Question 1

Find the exact value of the real number y.
-$$y = \cos ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)$$

Accepted Answer

A)  $\frac { 7 \pi } { 4 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { 11 \pi } { 6 }$ 
A)  $\frac { 7 \pi } { 4 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { 11 \pi } { 6 }$

Question 2

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

Accepted Answer

A)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
B)  $\frac { \sqrt { 3 } + 2 } { 4 }$ 
C)  $\frac { \sqrt { 3 } - 2 } { 4 }$ 
D)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
A)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
B)  $\frac { \sqrt { 3 } + 2 } { 4 }$ 
C)  $\frac { \sqrt { 3 } - 2 } { 4 }$ 
D)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$

Question 3

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

Accepted Answer

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

Question 4

Verify that each equation is an identity.
-$$\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \theta$$

Accepted Answer

\[\begin{array} { l } 
\cos ( 4 \theta )

Question 5

Determine all solutions of the equation in radians.
-Find $\sin \frac { \theta } { 2 }$, given that $\sin \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $270 ^ { \circ } < \theta < 360 ^ { \circ }$.

Accepted Answer

A)  $- \frac { \sqrt { 30 } } { 10 }$ 
B)  $\frac { \sqrt { 5 } } { 5 }$ 
C)  $- \frac { \sqrt { 5 } } { 5 }$ 
D)  $\frac { \sqrt { 10 } } { 10 }$ 
A)  $- \frac { \sqrt { 30 } } { 10 }$ 
B)  $\frac { \sqrt { 5 } } { 5 }$ 
C)  $- \frac { \sqrt { 5 } } { 5 }$ 
D)  $\frac { \sqrt { 10 } } { 10 }$

Question 6

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

Accepted Answer

A)  $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$ 
B)  $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$ 
C)  $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$ 
D)  $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$ 
A)  $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$ 
B)  $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$ 
C)  $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$ 
D)  $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$

Question 7

Use Identities to find the exact value.
-$\cos 165 ^ { \circ }$

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

Use a sum or difference identity to find the exact value.
-$\sin 15 ^ { \circ }$

Accepted Answer

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

Question 10

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

Question 11

Use a sum or difference identity to find the exact value.
-$\tan 345 ^ { \circ }$

Accepted Answer

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

Question 12

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$$\sin 2 x = 2 \sin x \cos x$$

Accepted Answer

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

Question 13

Solve the equation for x, where x is restricted to the given interval.
-$y = - \sec \frac { x } { 4 }$, for $x$ in $[ 0,2 \pi ) \cup ( 2 \pi , 4 \pi ]$

Accepted Answer

A)  $x = - \frac { 1 } { 4 } \operatorname { arcsec } y$ 
B)  $x = 4 \operatorname { arcsec } ( - y )$ 
C)  $x = - 4 \operatorname { arcsec } y$ 
D)  $x = \frac { 1 } { 4 } \operatorname { arcsec } ( - y )$ 
A)  $x = - \frac { 1 } { 4 } \operatorname { arcsec } y$ 
B)  $x = 4 \operatorname { arcsec } ( - y )$ 
C)  $x = - 4 \operatorname { arcsec } y$ 
D)  $x = \frac { 1 } { 4 } \operatorname { arcsec } ( - y )$

Question 14

Use identities to find the indicated value for each angle measure.
-$\sin \theta = \frac { 21 } { 29 } , \quad \cos \theta > 0 \quad$ Find $\cos ( 2 \theta )$

Accepted Answer

A)  $\frac { 41 } { 841 }$ 
B)  $- \frac { 42 } { 841 }$ 
C)  $- \frac { 41 } { 841 }$ 
D)  $\frac { 840 } { 841 }$ 
A)  $\frac { 41 } { 841 }$ 
B)  $- \frac { 42 } { 841 }$ 
C)  $- \frac { 41 } { 841 }$ 
D)  $\frac { 840 } { 841 }$

Question 15

Verify that each equation is an identity.
-$$( \sec \alpha - \tan \alpha ) ( \sec \alpha + \tan \alpha ) = 1$$

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-$\tan 3 x - \sec 3 x = 5$

Accepted Answer

A)  $\{ 1.9628,4.5808 \}$ 
B)  $\{ 4.3176,10.6008,16.8840 \}$ 
C)  $\{ 5.8884,13.7424 \}$ 
D)  $\{ 1.4392,3.5336,5.6280 \}$ 
A)  $\{ 1.9628,4.5808 \}$ 
B)  $\{ 4.3176,10.6008,16.8840 \}$ 
C)  $\{ 5.8884,13.7424 \}$ 
D)  $\{ 1.4392,3.5336,5.6280 \}$

Question 18

Provide an appropriate response.
-Graph the expression $\sec ^ { 2 } x - \sec ^ { 2 } x \sin ^ { 2 } x$ 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 \(\sec...

