Question 1

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 2

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

Accepted Answer

The answer of Prove the identity.
-\[\sec ^ { 4 }...

Question 3

Prove the identity.
-$$\frac { 5 \csc ^ { 2 } x + 4 \csc x - 1 } { \cot ^ { 2 } x } = \frac { 5 \csc x - 1 } { \csc x - 1 }$$

Accepted Answer

The answer of Prove the identity.
-\[\frac { 5 \csc ^...

Question 4

Write the expression as the sine, cosine, or tangent of an angle.
-$\cos 130 ^ { \circ } \cos 59 ^ { \circ } + \sin 130 ^ { \circ } \sin 59 ^ { \circ }$

Accepted Answer

A)  $\sin 71 ^ { \circ }$ 
B)  $\cos 189 ^ { \circ }$ 
C)  $\cos 71 ^ { \circ }$ 
D)  $\sin 189 ^ { \circ }$ 
A)  $\sin 71 ^ { \circ }$ 
B)  $\cos 189 ^ { \circ }$ 
C)  $\cos 71 ^ { \circ }$ 
D)  $\sin 189 ^ { \circ }$

Question 5

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

Accepted Answer

\[\begin{array} { l } 
\frac { \cos x }

Question 6

Prove the identity.
-$$\tan 6 u \tan 4 u = \frac { \tan ^ { 2 } 5 u - \tan ^ { 2 } u } { 1 - \tan ^ { 2 } 5 u \tan ^ { 2 } u }$$

Accepted Answer

The answer of Prove the identity.
-\[\tan 6 u \tan 4...

Question 7

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

Accepted Answer

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

Question 8

Prove the identity.
-$$\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$$

Accepted Answer

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

Question 9

Express the function as a sinusoid of the form y = a sin (bx + c).
-$$y = 4 \sin ( 6 x ) - \cos ( 6 x )$$

Accepted Answer

A)  $y = \sqrt { 17 } \sin \left( 6 x + \cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 17 } } \right) \right)$ 
B)  $y = \sqrt { 17 } \sin \left( 6 x - \sin ^ { - 1 } \left( \frac { 1 } { \sqrt { 17 } } \right) \right)$ 
C)  $y = \sqrt { 17 } \sin \left( 6 x + \sin ^ { - 1 } \left( \frac { 1 } { \sqrt { 17 } } \right) \right)$ 
D)  $y = \sqrt { 17 } \sin \left( 6 x - \cos - 1 \left( \frac { 1 } { \sqrt { 17 } } \right) \right\}$ 
A)  $y = \sqrt { 17 } \sin \left( 6 x + \cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 17 } } \right) \right)$ 
B)  $y = \sqrt { 17 } \sin \left( 6 x - \sin ^ { - 1 } \left( \frac { 1 } { \sqrt { 17 } } \right) \right)$ 
C)  $y = \sqrt { 17 } \sin \left( 6 x + \sin ^ { - 1 } \left( \frac { 1 } { \sqrt { 17 } } \right) \right)$ 
D)  $y = \sqrt { 17 } \sin \left( 6 x - \cos - 1 \left( \frac { 1 } { \sqrt { 17 } } \right) \right\}$

Question 10

Determine if the following is an identity.
-$$\frac { 1 + \csc x } { \sec x } = \cos x + \cot x$$

Accepted Answer

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

Question 11

Prove the identity.
-$$\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }$$

Accepted Answer

The answer of Prove the identity.
-\[\cot ( x + y...

Question 12

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

Accepted Answer

A)  $\cot x$ 
B)  $\sec x$ 
C)  $\cos x$ 
D)  $\sin x$ 
A)  $\cot x$ 
B)  $\sec x$ 
C)  $\cos x$ 
D)  $\sin x$

Question 13

Simplify the expression.
-$$\frac { 1 } { 1 - \cos x } + \frac { 1 } { 1 + \cos x }$$

