Question 1

Simplify the expression.
-$$\frac { \cos x } { \tan ^ { 2 } x } - \frac { \cos x } { \sin ^ { 2 } x }$$

Accepted Answer

A)  $- \cos x$ 
B)  $- \sin x$ 
C)  $\sin x$ 
D)  $\cos x$ 
A)  $- \cos x$ 
B)  $- \sin x$ 
C)  $\sin x$ 
D)  $\cos x$

Question 2

Match the graph with the correct equation.
-

Accepted Answer

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

Question 3

Solve the triangle.
-$$\mathrm { A } = 33 ^ { \circ } , \mathrm { a } = 19 , \mathrm {~b} = 14$$

Accepted Answer

A)  Cannot be solved 
B)  $\mathrm { B } = 23.7 ^ { \circ } , \mathrm { C } = 143.3 ^ { \circ } , \mathrm { c } \approx 23.3$ 
C)  $\mathrm { B } = 23.7 ^ { \circ } , \mathrm { C } = 123.3 ^ { \circ } , \mathrm { C } \approx 29.2$ 
D)  $\mathrm { B } = 23.7 ^ { \circ } , \mathrm { C } = 123.3 ^ { \circ } , \mathrm { c } \approx 17.5$ 
A)  Cannot be solved 
B)  $\mathrm { B } = 23.7 ^ { \circ } , \mathrm { C } = 143.3 ^ { \circ } , \mathrm { c } \approx 23.3$ 
C)  $\mathrm { B } = 23.7 ^ { \circ } , \mathrm { C } = 123.3 ^ { \circ } , \mathrm { C } \approx 29.2$ 
D)  $\mathrm { B } = 23.7 ^ { \circ } , \mathrm { C } = 123.3 ^ { \circ } , \mathrm { c } \approx 17.5$

Question 4

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

Accepted Answer

A)  $x = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
B)  $x = 0 , \pi , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
C)  $x = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
D)  $x = \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
A)  $x = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
B)  $x = 0 , \pi , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
C)  $x = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
D)  $x = \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$

Question 5

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

Accepted Answer

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

Question 6

State whether the given measurements determine zero, one, or two triangles.
-$$\mathrm { A } = 87 ^ { \circ } , \mathrm { a } = 24 , \mathrm {~b} = 54$$

Accepted Answer

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

Question 7

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.
-$$\mathrm { C } = - 3 ^ { \circ } , \mathrm { a } = 46 , \mathrm { c } = 22$$

Accepted Answer

A)  No triangle is formed. 
B)  $\mathrm { A } = 5.2 ^ { \circ } , \mathrm { B } = 177.8 ^ { \circ } , \mathrm { b } \approx - 15.8$ 
C)  $\mathrm { A } = 177.8 ^ { \circ } , \mathrm { B } = 5.2 ^ { \circ } , \mathrm { b } \approx - 15.8$ 
D)  The triangle cannot be solved with the Law of Sines. 
A)  No triangle is formed. 
B)  $\mathrm { A } = 5.2 ^ { \circ } , \mathrm { B } = 177.8 ^ { \circ } , \mathrm { b } \approx - 15.8$ 
C)  $\mathrm { A } = 177.8 ^ { \circ } , \mathrm { B } = 5.2 ^ { \circ } , \mathrm { b } \approx - 15.8$ 
D)  The triangle cannot be solved with the Law of Sines.

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

Express the function as a sinusoid of the form y = a sin (bx + c).
-$$y = 5 \sin ( 5 x ) - 3 \cos ( 5 x )$$

Accepted Answer

A)  $y = \sqrt { 34 } \sin \left( 5 x - \sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right\}$ 
B)  $y = \sqrt { 34 } \sin \left( 5 x + \cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right)$ 
C)  $y = \sqrt { 34 } \sin \left( 5 x + \sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right\}$ 
D)  $y = \sqrt { 34 } \sin \left( 5 x - \cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right)$ 
A)  $y = \sqrt { 34 } \sin \left( 5 x - \sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right\}$ 
B)  $y = \sqrt { 34 } \sin \left( 5 x + \cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right)$ 
C)  $y = \sqrt { 34 } \sin \left( 5 x + \sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right\}$ 
D)  $y = \sqrt { 34 } \sin \left( 5 x - \cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right)$

