Question 1

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

Accepted Answer

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

Question 2

Determine if the following is an identity.
-$\sin \theta \sec \theta = \cos \theta \csc \theta$

Accepted Answer

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

Question 3

Solve.
-A guy wire to a tower makes a $75 ^ { \circ }$ angle with level ground. At a point $40 \mathrm { ft }$ farther from the tower than the wire but on the same side as the base of the wire, the angle of elevation to the top of the tower is $37 ^ { \circ }$. Find the length of the wire (to the nearest foot).

Accepted Answer

A)  $44 \mathrm { ft }$ 
B)  $83 \mathrm { ft }$ 
C)  $39 \mathrm { ft }$ 
D)  $78 \mathrm { ft }$ 
A)  $44 \mathrm { ft }$ 
B)  $83 \mathrm { ft }$ 
C)  $39 \mathrm { ft }$ 
D)  $78 \mathrm { ft }$

Question 4

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

Accepted Answer

\[\begin{array} { l } 
\frac { \sin x +

Question 5

Solve the triangle.
-$$\mathrm { A } = 48 ^ { \circ } , \mathrm { a } = 33 , \mathrm {~b} = 26$$

Accepted Answer

A)  $\mathrm { B } = 35.8 ^ { \circ } , \mathrm { C } = 116.2 ^ { \circ } , \mathrm { C } \approx 35.3$ 
B)  $\mathrm { B } = 35.8 ^ { \circ } , \mathrm { C } = 96.2 ^ { \circ } , \mathrm { c } \approx 26.5$ 
C)  $\mathrm { B } = 35.8 ^ { \circ } , \mathrm { C } = 96.2 ^ { \circ } , \mathrm { C } \approx 44.1$ 
D)  Cannot be solved 
A)  $\mathrm { B } = 35.8 ^ { \circ } , \mathrm { C } = 116.2 ^ { \circ } , \mathrm { C } \approx 35.3$ 
B)  $\mathrm { B } = 35.8 ^ { \circ } , \mathrm { C } = 96.2 ^ { \circ } , \mathrm { c } \approx 26.5$ 
C)  $\mathrm { B } = 35.8 ^ { \circ } , \mathrm { C } = 96.2 ^ { \circ } , \mathrm { C } \approx 44.1$ 
D)  Cannot be solved

Question 6

Find the area. Round your answer to the nearest hundredth if necessary.
-Find the area of a regular decagon (10 sides) inscribed in a circle of radius 10 inches.

Accepted Answer

A)  $500 \sin 45 ^ { \circ } \approx 353.55$ square inches 
B)  $500 \sin 36 ^ { \circ } \approx 293.89$ square inches 
C)  $1000 \sin 36 ^ { \circ } \approx 587.79$ square inches 
D)  $500 \cos 36 ^ { \circ } \approx 404.51$ square inches 
A)  $500 \sin 45 ^ { \circ } \approx 353.55$ square inches 
B)  $500 \sin 36 ^ { \circ } \approx 293.89$ square inches 
C)  $1000 \sin 36 ^ { \circ } \approx 587.79$ square inches 
D)  $500 \cos 36 ^ { \circ } \approx 404.51$ square inches

Question 7

Simplify the expression.
-$\cot x \sin x - \sin \left( \frac { \pi } { 2 } - x \right) + \cos x$

Accepted Answer

A)  $\sin x$ 
B)  $2 \sin x$ 
C)  $2 \cos x$ 
D)  $\cos x$ 
A)  $\sin x$ 
B)  $2 \sin x$ 
C)  $2 \cos x$ 
D)  $\cos x$

Question 8

Prove the identity.
-$$\tan ^ { 2 } u ( 1 + \cos 2 u ) = 1 - \cos 2 u$$

Accepted Answer

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

Question 9

Simplify the expression to either 1 or -1.
-$$\sin ( - x ) \csc x$$

Accepted Answer

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

Question 10

Prove the identity.
-$$\csc ^ { 3 } x \tan ^ { 2 } x = \csc x \left( 1 + \tan ^ { 2 } x \right)$$

Accepted Answer

The answer of Prove the identity.
-\[\csc ^ { 3 }...

Question 11

Find an exact value.
-$$\sin \frac { 17 \pi } { 12 }$$

Accepted Answer

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

Question 12

Prove the identity.
-4 cot 4x = 2 cot 2u - 2 tan 2u

Accepted Answer

The answer of Prove the identity.
-4 cot 4x = 2...

