Question 1

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the
triangle.
-a = 35 b = 20.2
C = 12.8

Accepted Answer

A)  No triangle is formed. 
B)  101.19 
C)  98.26 
D)  99.73 
A)  No triangle is formed. 
B)  101.19 
C)  98.26 
D)  99.73

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

A)  $2 \sin ^ { 3 } x \cos ^ { 4 } x + 1$ 
B)  $2 \sin x \cos ^ { 2 } x + \sin x - 2 \sin ^ { 3 } x - \cos x$ 
C)  $2 \sin x - \sin ^ { 3 } x - \cos x$ 
D)  $3 \sin x \cos x - \sin ^ { 3 } x - \cos x$ 
A)  $2 \sin ^ { 3 } x \cos ^ { 4 } x + 1$ 
B)  $2 \sin x \cos ^ { 2 } x + \sin x - 2 \sin ^ { 3 } x - \cos x$ 
C)  $2 \sin x - \sin ^ { 3 } x - \cos x$ 
D)  $3 \sin x \cos x - \sin ^ { 3 } x - \cos x$

Question 4

Match the graph with the correct equation.
-

Accepted Answer

A)  y = sin 3x cos (1)- cos 3x sin (1) 
B)  y = cos 3x cos (1) + sin 3x sin (1) 
C)  y = cos 3x cos (1) - sin 3x sin (1) 
D)  y = sin 3x cos (1) + cos 3x sin (1) 
A)  y = sin 3x cos (1)- cos 3x sin (1) 
B)  y = cos 3x cos (1) + sin 3x sin (1) 
C)  y = cos 3x cos (1) - sin 3x sin (1) 
D)  y = sin 3x cos (1) + cos 3x sin (1)

Question 5

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the
triangle.
-a = 62 b = 70
C = 83.6

Accepted Answer

A)  2939.52 
B)  2126.98 
C)  2125.18 
D)  No triangle is formed. 
A)  2939.52 
B)  2126.98 
C)  2125.18 
D)  No triangle is formed.

Question 6

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

Accepted Answer

\[\begin{array} { l } 
\cos \left( x + \

Question 7

Prove the identity.
-$$\sin \left( \frac { \pi } { 4 } + x \right) = 2 \sqrt { 2 } ( \cos x + \sin x )$$

Accepted Answer

The answer of Prove the identity.
-\[\sin \left( \frac { \pi...

Question 8

Explain the condition that must exist to determine that there may be two triangles satisfying given values of a, $b$, and $A$, once the value of $\sin \mathrm { A }$ is found. What kind of triangle will this be?

Accepted Answer

There may be two triangles if

Question 9

Find all solutions to the equation in the interval [0, 2).
-$\sin 2 x - \sin 4 x = 0$

Accepted Answer

A)  $\frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
B)  $0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
C)  $0 , \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \pi , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
D)  $0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
A)  $\frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
B)  $0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
C)  $0 , \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \pi , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
D)  $0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$

Question 10

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

Accepted Answer

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

Question 11

$$\text { What happens if } C = 90 ^ { \circ } \text { when the law of cosines is applied in the form } c ^ { 2 } = a ^ { 2 } + b ^ { 2 } - 2 a b \cos C \text { ? }$$

Accepted Answer

The answer of \[\text { What happens if } C...

Question 12

Solve the triangle.
-$$\mathrm { a } = 3 , \mathrm {~b} = 11 , \mathrm { c } = 6$$

Accepted Answer

A)  No triangles possible 
B)  $\mathrm { A } \approx 31.8 ^ { \circ } , \mathrm { B } \approx 63.6 ^ { \circ } , \mathrm { C } \approx 84.6 ^ { \circ }$ 
C)  $\mathrm { A } \approx 63.6 ^ { \circ } , \mathrm { B } \approx 84.6 ^ { \circ } , \mathrm { C } \approx 11 ^ { \circ }$ 
D)  $\mathrm { A } \approx 31.8 ^ { \circ } , \mathrm { B } \approx 43.9 ^ { \circ } , \mathrm { C } \approx 104.3 ^ { \circ }$ 
A)  No triangles possible 
B)  $\mathrm { A } \approx 31.8 ^ { \circ } , \mathrm { B } \approx 63.6 ^ { \circ } , \mathrm { C } \approx 84.6 ^ { \circ }$ 
C)  $\mathrm { A } \approx 63.6 ^ { \circ } , \mathrm { B } \approx 84.6 ^ { \circ } , \mathrm { C } \approx 11 ^ { \circ }$ 
D)  $\mathrm { A } \approx 31.8 ^ { \circ } , \mathrm { B } \approx 43.9 ^ { \circ } , \mathrm { C } \approx 104.3 ^ { \circ }$

