Question 1

Provide an appropriate response.
-$$\text { Show that } \csc ( A + B ) = \frac { \csc A \csc B } { \cot B + \cot A } \text {. }$$

Accepted Answer

To show that \csc(A + B) = \frac{\csc A

Question 2

Find all solutions to the equation in the interval [0, 2).
-$\sin 2 x + \cos 3 x = 0$

Accepted Answer

A)  $0.3 \pi , 0.5 \pi , 0.7 \pi , 1.9 \pi$ 
B)  $0.3 \pi , 0.5 \pi , 0.7 \pi , 1.1 \pi , 1.5 \pi , 1.9 \pi$ 
C)  $0.3 \pi , 0.7 \pi , 1.1 \pi , 1.9 \pi$ 
D)  $0.5 \pi , 1.5 \pi$ 
A)  $0.3 \pi , 0.5 \pi , 0.7 \pi , 1.9 \pi$ 
B)  $0.3 \pi , 0.5 \pi , 0.7 \pi , 1.1 \pi , 1.5 \pi , 1.9 \pi$ 
C)  $0.3 \pi , 0.7 \pi , 1.1 \pi , 1.9 \pi$ 
D)  $0.5 \pi , 1.5 \pi$

Question 3

State whether the given measurements determine zero, one, or two triangles.
-$$B = 87 ^ { \circ } , b = 27 , c = 28$$

Accepted Answer

A)  Zero 
B)  Two 
C)  One 
A)  Zero 
B)  Two 
C)  One

Question 4

Find an exact value.
-$\sin 105 ^ { \circ }$

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 5

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)  $\mathrm { B } = 57 ^ { \circ } , \mathrm { a } \approx 5.4$ 
B)  $\mathrm { B } = 57 ^ { \circ } , \mathrm { a } \approx 53.1$ 
C)  $B = 33 ^ { \circ } , a \approx 53.1$ 
D)  No triangle is formed. 
A)  $\mathrm { B } = 57 ^ { \circ } , \mathrm { a } \approx 5.4$ 
B)  $\mathrm { B } = 57 ^ { \circ } , \mathrm { a } \approx 53.1$ 
C)  $B = 33 ^ { \circ } , a \approx 53.1$ 
D)  No triangle is formed.

Question 6

Solve the problem.
-A building has a ramp to its front doors to accommodate the handicapped. If the distance from the building to the end of the ramp is 13 feet and the height from the ground to the front doors is 4 feet, how long is the ramp?
(Round to the nearest tenth.)

Accepted Answer

A)  13.6 ft 
B)  12.4 ft 
C)  4.1 ft 
D)  5.7 ft 
A)  13.6 ft 
B)  12.4 ft 
C)  4.1 ft 
D)  5.7 ft

Question 7

Find an exact value.
-$\cos 165 ^ { \circ }$

Accepted Answer

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

Question 8

Write the expression as the sine, cosine, or tangent of an angle.
-$\cos \frac { \pi } { 7 } \cos \frac { \pi } { 11 } + \sin \frac { \pi } { 7 } \sin \frac { \pi } { 11 }$

Accepted Answer

A)  $\sin \frac { 18 \pi } { 77 }$ 
B)  $\cos \frac { 4 \pi } { 77 }$ 
C)  $\sin \frac { 4 \pi } { 77 }$ 
D)  $\cos \frac { 18 \pi } { 77 }$ 
A)  $\sin \frac { 18 \pi } { 77 }$ 
B)  $\cos \frac { 4 \pi } { 77 }$ 
C)  $\sin \frac { 4 \pi } { 77 }$ 
D)  $\cos \frac { 18 \pi } { 77 }$

Question 9

Write each expression in factored form as an algebraic expression of a single trigonometric function.
-$\csc ^ { 2 } x - 1$

Accepted Answer

A)  $( \csc x + 1 ) ( \csc x - 1 )$ 
B)  $\cot x$ 
C)  $( \cot x + 1 ) ( \cot x - 1 )$ 
D)  $\csc x - 1$ 
A)  $( \csc x + 1 ) ( \csc x - 1 )$ 
B)  $\cot x$ 
C)  $( \cot x + 1 ) ( \cot x - 1 )$ 
D)  $\csc x - 1$

Question 10

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

Accepted Answer

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

Question 11

Find all solutions in the interval [0, 2π]
-$$\sin ^ { 2 } x + \sin x = 0$$

Accepted Answer

A)  $x = 0 , \pi , \frac { 3 \pi } { 2 }$ 
B)  $x = 0 , \pi , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
C)  $x = 0 , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
D)  $x = 0 , \pi , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
A)  $x = 0 , \pi , \frac { 3 \pi } { 2 }$ 
B)  $x = 0 , \pi , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
C)  $x = 0 , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
D)  $x = 0 , \pi , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$

Question 12

Prove the identity.
-$$\cos 4 x = 1 - 2 \sin ^ { 2 } 2 x$$

Accepted Answer

The answer of Prove the identity.
-\[\cos 4 x = 1...