Question 13

State whether the given measurements determine zero, one, or two triangles.
-$$\mathrm { C } = 30 ^ { \circ } , \mathrm { a } = 20 , \mathrm { c } = 10$$

Accepted Answer

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

Question 14

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

Accepted Answer

\[\begin{array} { l } 
\cos 4 x + \cos 2

Question 15

Match the graph with the correct equation.
-

Accepted Answer

A)  y = sin x cos (2) + cos x sin (2) 
B)  y = cos x cos (2) - sin x sin (2) 
C)  y = cos x cos (2) + sin x sin (2) 
D)  y = sin x cos (2) - cos x sin (2) 
A)  y = sin x cos (2) + cos x sin (2) 
B)  y = cos x cos (2) - sin x sin (2) 
C)  y = cos x cos (2) + sin x sin (2) 
D)  y = sin x cos (2) - cos x sin (2)

Question 16

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

Accepted Answer

The answer of Prove the identity.
-\[\frac { 1 + \csc...

Question 17

Solve the triangle.
-$$\mathrm { a } = 12 , \mathrm {~b} = 20 , \mathrm { C } = 95 ^ { \circ }$$

Accepted Answer

A)  $\mathrm { c } \approx 23.4 , \mathrm {~A} \approx 33.6 ^ { \circ } , \mathrm { B } \approx 51.4 ^ { \circ }$ 
B)  $\mathrm { C } \approx 24.2 , \mathrm {~A} \approx 52.4 ^ { \circ } , \mathrm { B } \approx 32.6 ^ { \circ }$ 
C)  $\mathrm { C } \approx 23.4 , \mathrm {~A} \approx 29.6 ^ { \circ } , \mathrm { B } \approx 55.4 ^ { \circ }$ 
D)  $\mathrm { c } \approx 24.2 , \mathrm {~A} \approx 29.6 ^ { \circ } , \mathrm { B } \approx 55.4 ^ { \circ }$ 
A)  $\mathrm { c } \approx 23.4 , \mathrm {~A} \approx 33.6 ^ { \circ } , \mathrm { B } \approx 51.4 ^ { \circ }$ 
B)  $\mathrm { C } \approx 24.2 , \mathrm {~A} \approx 52.4 ^ { \circ } , \mathrm { B } \approx 32.6 ^ { \circ }$ 
C)  $\mathrm { C } \approx 23.4 , \mathrm {~A} \approx 29.6 ^ { \circ } , \mathrm { B } \approx 55.4 ^ { \circ }$ 
D)  $\mathrm { c } \approx 24.2 , \mathrm {~A} \approx 29.6 ^ { \circ } , \mathrm { B } \approx 55.4 ^ { \circ }$

Question 18

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

Accepted Answer

A)  $0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
B)  $\frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
C)  $0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
D)  No solution 
A)  $0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
B)  $\frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
C)  $0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
D)  No solution

Question 19

Match the graph with the correct equation.
-

Accepted Answer

A)  y = cos 5x cos (-2) + sin 5x sin (-2) 
B)  y = cos 5x cos (-2) - sin 5x sin (-2) 
C)  y = sin 5x cos (-2) - cos 5x sin(-2) 
D)  y = sin 5x cos (-2) + cos 5x sin (-2) 
A)  y = cos 5x cos (-2) + sin 5x sin (-2) 
B)  y = cos 5x cos (-2) - sin 5x sin (-2) 
C)  y = sin 5x cos (-2) - cos 5x sin(-2) 
D)  y = sin 5x cos (-2) + cos 5x sin (-2)

Question 20

Simplify the expression.
-$\frac { \cos \left( \frac { \pi } { 2 } - x \right) \tan x } { \sin ^ { 2 } x }$

Accepted Answer

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

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

Determine if the following is an identity. - $\sin \theta \sec \theta = \cos \theta \csc \theta$

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

Solve the triangle. - $\mathrm { A } = 48 ^ { \circ } , \mathrm { a } = 33 , \mathrm {~b} = 26$

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of a regular decagon (10 sides) inscribed in a circle of radius 10 inches.

Simplify the expression. - $\cot x \sin x - \sin \left( \frac { \pi } { 2 } - x \right) + \cos x$

Prove the identity. - $\tan ^ { 2 } u ( 1 + \cos 2 u ) = 1 - \cos 2 u$

Simplify the expression to either 1 or -1. - $\sin ( - x ) \csc x$

Prove the identity. - $\csc ^ { 3 } x \tan ^ { 2 } x = \csc x \left( 1 + \tan ^ { 2 } x \right)$

Find an exact value. - $\sin \frac { 17 \pi } { 12 }$

Prove the identity. -4 cot 4x = 2 cot 2u - 2 tan 2u

State whether the given measurements determine zero, one, or two triangles. - $\mathrm { C } = 30 ^ { \circ } , \mathrm { a } = 20 , \mathrm { c } = 10$

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

Match the graph with the correct equation. -

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

Solve the triangle. - $\mathrm { a } = 12 , \mathrm {~b} = 20 , \mathrm { C } = 95 ^ { \circ }$

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

Match the graph with the correct equation. -

Simplify the expression. - $\frac { \cos \left( \frac { \pi } { 2 } - x \right) \tan x } { \sin ^ { 2 } x }$