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

A)  $\frac { - \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
B)  $\frac { - \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
D)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
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 15

Simplify the expression to either 1 or -1.
-$\tan \left( \frac { \pi } { 2 } - x \right) \tan x$

Accepted Answer

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

Question 16

Solve the triangle.
-$$\mathrm { A } = 39 ^ { \circ } , \mathrm { B } = 32 ^ { \circ } , \mathrm { b } = 11$$

Accepted Answer

A)  $\mathrm { C } = 19 ^ { \circ } , \mathrm { a } \approx 9.3 , \mathrm { c } \approx 16.5$ 
B)  $\mathrm { C } = 109 ^ { \circ } , \mathrm { a } \approx 13 , \mathrm { c } \approx 19.6$ 
C)  $\mathrm { C } = 109 ^ { \circ } , \mathrm { a } \approx 9.3 , \mathrm { c } \approx 19.6$ 
D)  $\mathrm { C } = 109 ^ { \circ } , \mathrm { a } \approx 9.3 , \mathrm { c } \approx 16.5$ 
A)  $\mathrm { C } = 19 ^ { \circ } , \mathrm { a } \approx 9.3 , \mathrm { c } \approx 16.5$ 
B)  $\mathrm { C } = 109 ^ { \circ } , \mathrm { a } \approx 13 , \mathrm { c } \approx 19.6$ 
C)  $\mathrm { C } = 109 ^ { \circ } , \mathrm { a } \approx 9.3 , \mathrm { c } \approx 19.6$ 
D)  $\mathrm { C } = 109 ^ { \circ } , \mathrm { a } \approx 9.3 , \mathrm { c } \approx 16.5$

Question 17

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$ if $\sin \theta = - \frac { 12 } { 13 }$ and $\tan \theta > 0$.

Accepted Answer

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

Question 18

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

Accepted Answer

A)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  $0 , \pi$ 
D)  No solution 
A)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  $0 , \pi$ 
D)  No solution

Question 19

Find the area. Round your answer to the nearest hundredth if necessary.
-Find the area of the triangle with the following measurements: $\mathrm { A } = 55 ^ { \circ } , \mathrm { b } = 10 \mathrm {~m} , \mathrm { c } = 22 \mathrm {~m}$

Accepted Answer

A)  $180.21 \mathrm {~m} ^ { 2 }$ 
B)  $63.09 \mathrm {~m} ^ { 2 }$ 
C)  $110 \mathrm {~m} ^ { 2 }$ 
D)  $90.11 \mathrm {~m} ^ { 2 }$ 
A)  $180.21 \mathrm {~m} ^ { 2 }$ 
B)  $63.09 \mathrm {~m} ^ { 2 }$ 
C)  $110 \mathrm {~m} ^ { 2 }$ 
D)  $90.11 \mathrm {~m} ^ { 2 }$

Question 20

Provide an appropriate response.
-A student claims that the equation $\tan \theta + 1 = \sec \theta$ is an identity, since by letting $\theta = 0 ^ { \circ }$ we get $0 + 1 = 1$, a true statement. Comment on this student's reasoning.

Accepted Answer

The student's reasoning is incorrect. An

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the triangle. -a = 35 b = 20.2 C = 12.8

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

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

Match the graph with the correct equation. -

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the triangle. -a = 62 b = 70 C = 83.6

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

Prove the identity. - $\sin \left( \frac { \pi } { 4 } + x \right) = 2 \sqrt { 2 } ( \cos x + \sin x )$

Explain the condition that must exist to determine that there may be two triangles satisfying given values of a, $b$ , and $A$ , once the value of $\sin \mathrm { A }$ is found. What kind of triangle will this be?

