Question 1

Use the fundamental identities to simplify the expression.
-$\sin \frac { 11 \pi } { 12 }$

Accepted Answer

A)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
B)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 3 } - 2 } { 4 }$ 
D)  $\frac { \sqrt { 3 } + 2 } { 4 }$ 
A)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
B)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 3 } - 2 } { 4 }$ 
D)  $\frac { \sqrt { 3 } + 2 } { 4 }$

Question 2

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-

Accepted Answer

A)  $f(-x)=-f(x)$ 
B)  $f(-x)=f(x)$ 
A)  $f(-x)=-f(x)$ 
B)  $f(-x)=f(x)$

Question 3

Use identities to write each expression as a function of .
-$$\cos \left( \theta + 270 ^ { \circ } \right)$$

Accepted Answer

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

Question 4

Provide an appropriate response.
-Graph the expression $\sec ^ { 2 } x - \sec ^ { 2 } x \sin ^ { 2 } x$ 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 \(\sec...

Question 5

Use identities to write each expression as a function of .
-Find $\cos ( s - t )$ given that $\cos s = \frac { \sqrt { 10 } } { 10 }$, with $s$ in quadrant $I$, and $\sin t = - \frac { \sqrt { 3 } } { 2 }$, with $t$ in quadrant IV.

Accepted Answer

A)  $\frac { \sqrt { 10 } - 3 \sqrt { 30 } } { 20 }$ 
B)  $- \frac { \sqrt { 10 } } { 20 }$ 
C)  $\frac { \sqrt { 10 } - \sqrt { 30 } } { 19 }$ 
D)  $\frac { \sqrt { 10 } + 3 \sqrt { 30 } } { 20 }$ 
A)  $\frac { \sqrt { 10 } - 3 \sqrt { 30 } } { 20 }$ 
B)  $- \frac { \sqrt { 10 } } { 20 }$ 
C)  $\frac { \sqrt { 10 } - \sqrt { 30 } } { 19 }$ 
D)  $\frac { \sqrt { 10 } + 3 \sqrt { 30 } } { 20 }$

Question 6

Solve the problem.
-$$\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta }$$

Accepted Answer

A)  $- \sec ^ { 2 } \theta$ 
B)  1 
C)  $1 - \sin \theta$ 
D)  0 
A)  $- \sec ^ { 2 } \theta$ 
B)  1 
C)  $1 - \sin \theta$ 
D)  0

Question 7

Verify that each equation is an identity.
-$$\cos ( 4 u ) = 2 \cos ^ { 2 } ( 2 u ) - 1$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\cos...

Question 8

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = \frac { 1 } { 3 }$, with $\mathrm { s }$ in quadrant $\mathrm { I }$, and $\sin \mathrm { t } = - \frac { 1 } { 2 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 3 }$ 
B)  $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 3 }$ 
C)  $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 5 }$ 
D)  $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 5 }$ 
A)  $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 3 }$ 
B)  $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 3 }$ 
C)  $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 5 }$ 
D)  $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 5 }$

Question 9

Write the product as a sum or difference of trigonometric functions.
-$\sin 15 ^ { \circ } \cos 105 ^ { \circ } + \cos 15 ^ { \circ } \sin 105 ^ { \circ }$

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { 1 } { 4 }$ 
A)  $- \frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { 1 } { 4 }$

Question 10

Use a sum or difference identity to find the exact value.
-$\tan \left( - \frac { 5 \pi } { 12 } \right)$

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 11

Complete the sentence so the result is an identity. Let x be any real number.
-Write $\cot x$ in terms of $\sin x$.