Accepted Answer

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

Question 14

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

Accepted Answer

A)  $\cos \left( \frac { \pi } { 7 } + x \right)$ 
B)  $\sin \left( \frac { \pi } { 7 } - x \right)$ 
C)  $\sin \left( \frac { \pi } { 7 } + x \right)$ 
D)  $\cos \left( \frac { \pi } { 7 } - x \right)$ 
A)  $\cos \left( \frac { \pi } { 7 } + x \right)$ 
B)  $\sin \left( \frac { \pi } { 7 } - x \right)$ 
C)  $\sin \left( \frac { \pi } { 7 } + x \right)$ 
D)  $\cos \left( \frac { \pi } { 7 } - x \right)$

Question 15

Use the fundamental identities to find the value of the trigonometric function.
-Find $\csc \theta$ if $\cot \theta = - \sqrt { 35 }$ and $\cos \theta < 0$.

Accepted Answer

A)  6 
B)  $\frac { 1 } { 6 }$ 
C)  $- 6$ 
D)  $- \frac { 1 } { 6 }$ 
A)  6 
B)  $\frac { 1 } { 6 }$ 
C)  $- 6$ 
D)  $- \frac { 1 } { 6 }$

Question 16

Complete the identity.
-The expression $\frac { \sec \theta + \csc \theta } { \tan \theta + 1 }$
is to be the left hand side of an equation that is an identity. Which one of the following four expressions can be u: the right hand side of the equation to complete the identity?

Accepted Answer

A)  $\sec \theta$ 
B)  $\csc \theta$ 
C)  $\tan \theta$ 
D)  $\cos \theta$ 
A)  $\sec \theta$ 
B)  $\csc \theta$ 
C)  $\tan \theta$ 
D)  $\cos \theta$

Question 17

Find the exact value by using a half-angle identity.
-$\tan 75 ^ { \circ }$

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

A)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
B)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
C)  $\frac { \pi } { 8 } , \frac { 9 \pi } { 8 }$ 
D)  No solution 
A)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
B)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
C)  $\frac { \pi } { 8 } , \frac { 9 \pi } { 8 }$ 
D)  No solution

Question 20

Prove the identity.
-$$\cos \left( x + \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 } \cos x - \frac { 1 } { 2 } \sin x$$

Accepted Answer

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

Match the graph with the correct equation. -

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

Prove the identity. - $\frac { 5 \csc ^ { 2 } x + 4 \csc x - 1 } { \cot ^ { 2 } x } = \frac { 5 \csc x - 1 } { \csc x - 1 }$

Write the expression as the sine, cosine, or tangent of an angle. - $\cos 130 ^ { \circ } \cos 59 ^ { \circ } + \sin 130 ^ { \circ } \sin 59 ^ { \circ }$

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

Prove the identity. - $\tan 6 u \tan 4 u = \frac { \tan ^ { 2 } 5 u - \tan ^ { 2 } u } { 1 - \tan ^ { 2 } 5 u \tan ^ { 2 } u }$

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

Prove the identity. - $\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$

Express the function as a sinusoid of the form y = a sin (bx + c). - $y = 4 \sin ( 6 x ) - \cos ( 6 x )$

Determine if the following is an identity. - $\frac { 1 + \csc x } { \sec x } = \cos x + \cot x$

Prove the identity. - $\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }$

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

Simplify the expression. - $\frac { 1 } { 1 - \cos x } + \frac { 1 } { 1 + \cos x }$

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

Use the fundamental identities to find the value of the trigonometric function. -Find $\csc \theta$ if $\cot \theta = - \sqrt { 35 }$ and $\cos \theta < 0$ .

Complete the identity. -The expression $\frac { \sec \theta + \csc \theta } { \tan \theta + 1 }$ is to be the left hand side of an equation that is an identity. Which one of the following four expressions can be u: the right hand side of the equation to complete the identity?

