Question 1

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\tan \theta = - \frac { 5 } { 12 }$ and $\cos \theta > 0$

Accepted Answer

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

Question 2

Write the expression as the sine, cosine, or tangent of an angle.
-$$\frac { \tan 46 ^ { \circ } - \tan 27 ^ { \circ } } { 1 + \tan 46 ^ { \circ } \tan 27 ^ { \circ } }$$

Accepted Answer

A)  $\tan 19 ^ { \circ }$ 
B)  $\tan 73 ^ { \circ }$ 
C)  $\tan 72$ 
D)  $\tan 36.5 ^ { \circ }$ 
A)  $\tan 19 ^ { \circ }$ 
B)  $\tan 73 ^ { \circ }$ 
C)  $\tan 72$ 
D)  $\tan 36.5 ^ { \circ }$ A

Question 3

Find all solutions to the equation.
-$$\cos ^ { 2 } x + 2 \cos x + 1 = 0$$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } + 2 \mathrm { n } \pi , \frac { 3 \pi } { 4 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
B)  $\left\{ \frac { 3 \pi } { 2 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
C)  $\{ \pi + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \}$ 
D)  $\{ \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \}$ 
A)  $\left\{ \frac { \pi } { 4 } + 2 \mathrm { n } \pi , \frac { 3 \pi } { 4 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
B)  $\left\{ \frac { 3 \pi } { 2 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
C)  $\{ \pi + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \}$ 
D)  $\{ \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \}$ C

Question 4

Find the exact value by using a half-angle identity.
-$\tan \frac { 7 \pi } { 8 }$

Accepted Answer

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

Question 5

Simplify the expression.
-$$\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }$$

Accepted Answer

A)  $- \sin x$ 
B)  $\sin 2 x$ 
C)  $\cos ^ { 2 } x$ 
D)  $- \cos x$ 
A)  $- \sin x$ 
B)  $\sin 2 x$ 
C)  $\cos ^ { 2 } x$ 
D)  $- \cos x$

Question 6

Prove the identity.
-$$\cos \left( x + \frac { \pi } { 2 } \right) = - \sin x$$

Accepted Answer

The answer of Prove the identity.
-\[\cos \left( x + \frac...

Question 7

Simplify the expression.
-$$\frac { \cos x } { 1 - \sin x } + \frac { 1 - \sin x } { \cos x }$$

Accepted Answer

A)  $\sec ^ { 2 } x$ 
B)  $2 \csc x$ 
C)  2 
D)  $2 \sec x$ 
A)  $\sec ^ { 2 } x$ 
B)  $2 \csc x$ 
C)  2 
D)  $2 \sec x$

Question 8

Complete the identity.
-The expression $\frac { \sin \theta } { \cos \theta + 1 } + \cot \theta$
is to be the left hand side of an equation that is an identity. Which one of the following four expressions can be used as the right hand side of the equation to complete the identity?

Accepted Answer

A)  $\csc \theta$ 
B)  $\sec \theta$ 
C)  $\cos \theta$ 
D)  $\tan \theta$ 
A)  $\csc \theta$ 
B)  $\sec \theta$ 
C)  $\cos \theta$ 
D)  $\tan \theta$

Question 9

Find an exact value.
-$$\sin \frac { 19 \pi } { 12 }$$

Accepted Answer

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

Question 10

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 26 } } { 5 }$ and $\sin \theta < 0$

Accepted Answer

A)  $- 5$ 
B)  $- \sqrt { 26 }$ 
C)  $- \frac { 1 } { 5 }$ 
D)  5 
A)  $- 5$ 
B)  $- \sqrt { 26 }$ 
C)  $- \frac { 1 } { 5 }$ 
D)  5

Question 11

Determine if the following is an identity.
-$$\cos ^ { 2 } x - \sin ^ { 2 } x = 1 - 2 \sin ^ { 2 } x$$

Accepted Answer

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

Question 12

Prove the identity.
-$$\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)$$

Accepted Answer

The answer of Prove the identity.
-\[\cot ^ { 3 }...

