Question 1

Prove the identity.
-$$\cos ^ { 4 } x = \frac { 1 } { 8 } ( 3 + 4 \cos 2 x + \cos 4 x )$$

Accepted Answer

\[\begin{array} { l } 
\cos ^ { 4 } x =

Question 2

Explain, in your own words, the situation called "the ambiguous case of the law of sines."

Accepted Answer

The ambiguous case of the law of sines o

Question 3

Provide an appropriate response.
-Graph the expression cos x + sin x tan 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 cos...

Question 4

Solve the triangle.
-

Accepted Answer

A)  $\mathrm { B } = 34.7 ^ { \circ } , \mathrm { C } = 20.3 ^ { \circ } , \mathrm { a } = 21.5$ 
B)  $\mathrm { B } = 20.3 ^ { \circ } , \mathrm { C } = 34.7 ^ { \circ } , \mathrm { a } = 461$ 
C)  $\mathrm { B } = 34.7 ^ { \circ } , \mathrm { C } = 20.3 ^ { \circ } \mathrm { a } = 461$ 
D)  $B = 20.3 ^ { \circ } , \mathrm { C } = 34.7 ^ { \circ } , \mathrm { a } = 21.5$ 
A)  $\mathrm { B } = 34.7 ^ { \circ } , \mathrm { C } = 20.3 ^ { \circ } , \mathrm { a } = 21.5$ 
B)  $\mathrm { B } = 20.3 ^ { \circ } , \mathrm { C } = 34.7 ^ { \circ } , \mathrm { a } = 461$ 
C)  $\mathrm { B } = 34.7 ^ { \circ } , \mathrm { C } = 20.3 ^ { \circ } \mathrm { a } = 461$ 
D)  $B = 20.3 ^ { \circ } , \mathrm { C } = 34.7 ^ { \circ } , \mathrm { a } = 21.5$

Question 5

Prove the identity.
-$$\sin 6 x = 2 \sin x \cos x \left( 3 - 4 \sin ^ { 2 } x \right) \left( 1 - 4 \sin ^ { 2 } x \right)$$

Accepted Answer

\[\begin{array} { l } 
\sin 6 x = 2 \sin

Question 6

Use basic identities to simplify the expression.
-$$\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta$$

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

Find an exact value.
-$\cos 15 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
B)  $\frac { - \sqrt { 6 } + 1 } { 4 }$ 
C)  $\frac { - \sqrt { 2 } + \sqrt { 6 } } { 4 }$ 
D)  $\frac { - \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
A)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
B)  $\frac { - \sqrt { 6 } + 1 } { 4 }$ 
C)  $\frac { - \sqrt { 2 } + \sqrt { 6 } } { 4 }$ 
D)  $\frac { - \sqrt { 2 } - \sqrt { 6 } } { 4 }$

Question 9

Solve the triangle.
-$$B = 72 ^ { \circ } , b = 12 , c = 10$$

Accepted Answer

A)  $\mathrm { C } = 52.4 ^ { \circ } , \mathrm { A } = 55.6 ^ { \circ } , \mathrm { a } \approx 10.4$ 
B)  Cannot be solved 
C)  $\mathrm { C } = 57.6 ^ { \circ } , \mathrm { A } = 50.6 ^ { \circ } , \mathrm { a } \approx 10.4$ 
D)  $\mathrm { C } = 52.4 ^ { \circ } , \mathrm { A } = 55.6 ^ { \circ } , \mathrm { a } \approx 14.6$ 
A)  $\mathrm { C } = 52.4 ^ { \circ } , \mathrm { A } = 55.6 ^ { \circ } , \mathrm { a } \approx 10.4$ 
B)  Cannot be solved 
C)  $\mathrm { C } = 57.6 ^ { \circ } , \mathrm { A } = 50.6 ^ { \circ } , \mathrm { a } \approx 10.4$ 
D)  $\mathrm { C } = 52.4 ^ { \circ } , \mathrm { A } = 55.6 ^ { \circ } , \mathrm { a } \approx 14.6$