Find the exact value by using a half-angle identity. - $\tan 75 ^ { \circ }$

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

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

Prove the identity. - $\cos \left( x + \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 } \cos x - \frac { 1 } { 2 } \sin x$

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

Match the graph with the correct equation. -

Prove the identity. - sec⁡4x−tan⁡4x=sec⁡2x+tan⁡2x\sec ^ { 4 } x - \tan ^ { 4 } x = \sec ^ { 2 } x + \tan ^ { 2 } xsec4x−tan4x=sec2x+tan2x

Prove the identity. - 5csc⁡2x+4csc⁡x−1cot⁡2x=5csc⁡x−1csc⁡x−1\frac { 5 \csc ^ { 2 } x + 4 \csc x - 1 } { \cot ^ { 2 } x } = \frac { 5 \csc x - 1 } { \csc x - 1 }cot2x5csc2x+4cscx−1​=cscx−15cscx−1​

Write the expression as the sine, cosine, or tangent of an angle. - cos⁡130∘cos⁡59∘+sin⁡130∘sin⁡59∘\cos 130 ^ { \circ } \cos 59 ^ { \circ } + \sin 130 ^ { \circ } \sin 59 ^ { \circ }cos130∘cos59∘+sin130∘sin59∘

Prove the identity. - cos⁡xsec⁡x−1−cos⁡xsec⁡x+1=2cos⁡xtan⁡2x\frac { \cos x } { \sec x - 1 } - \frac { \cos x } { \sec x + 1 } = \frac { 2 \cos x } { \tan ^ { 2 } x }secx−1cosx​−secx+1cosx​=tan2x2cosx​

Prove the identity. - tan⁡6utan⁡4u=tan⁡25u−tan⁡2u1−tan⁡25utan⁡2u\tan 6 u \tan 4 u = \frac { \tan ^ { 2 } 5 u - \tan ^ { 2 } u } { 1 - \tan ^ { 2 } 5 u \tan ^ { 2 } u }tan6utan4u=1−tan25utan2utan25u−tan2u​

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

Prove the identity. - tan⁡(π2+x)=−cot⁡x\tan \left( \frac { \pi } { 2 } + x \right) = - \cot xtan(2π​+x)=−cotx

Express the function as a sinusoid of the form y = a sin (bx + c). - y=4sin⁡(6x)−cos⁡(6x)y = 4 \sin ( 6 x ) - \cos ( 6 x )y=4sin(6x)−cos(6x)

Determine if the following is an identity. - 1+csc⁡xsec⁡x=cos⁡x+cot⁡x\frac { 1 + \csc x } { \sec x } = \cos x + \cot xsecx1+cscx​=cosx+cotx

Prove the identity. - cot⁡(x+y)cot⁡(x−y)=1−tan⁡2xtan⁡2ytan⁡2x−tan⁡2y\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }cot(x+y)cot(x−y)=tan2x−tan2y1−tan2xtan2y​

Simplify the expression. - csc⁡2xsec⁡xsec⁡2x+csc⁡2x\frac { \csc ^ { 2 } x \sec x } { \sec ^ { 2 } x + \csc ^ { 2 } x }sec2x+csc2xcsc2xsecx​

Simplify the expression. - 11−cos⁡x+11+cos⁡x\frac { 1 } { 1 - \cos x } + \frac { 1 } { 1 + \cos x }1−cosx1​+1+cosx1​

Write the expression as the sine, cosine, or tangent of an angle. - cos⁡π7cos⁡x−sin⁡π7sin⁡x\cos \frac { \pi } { 7 } \cos x - \sin \frac { \pi } { 7 } \sin xcos7π​cosx−sin7π​sinx

Use the fundamental identities to find the value of the trigonometric function. -Find csc⁡θ\csc \thetacscθ if cot⁡θ=−35\cot \theta = - \sqrt { 35 }cotθ=−35​ and cos⁡θ<0\cos \theta < 0cosθ<0 .

Find the exact value by using a half-angle identity. - tan⁡75∘\tan 75 ^ { \circ }tan75∘

Rewrite with only sin x and cos x. - cos⁡2x−sin⁡x\cos 2 x - \sin xcos2x−sinx

Find all solutions to the equation in the interval [0, 2). - sin⁡2x=−sin⁡x\sin 2 x = - \sin xsin2x=−sinx

Prove the identity. - cos⁡(x+π6)=32cos⁡x−12sin⁡x\cos \left( x + \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 } \cos x - \frac { 1 } { 2 } \sin xcos(x+6π​)=23​​cosx−21​sinx