Question 13

Find all solutions to the equation.
-$$\sin ^ { 2 } x + \sin x = 0$$

Accepted Answer

A)  $\left\{ n \pi , \frac { \pi } { 2 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
B)  $\left\{ 2 n \pi , \frac { 3 \pi } { 2 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
C)  $\left\{ \frac { \pi } { 3 } + 2 n \pi , \frac { 2 \pi } { 3 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . . \right.$ 
D)  $\left\{ \mathrm { n } \pi , \frac { 3 \pi } { 2 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
A)  $\left\{ n \pi , \frac { \pi } { 2 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
B)  $\left\{ 2 n \pi , \frac { 3 \pi } { 2 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
C)  $\left\{ \frac { \pi } { 3 } + 2 n \pi , \frac { 2 \pi } { 3 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . . \right.$ 
D)  $\left\{ \mathrm { n } \pi , \frac { 3 \pi } { 2 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$

Question 14

Two triangles can be formed using the given measurements. Solve both triangles.
-$$\mathrm { B } = 45 ^ { \circ } , \mathrm { a } = 14 , \mathrm {~b} = 10$$

Accepted Answer

A)  $\mathrm { A } = 8.1 ^ { \circ } , \mathrm { C } = 126.9 ^ { \circ } , \mathrm { c } = 8.8 ; \mathrm { A } = 171.9 ^ { \circ } , \mathrm { C } = 53.1 ^ { \circ } , \mathrm { c } = 8.8$ 
B)  $\mathrm { A } = 81.9 ^ { \circ } , \mathrm { C } = 53.1 ^ { \circ } , \mathrm { C } = 8.8 ; \mathrm { A } = 98.1 ^ { \circ } , \mathrm { C } = 36.9 ^ { \circ } , \mathrm { c } = 8.8$ 
C)  $\mathrm { A } = 8.1 ^ { \circ } , \mathrm { C } = 126.9 ^ { \circ } , \mathrm { C } = 11.3 ; \mathrm { A } = 171.9 ^ { \circ } , \mathrm { C } = 53.1 ^ { \circ } , \mathrm { C } = 11.3$ 
D)  $\mathrm { A } = 81.9 ^ { \circ } , \mathrm { C } = 53.1 ^ { \circ } , \mathrm { C } = 11.3 ; \mathrm { A } = 98.1 ^ { \circ } , \mathrm { C } = 36.9 ^ { \circ } , \mathrm { c } = 8.5$ 
A)  $\mathrm { A } = 8.1 ^ { \circ } , \mathrm { C } = 126.9 ^ { \circ } , \mathrm { c } = 8.8 ; \mathrm { A } = 171.9 ^ { \circ } , \mathrm { C } = 53.1 ^ { \circ } , \mathrm { c } = 8.8$ 
B)  $\mathrm { A } = 81.9 ^ { \circ } , \mathrm { C } = 53.1 ^ { \circ } , \mathrm { C } = 8.8 ; \mathrm { A } = 98.1 ^ { \circ } , \mathrm { C } = 36.9 ^ { \circ } , \mathrm { c } = 8.8$ 
C)  $\mathrm { A } = 8.1 ^ { \circ } , \mathrm { C } = 126.9 ^ { \circ } , \mathrm { C } = 11.3 ; \mathrm { A } = 171.9 ^ { \circ } , \mathrm { C } = 53.1 ^ { \circ } , \mathrm { C } = 11.3$ 
D)  $\mathrm { A } = 81.9 ^ { \circ } , \mathrm { C } = 53.1 ^ { \circ } , \mathrm { C } = 11.3 ; \mathrm { A } = 98.1 ^ { \circ } , \mathrm { C } = 36.9 ^ { \circ } , \mathrm { c } = 8.5$

Question 15

Prove the identity.
-$$( \sin x ) ( \tan x \cos x - \cot x \cos x ) = 1 - 2 \cos ^ { 2 } x$$

Accepted Answer

\[\begin{array} { l } 
( \sin x ) ( \tan

Question 16

Prove the identity.
-$$\frac { \tan t } { 1 - \cot t } + \frac { 1 + \cot t } { \tan t } = \frac { \tan ^ { 2 } t + 2 - \csc ^ { 2 } t } { \tan t - 1 }$$

Accepted Answer

The answer of Prove the identity.
-\[\frac { \tan t }...