Find all solutions to the equation in the interval [0, 2). - $\sin 2 x - \sin 4 x = 0$

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

$\text { What happens if } C = 90 ^ { \circ } \text { when the law of cosines is applied in the form } c ^ { 2 } = a ^ { 2 } + b ^ { 2 } - 2 a b \cos C \text { ? }$

Solve the triangle. - $\mathrm { a } = 3 , \mathrm {~b} = 11 , \mathrm { c } = 6$

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

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

Simplify the expression to either 1 or -1. - $\tan \left( \frac { \pi } { 2 } - x \right) \tan x$

Solve the triangle. - $\mathrm { A } = 39 ^ { \circ } , \mathrm { B } = 32 ^ { \circ } , \mathrm { b } = 11$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\sin \theta = - \frac { 12 } { 13 }$ and $\tan \theta > 0$ .

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

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: $\mathrm { A } = 55 ^ { \circ } , \mathrm { b } = 10 \mathrm {~m} , \mathrm { c } = 22 \mathrm {~m}$

Provide an appropriate response. -A student claims that the equation $\tan \theta + 1 = \sec \theta$ is an identity, since by letting $\theta = 0 ^ { \circ }$ we get $0 + 1 = 1$ , a true statement. Comment on this student's reasoning.

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

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Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

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Exam 5: Analytic Trigonometry

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the triangle. -a = 35 b = 20.2 C = 12.8

Prove the identity. - cot⁡2x+csc⁡2x=2csc⁡2x−1\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1cot2x+csc2x=2csc2x−1

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

Match the graph with the correct equation. -

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the triangle. -a = 62 b = 70 C = 83.6

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

Prove the identity. - sin⁡(π4+x)=22(cos⁡x+sin⁡x)\sin \left( \frac { \pi } { 4 } + x \right) = 2 \sqrt { 2 } ( \cos x + \sin x )sin(4π​+x)=22​(cosx+sinx)

Explain the condition that must exist to determine that there may be two triangles satisfying given values of a, bbb , and AAA , once the value of sin⁡A\sin \mathrm { A }sinA is found. What kind of triangle will this be?

Find all solutions to the equation in the interval [0, 2). - sin⁡2x−sin⁡4x=0\sin 2 x - \sin 4 x = 0sin2x−sin4x=0

Prove the identity. - sin⁡x1−cos⁡x+sin⁡x1+cos⁡x=2csc⁡x\frac { \sin x } { 1 - \cos x } + \frac { \sin x } { 1 + \cos x } = 2 \csc x1−cosxsinx​+1+cosxsinx​=2cscx

Solve the triangle. - a=3, b=11,c=6\mathrm { a } = 3 , \mathrm {~b} = 11 , \mathrm { c } = 6a=3, b=11,c=6

Find an exact value. - tan⁡−7π12\tan \frac { - 7 \pi } { 12 }tan12−7π​

Find an exact value. - cos⁡π12\cos \frac { \pi } { 12 }cos12π​

Simplify the expression to either 1 or -1. - tan⁡(π2−x)tan⁡x\tan \left( \frac { \pi } { 2 } - x \right) \tan xtan(2π​−x)tanx

Solve the triangle. - A=39∘,B=32∘,b=11\mathrm { A } = 39 ^ { \circ } , \mathrm { B } = 32 ^ { \circ } , \mathrm { b } = 11A=39∘,B=32∘,b=11

Use the fundamental identities to find the value of the trigonometric function. -Find cos⁡θ\cos \thetacosθ if sin⁡θ=−1213\sin \theta = - \frac { 12 } { 13 }sinθ=−1312​ and tan⁡θ>0\tan \theta > 0tanθ>0 .

Find all solutions to the equation in the interval [0, 2). - cos⁡x−cot⁡x=0\cos x - \cot x = 0cosx−cotx=0

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of the triangle with the following measurements: A=55∘,b=10 m,c=22 m\mathrm { A } = 55 ^ { \circ } , \mathrm { b } = 10 \mathrm {~m} , \mathrm { c } = 22 \mathrm {~m}A=55∘,b=10 m,c=22 m

Provide an appropriate response. -A student claims that the equation tan⁡θ+1=sec⁡θ\tan \theta + 1 = \sec \thetatanθ+1=secθ is an identity, since by letting θ=0∘\theta = 0 ^ { \circ }θ=0∘ we get 0+1=10 + 1 = 10+1=1 , a true statement. Comment on this student's reasoning.