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

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Exam 5: Analytic Trigonometry

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

Determine if the following is an identity. - sin⁡θsec⁡θ=cos⁡θcsc⁡θ\sin \theta \sec \theta = \cos \theta \csc \thetasinθsecθ=cosθcscθ

Prove the identity. - sin⁡x+cos⁡xsin⁡x−cos⁡x=1+2sin⁡xcos⁡x2sin⁡2x−1\frac { \sin x + \cos x } { \sin x - \cos x } = \frac { 1 + 2 \sin x \cos x } { 2 \sin ^ { 2 } x - 1 }sinx−cosxsinx+cosx​=2sin2x−11+2sinxcosx​

Solve the triangle. - A=48∘,a=33, b=26\mathrm { A } = 48 ^ { \circ } , \mathrm { a } = 33 , \mathrm {~b} = 26A=48∘,a=33, b=26

Find the area. Round your answer to the nearest hundredth if necessary. -Find the area of a regular decagon (10 sides) inscribed in a circle of radius 10 inches.

Simplify the expression. - cot⁡xsin⁡x−sin⁡(π2−x)+cos⁡x\cot x \sin x - \sin \left( \frac { \pi } { 2 } - x \right) + \cos xcotxsinx−sin(2π​−x)+cosx

Prove the identity. - tan⁡2u(1+cos⁡2u)=1−cos⁡2u\tan ^ { 2 } u ( 1 + \cos 2 u ) = 1 - \cos 2 utan2u(1+cos2u)=1−cos2u

Simplify the expression to either 1 or -1. - sin⁡(−x)csc⁡x\sin ( - x ) \csc xsin(−x)cscx

Prove the identity. - csc⁡3xtan⁡2x=csc⁡x(1+tan⁡2x)\csc ^ { 3 } x \tan ^ { 2 } x = \csc x \left( 1 + \tan ^ { 2 } x \right)csc3xtan2x=cscx(1+tan2x)

Find an exact value. - sin⁡17π12\sin \frac { 17 \pi } { 12 }sin1217π​

Prove the identity. -4 cot 4x = 2 cot 2u - 2 tan 2u

State whether the given measurements determine zero, one, or two triangles. - C=30∘,a=20,c=10\mathrm { C } = 30 ^ { \circ } , \mathrm { a } = 20 , \mathrm { c } = 10C=30∘,a=20,c=10

Prove the identity. - cos⁡4x+cos⁡2x=2−2sin⁡22x−2sin⁡2x\cos 4 x + \cos 2 x = 2 - 2 \sin ^ { 2 } 2 x - 2 \sin ^ { 2 } xcos4x+cos2x=2−2sin22x−2sin2x

Match the graph with the correct equation. -

Prove the identity. - 1+csc⁡xsec⁡x=cos⁡x+cot⁡x\frac { 1 + \csc x } { \sec x } = \cos x + \cot xsecx1+cscx​=cosx+cotx

Solve the triangle. - a=12, b=20,C=95∘\mathrm { a } = 12 , \mathrm {~b} = 20 , \mathrm { C } = 95 ^ { \circ }a=12, b=20,C=95∘

Find all solutions to the equation in the interval [0, 2). - cos⁡4x−cos⁡2x=0\cos 4 x - \cos 2 x = 0cos4x−cos2x=0

Match the graph with the correct equation. -

Simplify the expression. - cos⁡(π2−x)tan⁡xsin⁡2x\frac { \cos \left( \frac { \pi } { 2 } - x \right) \tan x } { \sin ^ { 2 } x }sin2xcos(2π​−x)tanx​

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

Determine if the following is an identity. - $\sin \theta \sec \theta = \cos \theta \csc \theta$

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

Solve the triangle. - $\mathrm { A } = 48 ^ { \circ } , \mathrm { a } = 33 , \mathrm {~b} = 26$

Simplify the expression. - $\cot x \sin x - \sin \left( \frac { \pi } { 2 } - x \right) + \cos x$

Prove the identity. - $\tan ^ { 2 } u ( 1 + \cos 2 u ) = 1 - \cos 2 u$

Simplify the expression to either 1 or -1. - $\sin ( - x ) \csc x$

Prove the identity. - $\csc ^ { 3 } x \tan ^ { 2 } x = \csc x \left( 1 + \tan ^ { 2 } x \right)$

Find an exact value. - $\sin \frac { 17 \pi } { 12 }$

State whether the given measurements determine zero, one, or two triangles. - $\mathrm { C } = 30 ^ { \circ } , \mathrm { a } = 20 , \mathrm { c } = 10$

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

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

Solve the triangle. - $\mathrm { a } = 12 , \mathrm {~b} = 20 , \mathrm { C } = 95 ^ { \circ }$

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

Simplify the expression. - $\frac { \cos \left( \frac { \pi } { 2 } - x \right) \tan x } { \sin ^ { 2 } x }$