Question 13

Prove the identity.
-$$4 \csc 2 x = 2 \csc ^ { 2 } x \tan x$$

Accepted Answer

The answer of Prove the identity.
-\[4 \csc 2 x =...

Question 14

Find the area. Round your answer to the nearest hundredth if necessary.
-Find the area of the triangle with the following measurements: $\mathrm { A } = 50 ^ { \circ } , \mathrm { b } = 25 \mathrm { ft } , \mathrm { c } = 15 \mathrm { ft }$

Accepted Answer

A)  $120.52 \mathrm { ft } ^ { 2 }$ 
B)  $375 \mathrm { ft } ^ { 2 }$ 
C)  $287.27 \mathrm { ft } ^ { 2 }$ 
D)  $143.63 \mathrm { ft } ^ { 2 }$ 
A)  $120.52 \mathrm { ft } ^ { 2 }$ 
B)  $375 \mathrm { ft } ^ { 2 }$ 
C)  $287.27 \mathrm { ft } ^ { 2 }$ 
D)  $143.63 \mathrm { ft } ^ { 2 }$

Question 15

Two triangles can be formed using the given measurements. Solve both triangles.
-$$\mathrm { B } = 35 ^ { \circ } , \mathrm { b } = 23 , \mathrm { c } = 24$$

Accepted Answer

A)  $\mathrm { A } = 91.8 ^ { \circ } , \mathrm { C } = 53.2 ^ { \circ } , \mathrm { a } = 13.2 ; \mathrm { A } = 88.2 ^ { \circ } , \mathrm { C } = 126.8 ^ { \circ } , \mathrm { a } = 13.2$ 
B)  $\mathrm { A } = 91.8 ^ { \circ } , \mathrm { C } = 53.2 ^ { \circ } , \mathrm { a } = 40.1 ; \mathrm { A } = 88.2 ^ { \circ } , \mathrm { C } = 126.8 ^ { \circ } , \mathrm { a } = 40.1$ 
C)  $\mathrm { A } = 108.2 ^ { \circ } , \mathrm { C } = 36.8 ^ { \circ } , \mathrm { a } = 13.9 ; \mathrm { A } = 1.8 ^ { \circ } , \mathrm { C } = 143.2 ^ { \circ } , \mathrm { a } = 13.9$ 
D)  $\mathrm { A } = 108.2 ^ { \circ } , \mathrm { C } = 36.8 ^ { \circ } , \mathrm { a } = 38.1 ; \mathrm { A } = 1.8 ^ { \circ } , \mathrm { C } = 143.2 ^ { \circ } , \mathrm { a } = 1.3$ 
A)  $\mathrm { A } = 91.8 ^ { \circ } , \mathrm { C } = 53.2 ^ { \circ } , \mathrm { a } = 13.2 ; \mathrm { A } = 88.2 ^ { \circ } , \mathrm { C } = 126.8 ^ { \circ } , \mathrm { a } = 13.2$ 
B)  $\mathrm { A } = 91.8 ^ { \circ } , \mathrm { C } = 53.2 ^ { \circ } , \mathrm { a } = 40.1 ; \mathrm { A } = 88.2 ^ { \circ } , \mathrm { C } = 126.8 ^ { \circ } , \mathrm { a } = 40.1$ 
C)  $\mathrm { A } = 108.2 ^ { \circ } , \mathrm { C } = 36.8 ^ { \circ } , \mathrm { a } = 13.9 ; \mathrm { A } = 1.8 ^ { \circ } , \mathrm { C } = 143.2 ^ { \circ } , \mathrm { a } = 13.9$ 
D)  $\mathrm { A } = 108.2 ^ { \circ } , \mathrm { C } = 36.8 ^ { \circ } , \mathrm { a } = 38.1 ; \mathrm { A } = 1.8 ^ { \circ } , \mathrm { C } = 143.2 ^ { \circ } , \mathrm { a } = 1.3$

Question 16

Find all solutions to the equation.
-$\cos x = \sin x$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } + 2 \mathrm { n } \pi , \frac { 7 \pi } { 4 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
B)  $\left\{ \frac { \pi } { 4 } + n \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
C)  $\left\{ \frac { \pi } { 2 } + \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
D)  $\left\{ \frac { 3 \pi } { 4 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 4 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
A)  $\left\{ \frac { \pi } { 4 } + 2 \mathrm { n } \pi , \frac { 7 \pi } { 4 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
B)  $\left\{ \frac { \pi } { 4 } + n \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
C)  $\left\{ \frac { \pi } { 2 } + \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
D)  $\left\{ \frac { 3 \pi } { 4 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 4 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$