Question 13

Solve the triangle.
-$$\mathrm { b } = 20 , \mathrm { c } = 28 , \mathrm {~A} = 81 ^ { \circ }$$

Accepted Answer

A)  $\mathrm { a } \approx 34.3 , \mathrm {~B} \approx 38.5 ^ { \circ } , \mathrm { C } \approx 60.5 ^ { \circ }$ 
B)  $\mathrm { a } \approx 34.3 , \mathrm {~B} \approx 42.5 ^ { \circ } , \mathrm { C } \approx 56.5 ^ { \circ }$ 
C)  No triangles possible 
D)  $\mathrm { a } \approx 31.8 , \mathrm {~B} \approx 38.5 ^ { \circ } , \mathrm { C } \approx 60.5 ^ { \circ }$ 
A)  $\mathrm { a } \approx 34.3 , \mathrm {~B} \approx 38.5 ^ { \circ } , \mathrm { C } \approx 60.5 ^ { \circ }$ 
B)  $\mathrm { a } \approx 34.3 , \mathrm {~B} \approx 42.5 ^ { \circ } , \mathrm { C } \approx 56.5 ^ { \circ }$ 
C)  No triangles possible 
D)  $\mathrm { a } \approx 31.8 , \mathrm {~B} \approx 38.5 ^ { \circ } , \mathrm { C } \approx 60.5 ^ { \circ }$

Question 14

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

Accepted Answer

A)  $\cot x$ 
B)  $\csc x$ 
C)  $\tan x$ 
D)  $- \cot x$ 
A)  $\cot x$ 
B)  $\csc x$ 
C)  $\tan x$ 
D)  $- \cot x$

Question 15

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

Accepted Answer

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

Question 16

Find the exact value by using a half-angle identity.
-$\cos 165 ^ { \circ }$

Accepted Answer

A)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
B)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
C)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
A)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
B)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
C)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$

Question 17

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

Accepted Answer

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

Question 18

Write each expression in factored form as an algebraic expression of a single trigonometric function.
-$1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x$

Accepted Answer

A)  $\sin ^ { 2 } x$ 
B)  $\left( 1 - \sin ^ { 2 } x \right)$ 
C)  $\cos ^ { 4 } x$ 
D)  $\left( 1 + \tan ^ { 2 } x \right)$ 
A)  $\sin ^ { 2 } x$ 
B)  $\left( 1 - \sin ^ { 2 } x \right)$ 
C)  $\cos ^ { 4 } x$ 
D)  $\left( 1 + \tan ^ { 2 } x \right)$

Question 19

Simplify the expression.
-$\frac { ( \csc y + \cot y ) ( \csc y - \cot y ) } { \csc y }$

Accepted Answer

A)  $\cos y$ 
B)  $- \sin y$ 
C)  $\sin y$ 
D)  $\csc y$ 
A)  $\cos y$ 
B)  $- \sin y$ 
C)  $\sin y$ 
D)  $\csc y$

Question 20

Prove the identity.
-$$\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$$

Accepted Answer

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

Simplify the expression. - $\frac { \cos x } { \tan ^ { 2 } x } - \frac { \cos x } { \sin ^ { 2 } x }$

Match the graph with the correct equation. -

Solve the triangle. - $\mathrm { A } = 33 ^ { \circ } , \mathrm { a } = 19 , \mathrm {~b} = 14$

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

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

State whether the given measurements determine zero, one, or two triangles. - $\mathrm { A } = 87 ^ { \circ } , \mathrm { a } = 24 , \mathrm {~b} = 54$

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

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

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

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

Express the function as a sinusoid of the form y = a sin (bx + c). - $y = 5 \sin ( 5 x ) - 3 \cos ( 5 x )$