Question 10

Find an exact value.
-$\tan 15 ^ { \circ }$

Accepted Answer

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

Question 11

Find the area. Round your answer to the nearest hundredth if necessary.
-Find the area of the triangle with the following measurements:
$$\mathrm { B } = 108 ^ { \circ } , \mathrm { a } = 12 \mathrm {~cm} , \mathrm { c } = 20 \mathrm {~cm}$$

Accepted Answer

A)  $120 \mathrm {~cm} ^ { 2 }$ 
B)  $228.25 \mathrm {~cm} ^ { 2 }$ 
C)  $37.08 \mathrm {~cm} ^ { 2 }$ 
D)  $114.13 \mathrm {~cm} ^ { 2 }$ 
A)  $120 \mathrm {~cm} ^ { 2 }$ 
B)  $228.25 \mathrm {~cm} ^ { 2 }$ 
C)  $37.08 \mathrm {~cm} ^ { 2 }$ 
D)  $114.13 \mathrm {~cm} ^ { 2 }$

Question 12

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

Accepted Answer

A)  $1 - \cos \theta$ 
B)  $2 \sec ^ { 2 } \theta$ 
C)  $\tan ^ { 2 } \theta$ 
D)  $3 \sec \theta$ as 
A)  $1 - \cos \theta$ 
B)  $2 \sec ^ { 2 } \theta$ 
C)  $\tan ^ { 2 } \theta$ 
D)  $3 \sec \theta$ as

Question 13

Find all solutions in the interval [0, 2π]
-$\sin ( \cos x ) = 0$

Accepted Answer

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

Question 14

Solve the problem.
-A tower is supported by a guy wire 473 ft long. If the wire makes an angle of $$39 ^ { \circ }$$ with respect to the ground and the distance from the point where the wire is attached to the ground and the tower is 129 ft, how tall is the
Tower? Round your answer to the nearest tenth.

Accepted Answer

A)  579.0 ft 
B)  536.5 ft 
C)  381.5 ft 
D)  439.3 ft 
A)  579.0 ft 
B)  536.5 ft 
C)  381.5 ft 
D)  439.3 ft

Question 15

Find all solutions to the equation.
-$\cos x = - \frac { 1 } { 2 }$ (Express your answer in radians, in exact form.)

Accepted Answer

A)  $\left\{ \frac { 5 \pi } { 6 } + \mathrm { n } \pi \quad \mathrm { |n } = 0 , \pm 1 , \pm 2 , \ldots . \cdot \right\}$ 
B)  $\left\{ \frac { 2 \pi } { 3 } + 2 \mathrm { n } \pi , \frac { 4 \pi } { 3 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
C)  $\left\{ \frac { 2 \pi } { 3 } + n \pi \quad \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
D)  $\left\{ \frac { 5 \pi } { 6 } + 2 \mathrm { n } \pi , \frac { 7 \pi } { 6 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
A)  $\left\{ \frac { 5 \pi } { 6 } + \mathrm { n } \pi \quad \mathrm { |n } = 0 , \pm 1 , \pm 2 , \ldots . \cdot \right\}$ 
B)  $\left\{ \frac { 2 \pi } { 3 } + 2 \mathrm { n } \pi , \frac { 4 \pi } { 3 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
C)  $\left\{ \frac { 2 \pi } { 3 } + n \pi \quad \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$ 
D)  $\left\{ \frac { 5 \pi } { 6 } + 2 \mathrm { n } \pi , \frac { 7 \pi } { 6 } + 2 \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots \right\}$