Question 17

Find an exact value.
-$$\cos \frac { 19 \pi } { 12 }$$

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)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$

Question 18

Prove the identity.
-sin 4u = 2 sin 2u cos 2u

Accepted Answer

sin 4u = s

Question 19

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is
formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used.
-

Accepted Answer

A)  The triangle cannot be solved with the Law of Sines. 
B)  $\mathrm { B } = 59 ^ { \circ } , \mathrm { a } \approx 5.7 , \mathrm {~b} \approx 15.8$ 
C)  $B = 59 ^ { \circ } , a \approx 56.8 , b \approx 15.8$ 
D)  $B = 31 ^ { \circ } , a \approx 56.8 , b \approx 15.8$ 
A)  The triangle cannot be solved with the Law of Sines. 
B)  $\mathrm { B } = 59 ^ { \circ } , \mathrm { a } \approx 5.7 , \mathrm {~b} \approx 15.8$ 
C)  $B = 59 ^ { \circ } , a \approx 56.8 , b \approx 15.8$ 
D)  $B = 31 ^ { \circ } , a \approx 56.8 , b \approx 15.8$

Question 20

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is
formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used.
-$$\mathrm { B } = 147 ^ { \circ } , \mathrm { c } = 7 , \mathrm {~b} = 11$$

Accepted Answer

A)  The triangle cannot be solved with the Law of Sines. 
B)  No triangle is formed. 
C)  $\mathrm { C } = 20.3 ^ { \circ } , \mathrm { A } = 12.7 ^ { \circ } , \mathrm { a } \approx 4.4$ 
D)  $\mathrm { C } = 12.7 ^ { \circ } , \mathrm { A } = 20.3 ^ { \circ } , \mathrm { a } \approx 4.4$ 
A)  The triangle cannot be solved with the Law of Sines. 
B)  No triangle is formed. 
C)  $\mathrm { C } = 20.3 ^ { \circ } , \mathrm { A } = 12.7 ^ { \circ } , \mathrm { a } \approx 4.4$ 
D)  $\mathrm { C } = 12.7 ^ { \circ } , \mathrm { A } = 20.3 ^ { \circ } , \mathrm { a } \approx 4.4$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\tan \theta = - \frac { 5 } { 12 }$ and $\cos \theta > 0$

Write the expression as the sine, cosine, or tangent of an angle. - $\frac { \tan 46 ^ { \circ } - \tan 27 ^ { \circ } } { 1 + \tan 46 ^ { \circ } \tan 27 ^ { \circ } }$

Find all solutions to the equation. - $\cos ^ { 2 } x + 2 \cos x + 1 = 0$

Find the exact value by using a half-angle identity. - $\tan \frac { 7 \pi } { 8 }$

Simplify the expression. - $\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }$

Prove the identity. - $\cos \left( x + \frac { \pi } { 2 } \right) = - \sin x$

Simplify the expression. - $\frac { \cos x } { 1 - \sin x } + \frac { 1 - \sin x } { \cos x }$

Complete the identity. -The expression $\frac { \sin \theta } { \cos \theta + 1 } + \cot \theta$ is to be the left hand side of an equation that is an identity. Which one of the following four expressions can be used as the right hand side of the equation to complete the identity?

