frac { x } { 2 }cos2x​ , given that sec⁡x=4\sec x = 4secx=4 and xxx terminates in 0<x<π20 < x < \frac { \pi } { 2 }0<x<2π​ .

Verify that each equation is an identity. - cot⁡(2θ)=csc⁡2θ−22cot⁡θ\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }cot(2θ)=2cotθcsc2θ−2​

Solve the equation for exact solutions over the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - cos⁡2x+2cos⁡x+1=0\cos ^ { 2 } x + 2 \cos x + 1 = 0cos2x+2cosx+1=0

The statement tan-1(tan x) = x for all real numbers in the interval -Explain why you can find sin 1.262 on your calculator, but sin⁡−11.262\sin ^ { - 1 } 1.262sin−11.262 results in an erro message.

Write the following as an algebraic expression in u, u > 0. - tan⁡(cos⁡−1u3)\tan \left(\cos ^{-1} \frac{\mathrm{u}}{3}\right)tan(cos−13u​)

Verify that each equation is an identity. - 1−cot⁡2θ1+cot⁡2θ+1=2sin⁡2θ\frac { 1 - \cot ^ { 2 } \theta } { 1 + \cot ^ { 2 } \theta } + 1 = 2 \sin ^ { 2 } \theta1+cot2θ1−cot2θ​+1=2sin2θ

Solve the problem. - (cos⁡x2−sin⁡x2)2=1−sin⁡x\left( \cos \frac { x } { 2 } - \sin \frac { x } { 2 } \right) ^ { 2 } = 1 - \sin x(cos2x​−sin2x​)2=1−sinx

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

Solve the equation for solutions over the interval [0, [0,2π)[ 0,2 \pi )[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 an identity to write the expression as a single trigonometric function or as a single number. - 2tan⁡30∘1−tan⁡230∘\frac { 2 \tan 30 ^ { \circ } } { 1 - \tan ^ { 2 } 30 ^ { \circ } }1−tan230∘2tan30∘​

Use identities to fill in the blank with the appropriate trigonometric function name. - tan⁡14∘=1.......76∘\tan 14 ^ { \circ } = \frac { 1 } {....... 76 ^ { \circ } }tan14∘=.......76∘1​

Use an identity to write the expression as a single trigonometric function or as a single number. - 4sin⁡2xcos⁡2x4 \sin 2 x \cos 2 x4sin2xcos2x

Solve the equation for x, where x is restricted to the given interval. - y=cos⁡(x+12)y = \cos ( x + 12 )y=cos(x+12) , for xxx in [−12,π−12][ - 12 , \pi - 12 ][−12,π−12]

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=sin⁡−1(0.6561)y = \sin ^ { - 1 } ( 0.6561 )y=sin−1(0.6561)

Solve the equation for exact solutions. - tan⁡−1(x+2)+tan⁡−1x=tan⁡−114\tan ^ { - 1 } ( x + 2 ) + \tan ^ { - 1 } x = \tan ^ { - 1 } \frac { 1 } { 4 }tan−1(x+2)+tan−1x=tan−141​

Give the degree measure of θ\thetaθ - θ=sin⁡−1(0.5)\theta = \sin ^ { - 1 } ( 0.5 )θ=sin−1(0.5)

Use the fundamental identities to find the value of the trigonometric function. -Find cos⁡θ\cos \thetacosθ if sin⁡θ=−513\sin \theta = - \frac { 5 } { 13 }sinθ=−135​ and θ\thetaθ is in quadrant III.

Write the following as an algebraic expression in u, u > 0. - cos⁡(arcsin⁡u)\cos ( \arcsin u )cos(arcsinu)

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