Question 16

Find an exact value.
-$$\sin \frac { - 11 \pi } { 12 }$$

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 17

Rewrite with only sin x and cos x.
-$\sin 3 x - \cos 2 x$

Accepted Answer

A)  $2 \sin x \cos ^ { 2 } x - \sin x + 2 \sin ^ { 3 } x + 1 - 2 \sin ^ { 2 } x$ 
B)  $2 \sin x \cos ^ { 2 } x + \sin x - 2 \sin ^ { 3 } x - 1 - 2 \sin ^ { 2 } x$ 
C)  $2 \sin x \cos ^ { 2 } x + \sin x - 2 \sin ^ { 3 } x - 1 + 2 \sin ^ { 2 } x$ 
D)  $2 \sin x \cos ^ { 2 } x + \sin x - 2 \sin ^ { 2 } x - 1 + 2 \sin ^ { 2 } x$ 
A)  $2 \sin x \cos ^ { 2 } x - \sin x + 2 \sin ^ { 3 } x + 1 - 2 \sin ^ { 2 } x$ 
B)  $2 \sin x \cos ^ { 2 } x + \sin x - 2 \sin ^ { 3 } x - 1 - 2 \sin ^ { 2 } x$ 
C)  $2 \sin x \cos ^ { 2 } x + \sin x - 2 \sin ^ { 3 } x - 1 + 2 \sin ^ { 2 } x$ 
D)  $2 \sin x \cos ^ { 2 } x + \sin x - 2 \sin ^ { 2 } x - 1 + 2 \sin ^ { 2 } x$

Question 18

Provide an appropriate response.
-Suppose someone tells you that the cosine of the difference of two angles is the difference of their cosines. Write
in your own words how you would correct the person's statement.

Accepted Answer

Wording will vary, but an acce

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.
-$$\mathrm { A } = 39 ^ { \circ } , \mathrm { b } = 15 , \mathrm {~B} = 28 ^ { \circ }$$

Accepted Answer

A)  $\mathrm { C } = 113 ^ { \circ } , \mathrm { a } \approx 11.2 , \mathrm { c } \approx 7.7$ 
B)  The triangle cannot be solved with the Law of Sines. 
C)  No triangle is formed. 
D)  $\mathrm { C } = 113 ^ { \circ } , \mathrm { a } \approx 20.1 , \mathrm { c } \approx 29.4$ 
A)  $\mathrm { C } = 113 ^ { \circ } , \mathrm { a } \approx 11.2 , \mathrm { c } \approx 7.7$ 
B)  The triangle cannot be solved with the Law of Sines. 
C)  No triangle is formed. 
D)  $\mathrm { C } = 113 ^ { \circ } , \mathrm { a } \approx 20.1 , \mathrm { c } \approx 29.4$

Question 20

Solve.
-Two tracking stations are on the equator 140 miles apart. A weather balloon is located on a bearing of $\mathrm { N } ^ { \circ } \mathrm { E }$ from the western station and on a bearing of $\mathrm { N } \mathrm {} 13 ^ { \circ } \mathrm { E }$ from the eastern station. How far is the balloon from the western station?

Accepted Answer

A)  294 miles 
B)  408 miles 
C)  285 miles 
D)  399 miles 
A)  294 miles 
B)  408 miles 
C)  285 miles 
D)  399 miles

Prove the identity. - $\cos ^ { 4 } x = \frac { 1 } { 8 } ( 3 + 4 \cos 2 x + \cos 4 x )$

Explain, in your own words, the situation called "the ambiguous case of the law of sines."

Provide an appropriate response. -Graph the expression cos x + sin x tan x on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Solve the triangle. -

Prove the identity. - $\sin 6 x = 2 \sin x \cos x \left( 3 - 4 \sin ^ { 2 } x \right) \left( 1 - 4 \sin ^ { 2 } x \right)$

Use basic identities to simplify the expression. - $\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta$

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

Find an exact value. - $\cos 15 ^ { \circ }$

Solve the triangle. - $B = 72 ^ { \circ } , b = 12 , c = 10$

Find an exact value. - $\tan 15 ^ { \circ }$

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: $\mathrm { B } = 108 ^ { \circ } , \mathrm { a } = 12 \mathrm {~cm} , \mathrm { c } = 20 \mathrm {~cm}$

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

Find all solutions in the interval [0, 2π] - $\sin ( \cos x ) = 0$

Find all solutions to the equation. - $\cos x = - \frac { 1 } { 2 }$ (Express your answer in radians, in exact form.)