Question 19

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$\tan 2 x - \tan x = 0$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
C)  $\{ 0 \}$ 
D)  $\{ 0 , \pi \}$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
C)  $\{ 0 \}$ 
D)  $\{ 0 , \pi \}$

Question 20

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

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)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$

Find the exact value of the real number y. - $y = \cos ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)$

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

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

Verify that each equation is an identity. - $\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \theta$

Determine all solutions of the equation in radians. -Find $\sin \frac { \theta } { 2 }$ , given that $\sin \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $270 ^ { \circ } < \theta < 360 ^ { \circ }$ .

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

Use Identities to find the exact value. - $\cos 165 ^ { \circ }$

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

Use a sum or difference identity to find the exact value. - $\sin 15 ^ { \circ }$

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 sum or difference identity to find the exact value. - $\tan 345 ^ { \circ }$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\sin 2 x = 2 \sin x \cos x$

Solve the equation for x, where x is restricted to the given interval. - $y = - \sec \frac { x } { 4 }$ , for $x$ in $[ 0,2 \pi ) \cup ( 2 \pi , 4 \pi ]$

Use identities to find the indicated value for each angle measure. - $\sin \theta = \frac { 21 } { 29 } , \quad \cos \theta > 0 \quad$ Find $\cos ( 2 \theta )$

Verify that each equation is an identity. - $( \sec \alpha - \tan \alpha ) ( \sec \alpha + \tan \alpha ) = 1$

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

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. - $\tan 3 x - \sec 3 x = 5$

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

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\tan 2 x - \tan x = 0$

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

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

Find the exact value of the real number y. - y=cos⁡−1(22)y = \cos ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)y=cos−1(22​​)

Use a sum or difference identity to find the exact value. - sin⁡11π12\sin \frac { 11 \pi } { 12 }sin1211π​

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

Verify that each equation is an identity. - cos⁡(4θ)=cos⁡4θ−6sin⁡2θcos⁡2θ+sin⁡4θ\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \thetacos(4θ)=cos4θ−6sin2θcos2θ+sin4θ

Determine all solutions of the equation in radians. -Find sin⁡θ2\sin \frac { \theta } { 2 }sin2θ​ , given that sin⁡θ=−35\sin \theta = - \frac { 3 } { 5 }sinθ=−53​ and θ\thetaθ terminates in 270∘<θ<360∘270 ^ { \circ } < \theta < 360 ^ { \circ }270∘<θ<360∘ .

Use Identities to find the exact value. - cos⁡165∘\cos 165 ^ { \circ }cos165∘

Perform the indicated operations and simplify the result so there are no quotients. - sec⁡θ−1sec⁡θ\sec \theta - \frac { 1 } { \sec \theta }secθ−secθ1​

Use a sum or difference identity to find the exact value. - sin⁡15∘\sin 15 ^ { \circ }sin15∘

Use a sum or difference identity to find the exact value. - sin⁡π15cos⁡4π15+cos⁡π15sin⁡4π15\sin \frac { \pi } { 15 } \cos \frac { 4 \pi } { 15 } + \cos \frac { \pi } { 15 } \sin \frac { 4 \pi } { 15 }sin15π​cos154π​+cos15π​sin154π​

Use a sum or difference identity to find the exact value. - tan⁡345∘\tan 345 ^ { \circ }tan345∘

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - sin⁡2x=2sin⁡xcos⁡x\sin 2 x = 2 \sin x \cos xsin2x=2sinxcosx

Solve the equation for x, where x is restricted to the given interval. - y=−sec⁡x4y = - \sec \frac { x } { 4 }y=−sec4x​ , for xxx in [0,2π)∪(2π,4π][ 0,2 \pi ) \cup ( 2 \pi , 4 \pi ][0,2π)∪(2π,4π]

Use identities to find the indicated value for each angle measure. - sin⁡θ=2129,cos⁡θ>0\sin \theta = \frac { 21 } { 29 } , \quad \cos \theta > 0 \quadsinθ=2921​,cosθ>0 Find cos⁡(2θ)\cos ( 2 \theta )cos(2θ)

Verify that each equation is an identity. - (sec⁡α−tan⁡α)(sec⁡α+tan⁡α)=1( \sec \alpha - \tan \alpha ) ( \sec \alpha + \tan \alpha ) = 1(secα−tanα)(secα+tanα)=1

Write the following as an algebraic expression in u, u > 0. - sin⁡(arctan⁡u)\sin ( \arctan u )sin(arctanu)

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. - tan⁡3x−sec⁡3x=5\tan 3 x - \sec 3 x = 5tan3x−sec3x=5

Provide an appropriate response. -Graph the expression sec⁡2x−sec⁡2xsin⁡2x\sec ^ { 2 } x - \sec ^ { 2 } x \sin ^ { 2 } xsec2x−sec2xsin2x on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Solve the equation for solutions in the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - tan⁡2x−tan⁡x=0\tan 2 x - \tan x = 0tan2x−tanx=0

Find the exact value by using a half-angle identity. - sin⁡5π12\sin \frac { 5 \pi } { 12 }sin125π​