Question 1

Simplify the expression.
-$$\frac { 1 } { \csc x - \cot x } + \frac { 1 } { \csc x + \cot x }$$

Accepted Answer

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

Question 2

Find all solutions to the equation.
-$5 \sin ^ { 2 } x - 10 \sin x + 8 = 3$

Accepted Answer

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

Question 3

Write each expression in factored form as an algebraic expression of a single trigonometric function.
-$4 \cot ^ { 2 } x - \frac { 4 } { \tan x } + \cos x \sec x$

Accepted Answer

A)  $( 4 \tan x + 1 ) ( \tan x + 1 )$ 
B)  $( 2 \cot x - 1 ) ^ { 2 }$ 
C)  1 
D)  $( 2 \tan x - 1 ) ( 2 \tan x + 1 )$ 
A)  $( 4 \tan x + 1 ) ( \tan x + 1 )$ 
B)  $( 2 \cot x - 1 ) ^ { 2 }$ 
C)  1 
D)  $( 2 \tan x - 1 ) ( 2 \tan x + 1 )$

Question 4

Prove the identity.
-$$\frac { \cot x } { 1 + \csc x } = \frac { \csc x - 1 } { \cot x }$$

Accepted Answer

The answer of Prove the identity.
-\[\frac { \cot x }...

Question 5

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

Accepted Answer

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

Question 6

Solve the triangle.
-

Accepted Answer

A)  $\mathrm { B } = 30 ^ { \circ } , \mathrm { b } \approx 9.7 , \mathrm { c } \approx 10.5$ 
B)  $\mathrm { B } = 45 ^ { \circ } , \mathrm { b } \approx 7.7 , \mathrm { c } \approx 8.5$ 
C)  $B = 60 ^ { \circ } , b \approx 8.5 , c \approx 10.5$ 
D)  $B = 45 ^ { \circ } , b \approx 8.5 , c \approx 7.7$ 
A)  $\mathrm { B } = 30 ^ { \circ } , \mathrm { b } \approx 9.7 , \mathrm { c } \approx 10.5$ 
B)  $\mathrm { B } = 45 ^ { \circ } , \mathrm { b } \approx 7.7 , \mathrm { c } \approx 8.5$ 
C)  $B = 60 ^ { \circ } , b \approx 8.5 , c \approx 10.5$ 
D)  $B = 45 ^ { \circ } , b \approx 8.5 , c \approx 7.7$

Question 7

Use the fundamental identities to find the value of the trigonometric function.
-Given that $\cos \left( \theta - \frac { \pi } { 2 } \right) = - \frac { 4 } { 5 }$, find $\sin ( - \theta )$.

Accepted Answer

A)  $- \frac { 4 } { 5 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 3 } { 5 }$ 
D)  $- \frac { 3 } { 5 }$ 
A)  $- \frac { 4 } { 5 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 3 } { 5 }$ 
D)  $- \frac { 3 } { 5 }$

Question 8

Solve the triangle.
-$$\mathrm { A } = 52 ^ { \circ } , \mathrm { B } = 63 ^ { \circ } , \mathrm { a } = 8$$

Accepted Answer

A)  $\mathrm { C } = 65 ^ { \circ } , \mathrm { b } \approx 9 , \mathrm { c } \approx 9.2$ 
B)  $\mathrm { C } = 25 ^ { \circ } , \mathrm { b } \approx 7.1 , \mathrm { c } \approx 8.1$ 
C)  $\mathrm { C } = 65 ^ { \circ } , \mathrm { b } \approx 7.1 , \mathrm { c } \approx 8.1$ 
D)  $\mathrm { C } = 65 ^ { \circ } , \mathrm { b } \approx 9 , \mathrm { c } \approx 8.1$ 
A)  $\mathrm { C } = 65 ^ { \circ } , \mathrm { b } \approx 9 , \mathrm { c } \approx 9.2$ 
B)  $\mathrm { C } = 25 ^ { \circ } , \mathrm { b } \approx 7.1 , \mathrm { c } \approx 8.1$ 
C)  $\mathrm { C } = 65 ^ { \circ } , \mathrm { b } \approx 7.1 , \mathrm { c } \approx 8.1$ 
D)  $\mathrm { C } = 65 ^ { \circ } , \mathrm { b } \approx 9 , \mathrm { c } \approx 8.1$

Question 9

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

Accepted Answer

The answer of Prove the identity.
-\[\cot ^ { 2 }...

Question 10

Use basic identities to simplify the expression.
-$\cos \theta - \cos \theta \sin ^ { 2 } \theta$

Accepted Answer

A)  $\sec ^ { 2 } \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\cos ^ { 3 } \theta$ 
D)  $\sin \theta$ 
A)  $\sec ^ { 2 } \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\cos ^ { 3 } \theta$ 
D)  $\sin \theta$

Question 11

Simplify the expression to either 1 or -1.
-$$\cot ^ { 2 } x - \csc ^ { 2 } ( - x )$$

Accepted Answer

A)  1 
B)  $- 1$ 
A)  1 
B)  $- 1$

Question 12

Find an exact value.
-$$\cos \frac { - 7 \pi } { 12 }$$

Accepted Answer

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

Question 13

Prove the identity.
-$$\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } x$$

Accepted Answer

The answer of Prove the identity.
-\[\tan ^ { 2 }...