Find an exact value. - $\sin \frac { - 11 \pi } { 12 }$

Rewrite with only sin x and cos x. - $\sin 3 x - \cos 2 x$

Provide an appropriate response. -Suppose someone tells you that the cosine of the difference of two angles is the difference of their cosines. Write in your own words how you would correct the person's statement.

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

Prerequisites

Filters

Exam 5: Analytic Trigonometry

Prove the identity. - cos⁡4x=18(3+4cos⁡2x+cos⁡4x)\cos ^ { 4 } x = \frac { 1 } { 8 } ( 3 + 4 \cos 2 x + \cos 4 x )cos4x=81​(3+4cos2x+cos4x)

Explain, in your own words, the situation called "the ambiguous case of the law of sines."

Provide an appropriate response. -Graph the expression cos x + sin x tan x on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Solve the triangle. -

Prove the identity. - sin⁡6x=2sin⁡xcos⁡x(3−4sin⁡2x)(1−4sin⁡2x)\sin 6 x = 2 \sin x \cos x \left( 3 - 4 \sin ^ { 2 } x \right) \left( 1 - 4 \sin ^ { 2 } x \right)sin6x=2sinxcosx(3−4sin2x)(1−4sin2x)

Use basic identities to simplify the expression. - 1cot⁡2θ+sec⁡θcos⁡θ\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \thetacot2θ1​+secθcosθ

Find an exact value. - cos⁡105∘\cos 105 ^ { \circ }cos105∘

Find an exact value. - cos⁡15∘\cos 15 ^ { \circ }cos15∘

Solve the triangle. - B=72∘,b=12,c=10B = 72 ^ { \circ } , b = 12 , c = 10B=72∘,b=12,c=10

Find an exact value. - tan⁡15∘\tan 15 ^ { \circ }tan15∘

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: B=108∘,a=12 cm,c=20 cm\mathrm { B } = 108 ^ { \circ } , \mathrm { a } = 12 \mathrm {~cm} , \mathrm { c } = 20 \mathrm {~cm}B=108∘,a=12 cm,c=20 cm

Find all solutions in the interval [0, 2π] - sin⁡(cos⁡x)=0\sin ( \cos x ) = 0sin(cosx)=0

Find all solutions to the equation. - cos⁡x=−12\cos x = - \frac { 1 } { 2 }cosx=−21​ (Express your answer in radians, in exact form.)

Find an exact value. - sin⁡−11π12\sin \frac { - 11 \pi } { 12 }sin12−11π​

Rewrite with only sin x and cos x. - sin⁡3x−cos⁡2x\sin 3 x - \cos 2 xsin3x−cos2x

Provide an appropriate response. -Suppose someone tells you that the cosine of the difference of two angles is the difference of their cosines. Write in your own words how you would correct the person's statement.

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

Prerequisites

Filters

Prove the identity. - $\cos ^ { 4 } x = \frac { 1 } { 8 } ( 3 + 4 \cos 2 x + \cos 4 x )$

Prove the identity. - $\sin 6 x = 2 \sin x \cos x \left( 3 - 4 \sin ^ { 2 } x \right) \left( 1 - 4 \sin ^ { 2 } x \right)$

Use basic identities to simplify the expression. - $\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta$

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

Find an exact value. - $\cos 15 ^ { \circ }$

Solve the triangle. - $B = 72 ^ { \circ } , b = 12 , c = 10$

Find an exact value. - $\tan 15 ^ { \circ }$

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: $\mathrm { B } = 108 ^ { \circ } , \mathrm { a } = 12 \mathrm {~cm} , \mathrm { c } = 20 \mathrm {~cm}$

Find all solutions in the interval [0, 2π] - $\sin ( \cos x ) = 0$

Find all solutions to the equation. - $\cos x = - \frac { 1 } { 2 }$ (Express your answer in radians, in exact form.)

Find an exact value. - $\sin \frac { - 11 \pi } { 12 }$

Rewrite with only sin x and cos x. - $\sin 3 x - \cos 2 x$