Question 17

Find all solutions in the interval [0, 2π).
-$$\cot \left( \frac { x } { 2 } \right) = \frac { 1 - \cos x } { 1 + \cos x }$$

Accepted Answer

A)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
B)  0 
C)  $\frac { \pi } { 2 }$ 
D)  $\frac { 3 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
A)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
B)  0 
C)  $\frac { \pi } { 2 }$ 
D)  $\frac { 3 \pi } { 4 } , \frac { 7 \pi } { 4 }$

Question 18

Simplify the expression.
-$$\frac { \cot x } { \sec ^ { 2 } x } + \frac { \cot x } { \csc ^ { 2 } x }$$

Accepted Answer

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

Question 19

Given the SSS 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

Use the law of cosin

Question 20

Find all solutions to the equation.
-$\sin x = \frac { \sqrt { 3 } } { 2 } \quad$ (Express your answer in radians, in exact form.)

Accepted Answer

A)  $\left\{ \frac { \pi } { 3 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 3 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
B)  $\left\{ \frac { \pi } { 3 } + n \pi \quad \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
C)  $\left\{ \frac { \pi } { 6 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 6 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
D)  $\left\{ \frac { \pi } { 3 } + 2 \mathrm { n } \pi , \frac { 2 \pi } { 3 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
A)  $\left\{ \frac { \pi } { 3 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 3 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
B)  $\left\{ \frac { \pi } { 3 } + n \pi \quad \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
C)  $\left\{ \frac { \pi } { 6 } + 2 \mathrm { n } \pi , \frac { 5 \pi } { 6 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
D)  $\left\{ \frac { \pi } { 3 } + 2 \mathrm { n } \pi , \frac { 2 \pi } { 3 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$

Provide an appropriate response. - $\text { Show that } \csc ( A + B ) = \frac { \csc A \csc B } { \cot B + \cot A } \text {. }$

Find all solutions to the equation in the interval [0, 2). - $\sin 2 x + \cos 3 x = 0$

State whether the given measurements determine zero, one, or two triangles. - $B = 87 ^ { \circ } , b = 27 , c = 28$

Find an exact value. - $\sin 105 ^ { \circ }$

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. -

Solve the problem. -A building has a ramp to its front doors to accommodate the handicapped. If the distance from the building to the end of the ramp is 13 feet and the height from the ground to the front doors is 4 feet, how long is the ramp? (Round to the nearest tenth.)

Find an exact value. - $\cos 165 ^ { \circ }$

Write the expression as the sine, cosine, or tangent of an angle. - $\cos \frac { \pi } { 7 } \cos \frac { \pi } { 11 } + \sin \frac { \pi } { 7 } \sin \frac { \pi } { 11 }$

Write each expression in factored form as an algebraic expression of a single trigonometric function. - $\csc ^ { 2 } x - 1$

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

Find all solutions in the interval [0, 2π] - $\sin ^ { 2 } x + \sin x = 0$

Prove the identity. - $\cos 4 x = 1 - 2 \sin ^ { 2 } 2 x$

Prove the identity. - $4 \csc 2 x = 2 \csc ^ { 2 } x \tan x$

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: $\mathrm { A } = 50 ^ { \circ } , \mathrm { b } = 25 \mathrm { ft } , \mathrm { c } = 15 \mathrm { ft }$

Two triangles can be formed using the given measurements. Solve both triangles. - $\mathrm { B } = 35 ^ { \circ } , \mathrm { b } = 23 , \mathrm { c } = 24$

Find all solutions to the equation. - $\cos x = \sin x$

Find all solutions in the interval [0, 2π). - $\cot \left( \frac { x } { 2 } \right) = \frac { 1 - \cos x } { 1 + \cos x }$

Simplify the expression. - $\frac { \cot x } { \sec ^ { 2 } x } + \frac { \cot x } { \csc ^ { 2 } x }$

Given the SSS 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.

Find all solutions to the equation. - $\sin x = \frac { \sqrt { 3 } } { 2 } \quad$ (Express your answer in radians, in exact form.)

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

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Exam 5: Analytic Trigonometry

Provide an appropriate response. - Show that csc⁡(A+B)=csc⁡Acsc⁡Bcot⁡B+cot⁡A. \text { Show that } \csc ( A + B ) = \frac { \csc A \csc B } { \cot B + \cot A } \text {. } Show that csc(A+B)=cotB+cotAcscAcscB​.

Find all solutions to the equation in the interval [0, 2). - sin⁡2x+cos⁡3x=0\sin 2 x + \cos 3 x = 0sin2x+cos3x=0

State whether the given measurements determine zero, one, or two triangles. - B=87∘,b=27,c=28B = 87 ^ { \circ } , b = 27 , c = 28B=87∘,b=27,c=28

Find an exact value. - sin⁡105∘\sin 105 ^ { \circ }sin105∘