Question 14

Solve the triangle.
-$$\mathrm { A } = 19 ^ { \circ } , \mathrm { C } = 105 ^ { \circ } , \mathrm { c } = 6$$

Accepted Answer

A)  $B = 34 ^ { \circ } , \mathrm { a } \approx 17.8 , \mathrm {~b} \approx 15.3$ 
B)  $\mathrm { B } = 56 ^ { \circ } , \mathrm { a } \approx 17.8 , \mathrm {~b} \approx 15.3$ 
C)  $B = 56 ^ { \circ } , \mathrm { a } \approx 2 , \mathrm {~b} \approx 5.1$ 
D)  $B = 56 ^ { \circ } , a \approx 17.8 , b \approx 5.1$ 
A)  $B = 34 ^ { \circ } , \mathrm { a } \approx 17.8 , \mathrm {~b} \approx 15.3$ 
B)  $\mathrm { B } = 56 ^ { \circ } , \mathrm { a } \approx 17.8 , \mathrm {~b} \approx 15.3$ 
C)  $B = 56 ^ { \circ } , \mathrm { a } \approx 2 , \mathrm {~b} \approx 5.1$ 
D)  $B = 56 ^ { \circ } , a \approx 17.8 , b \approx 5.1$

Question 15

Prove the identity.
-sin (x + y) - sin (x - y) = 2 cos x sin y

Accepted Answer

sin (x + y) - sin (x

Question 16

Prove the identity.
-cos (x - y) - cos (x + y) = 2 sin x sin y

Accepted Answer

cos (x - y) - cos (x + y) = co

Question 17

Prove the identity.
-$$\cos x \csc x \tan x = 1$$

Accepted Answer

The answer of Prove the identity.
-\[\cos x \csc x \tan...

Question 18

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cot \theta$ if $\csc \theta = \frac { \sqrt { 17 } } { 4 }$ and $\tan \theta > 0$.

Accepted Answer

A)  $\frac { \sqrt { 17 } } { 17 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  4 
D)  $\sqrt { 17 }$ 
A)  $\frac { \sqrt { 17 } } { 17 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  4 
D)  $\sqrt { 17 }$

Question 19

Find $\sin \theta$ if $\cot \theta = - 4$ and $\cos \theta < 0$.

Accepted Answer

A)  - 17 
B)  $\frac { \sqrt { 17 } } { 17 }$ 
C)  $- \frac { 1 } { 4 }$ 
D)  $\frac { \sqrt { 17 } } { 4 }$ 
A)  - 17 
B)  $\frac { \sqrt { 17 } } { 17 }$ 
C)  $- \frac { 1 } { 4 }$ 
D)  $\frac { \sqrt { 17 } } { 4 }$

Question 20

Given the SAS parts of a triangle, is it better to use the law of sines or the law of cosines as the first step in
solving the triangle? Explain.

Accepted Answer

Must use the law of cosines, s

Simplify the expression. - $\frac { 1 } { \csc x - \cot x } + \frac { 1 } { \csc x + \cot x }$

Find all solutions to the equation. - $5 \sin ^ { 2 } x - 10 \sin x + 8 = 3$

Write each expression in factored form as an algebraic expression of a single trigonometric function. - $4 \cot ^ { 2 } x - \frac { 4 } { \tan x } + \cos x \sec x$

Prove the identity. - $\frac { \cot x } { 1 + \csc x } = \frac { \csc x - 1 } { \cot x }$

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

Solve the triangle. -

Use the fundamental identities to find the value of the trigonometric function. -Given that $\cos \left( \theta - \frac { \pi } { 2 } \right) = - \frac { 4 } { 5 }$ , find $\sin ( - \theta )$ .

Solve the triangle. - $\mathrm { A } = 52 ^ { \circ } , \mathrm { B } = 63 ^ { \circ } , \mathrm { a } = 8$

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

Use basic identities to simplify the expression. - $\cos \theta - \cos \theta \sin ^ { 2 } \theta$

Simplify the expression to either 1 or -1. - $\cot ^ { 2 } x - \csc ^ { 2 } ( - x )$

Find an exact value. - $\cos \frac { - 7 \pi } { 12 }$

Prove the identity. - $\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } x$

Solve the triangle. - $\mathrm { A } = 19 ^ { \circ } , \mathrm { C } = 105 ^ { \circ } , \mathrm { c } = 6$

Prove the identity. -sin (x + y) - sin (x - y) = 2 cos x sin y

Prove the identity. -cos (x - y) - cos (x + y) = 2 sin x sin y

Prove the identity. - $\cos x \csc x \tan x = 1$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cot \theta$ if $\csc \theta = \frac { \sqrt { 17 } } { 4 }$ and $\tan \theta > 0$ .

Find $\sin \theta$ if $\cot \theta = - 4$ and $\cos \theta < 0$ .

Given the SAS parts of a triangle, is it better to use the law of sines or the law of cosines as the first step in solving the triangle? Explain.

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Exam 5: Analytic Trigonometry

Simplify the expression. - 1csc⁡x−cot⁡x+1csc⁡x+cot⁡x\frac { 1 } { \csc x - \cot x } + \frac { 1 } { \csc x + \cot x }cscx−cotx1​+cscx+cotx1​

Find all solutions to the equation. - 5sin⁡2x−10sin⁡x+8=35 \sin ^ { 2 } x - 10 \sin x + 8 = 35sin2x−10sinx+8=3

Write each expression in factored form as an algebraic expression of a single trigonometric function. - 4cot⁡2x−4tan⁡x+cos⁡xsec⁡x4 \cot ^ { 2 } x - \frac { 4 } { \tan x } + \cos x \sec x4cot2x−tanx4​+cosxsecx

Prove the identity. - cot⁡x1+csc⁡x=csc⁡x−1cot⁡x\frac { \cot x } { 1 + \csc x } = \frac { \csc x - 1 } { \cot x }1+cscxcotx​=cotxcscx−1​

Determine if the following is an identity. - (sin⁡x+cos⁡x)2=1( \sin x + \cos x ) ^ { 2 } = 1(sinx+cosx)2=1