Solve the triangle. - $\mathrm { b } = 20 , \mathrm { c } = 28 , \mathrm {~A} = 81 ^ { \circ }$

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

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

Find the exact value by using a half-angle identity. - $\cos 165 ^ { \circ }$

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

Write each expression in factored form as an algebraic expression of a single trigonometric function. - $1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x$

Simplify the expression. - $\frac { ( \csc y + \cot y ) ( \csc y - \cot y ) } { \csc y }$

Prove the identity. - $\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$

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

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An Introduction to Calculus: Limits, Derivatives, and Integrals

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Exam 5: Analytic Trigonometry

Simplify the expression. - cos⁡xtan⁡2x−cos⁡xsin⁡2x\frac { \cos x } { \tan ^ { 2 } x } - \frac { \cos x } { \sin ^ { 2 } x }tan2xcosx​−sin2xcosx​

Match the graph with the correct equation. -

Solve the triangle. - A=33∘,a=19, b=14\mathrm { A } = 33 ^ { \circ } , \mathrm { a } = 19 , \mathrm {~b} = 14A=33∘,a=19, b=14

Find all solutions in the interval [0, 2π] - 2sin⁡2x=sin⁡x2 \sin ^ { 2 } x = \sin x2sin2x=sinx

Use basic identities to simplify the expression. - cos⁡2θsin⁡2θ+csc⁡θsin⁡θ\frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta } + \csc \theta \sin \thetasin2θcos2θ​+cscθsinθ

State whether the given measurements determine zero, one, or two triangles. - A=87∘,a=24, b=54\mathrm { A } = 87 ^ { \circ } , \mathrm { a } = 24 , \mathrm {~b} = 54A=87∘,a=24, b=54

Rewrite with only sin x and cos x. - sin⁡2x−cos⁡x\sin 2 x - \cos xsin2x−cosx

Prove the identity. - sin⁡xcsc⁡x−1+sin⁡xcsc⁡x+1=2tan⁡2x\frac { \sin x } { \csc x - 1 } + \frac { \sin x } { \csc x + 1 } = 2 \tan ^ { 2 } xcscx−1sinx​+cscx+1sinx​=2tan2x

Find all solutions in the interval [0, 2π). - cos⁡2(x2)=cos⁡2x\cos ^ { 2 } \left( \frac { x } { 2 } \right) = \cos ^ { 2 } xcos2(2x​)=cos2x

Find an exact value. - sin⁡11π12\sin \frac { 11 \pi } { 12 }sin1211π​

Express the function as a sinusoid of the form y = a sin (bx + c). - y=5sin⁡(5x)−3cos⁡(5x)y = 5 \sin ( 5 x ) - 3 \cos ( 5 x )y=5sin(5x)−3cos(5x)

Solve the triangle. - b=20,c=28, A=81∘\mathrm { b } = 20 , \mathrm { c } = 28 , \mathrm {~A} = 81 ^ { \circ }b=20,c=28, A=81∘

Simplify the expression. - 1−cot⁡xtan⁡x−1\frac { 1 - \cot x } { \tan x - 1 }tanx−11−cotx​

Rewrite with only sin x and cos x. - sin⁡2x−cos⁡2x\sin 2 x - \cos 2 xsin2x−cos2x

Find the exact value by using a half-angle identity. - cos⁡165∘\cos 165 ^ { \circ }cos165∘

Find all solutions in the interval [0, 2π] - cos⁡x=sin⁡x\cos x = \sin xcosx=sinx

Write each expression in factored form as an algebraic expression of a single trigonometric function. - 1−2sin⁡2x+sin⁡4x1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x1−2sin2x+sin4x

Simplify the expression. - (csc⁡y+cot⁡y)(csc⁡y−cot⁡y)csc⁡y\frac { ( \csc y + \cot y ) ( \csc y - \cot y ) } { \csc y }cscy(cscy+coty)(cscy−coty)​

Prove the identity. - tan⁡(x−π4)=tan⁡x−11+tan⁡x\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }tan(x−4π​)=1+tanxtanx−1​