Find an exact value. - $\sin \frac { 19 \pi } { 12 }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 26 } } { 5 }$ and $\sin \theta < 0$

Determine if the following is an identity. - $\cos ^ { 2 } x - \sin ^ { 2 } x = 1 - 2 \sin ^ { 2 } x$

Prove the identity. - $\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)$

Find all solutions to the equation. - $\sin ^ { 2 } x + \sin x = 0$

Two triangles can be formed using the given measurements. Solve both triangles. - $\mathrm { B } = 45 ^ { \circ } , \mathrm { a } = 14 , \mathrm {~b} = 10$

Prove the identity. - $( \sin x ) ( \tan x \cos x - \cot x \cos x ) = 1 - 2 \cos ^ { 2 } x$

Prove the identity. - $\frac { \tan t } { 1 - \cot t } + \frac { 1 + \cot t } { \tan t } = \frac { \tan ^ { 2 } t + 2 - \csc ^ { 2 } t } { \tan t - 1 }$

Find an exact value. - $\cos \frac { 19 \pi } { 12 }$

Prove the identity. -sin 4u = 2 sin 2u cos 2u

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used. -

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Exam 5: Analytic Trigonometry

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ if tan⁡θ=−512\tan \theta = - \frac { 5 } { 12 }tanθ=−125​ and cos⁡θ>0\cos \theta > 0cosθ>0

Write the expression as the sine, cosine, or tangent of an angle. - tan⁡46∘−tan⁡27∘1+tan⁡46∘tan⁡27∘\frac { \tan 46 ^ { \circ } - \tan 27 ^ { \circ } } { 1 + \tan 46 ^ { \circ } \tan 27 ^ { \circ } }1+tan46∘tan27∘tan46∘−tan27∘​

Find all solutions to the equation. - cos⁡2x+2cos⁡x+1=0\cos ^ { 2 } x + 2 \cos x + 1 = 0cos2x+2cosx+1=0

Find the exact value by using a half-angle identity. - tan⁡7π8\tan \frac { 7 \pi } { 8 }tan87π​

Simplify the expression. - 1−sin⁡2xsin⁡x−csc⁡x\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }sinx−cscx1−sin2x​

Prove the identity. - cos⁡(x+π2)=−sin⁡x\cos \left( x + \frac { \pi } { 2 } \right) = - \sin xcos(x+2π​)=−sinx

Simplify the expression. - cos⁡x1−sin⁡x+1−sin⁡xcos⁡x\frac { \cos x } { 1 - \sin x } + \frac { 1 - \sin x } { \cos x }1−sinxcosx​+cosx1−sinx​

Find an exact value. - sin⁡19π12\sin \frac { 19 \pi } { 12 }sin1219π​

Use the fundamental identities to find the value of the trigonometric function. -Find tan⁡θ\tan \thetatanθ if sec⁡θ=265\sec \theta = \frac { \sqrt { 26 } } { 5 }secθ=526​​ and sin⁡θ<0\sin \theta < 0sinθ<0

Determine if the following is an identity. - cos⁡2x−sin⁡2x=1−2sin⁡2x\cos ^ { 2 } x - \sin ^ { 2 } x = 1 - 2 \sin ^ { 2 } xcos2x−sin2x=1−2sin2x

Prove the identity. - cot⁡3x=cot⁡x(csc⁡2x−1)\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)cot3x=cotx(csc2x−1)

Find all solutions to the equation. - sin⁡2x+sin⁡x=0\sin ^ { 2 } x + \sin x = 0sin2x+sinx=0

Two triangles can be formed using the given measurements. Solve both triangles. - B=45∘,a=14, b=10\mathrm { B } = 45 ^ { \circ } , \mathrm { a } = 14 , \mathrm {~b} = 10B=45∘,a=14, b=10

Prove the identity. - (sin⁡x)(tan⁡xcos⁡x−cot⁡xcos⁡x)=1−2cos⁡2x( \sin x ) ( \tan x \cos x - \cot x \cos x ) = 1 - 2 \cos ^ { 2 } x(sinx)(tanxcosx−cotxcosx)=1−2cos2x

Prove the identity. - tan⁡t1−cot⁡t+1+cot⁡ttan⁡t=tan⁡2t+2−csc⁡2ttan⁡t−1\frac { \tan t } { 1 - \cot t } + \frac { 1 + \cot t } { \tan t } = \frac { \tan ^ { 2 } t + 2 - \csc ^ { 2 } t } { \tan t - 1 }1−cotttant​+tant1+cott​=tant−1tan2t+2−csc2t​

Find an exact value. - cos⁡19π12\cos \frac { 19 \pi } { 12 }cos1219π​