Accepted Answer

A)  $\frac { \pm \sqrt { 1 - \sin ^ { 2 } x } } { 1 - \sin ^ { 2 } x }$ 
B)  $\frac { \pm \sqrt { 1 - \sin ^ { 2 } x } } { \sin x }$ 
C)  $\pm \sqrt { 1 - \sin ^ { 2 } x }$ 
D)  $\frac { \pm \sin x \sqrt { 1 - \sin ^ { 2 } x } } { 1 - \sin ^ { 2 } x }$ 
A)  $\frac { \pm \sqrt { 1 - \sin ^ { 2 } x } } { 1 - \sin ^ { 2 } x }$ 
B)  $\frac { \pm \sqrt { 1 - \sin ^ { 2 } x } } { \sin x }$ 
C)  $\pm \sqrt { 1 - \sin ^ { 2 } x }$ 
D)  $\frac { \pm \sin x \sqrt { 1 - \sin ^ { 2 } x } } { 1 - \sin ^ { 2 } x }$

Question 12

Find the exact value of the expression using the provided information.
-$\frac { 2 \tan 30 ^ { \circ } } { 1 - \tan ^ { 2 } 30 ^ { \circ } }$

Accepted Answer

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

Question 13

Factor the trigonometric expression and simplify.
-Find $\sin \theta$ if $\cos \theta = \frac { 5 } { 6 }$ and $\theta$ is in quadrant IV.

Accepted Answer

A)  $\frac { 4 } { 5 }$ 
B)  $\frac { \sqrt { 33 } } { 4 }$ 
C)  $- \frac { \sqrt { 11 } } { 6 }$ 
D)  $\frac { 5 \sqrt { 29 } } { 29 }$ 
A)  $\frac { 4 } { 5 }$ 
B)  $\frac { \sqrt { 33 } } { 4 }$ 
C)  $- \frac { \sqrt { 11 } } { 6 }$ 
D)  $\frac { 5 \sqrt { 29 } } { 29 }$

Question 14

Using a sum or difference identity, write the following as an expression involving functions of x.
-$\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)$

Accepted Answer

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

Question 15

Decide whether the expression is or is not an identity.
-$\frac { \csc x \cot x } { \sec x }$

Accepted Answer

A)  $\csc ^ { 2 } x$ 
B)  $\sec ^ { 2 } x$ 
C)  $\cot ^ { 2 } x$ 
D)  1 
A)  $\csc ^ { 2 } x$ 
B)  $\sec ^ { 2 } x$ 
C)  $\cot ^ { 2 } x$ 
D)  1

Question 16

Express the function as a trigonometric function of x.
-$$\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$$

Accepted Answer

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

Question 17

Use the fundamental identities to find the value of the trigonometric function.
-

Accepted Answer

A)  $f(-x)=-f(x)$ 
B)  $f(-x)=f(x)$ 
A)  $f(-x)=-f(x)$ 
B)  $f(-x)=f(x)$

Question 18

Express the function as a trigonometric function of x.
-Find $\sin \theta$ if $\cos \theta = - \frac { \sqrt { 11 } } { 6 }$ and $\theta$ is in quadrant III.

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 12 }$ 
B)  $\frac { 5 } { 6 }$ 
C)  $- \frac { 25 } { 36 }$ 
D)  $- \frac { 5 } { 6 }$ 
A)  $- \frac { \sqrt { 3 } } { 12 }$ 
B)  $\frac { 5 } { 6 }$ 
C)  $- \frac { 25 } { 36 }$ 
D)  $- \frac { 5 } { 6 }$

Question 19

Complete the sentence so the result is an identity. Let x be any real number.
-$$\sqrt { 1 - \cos ^ { 2 } x } = 1 - \cos x$$

Accepted Answer

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

Question 20

Use a sum or difference identity to find the exact value.
-$\sin \frac { 7 \pi } { 24 } \cos \frac { \pi } { 8 } - \cos \frac { 7 \pi } { 24 } \sin \frac { \pi } { 8 }$

Accepted Answer

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

Use the fundamental identities to simplify the expression. - $\sin \frac { 11 \pi } { 12 }$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Use identities to write each expression as a function of . - $\cos \left( \theta + 270 ^ { \circ } \right)$

Provide an appropriate response. -Graph the expression $\sec ^ { 2 } x - \sec ^ { 2 } x \sin ^ { 2 } x$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Use identities to write each expression as a function of . -Find $\cos ( s - t )$ given that $\cos s = \frac { \sqrt { 10 } } { 10 }$ , with $s$ in quadrant $I$ , and $\sin t = - \frac { \sqrt { 3 } } { 2 }$ , with $t$ in quadrant IV.

Solve the problem. - $\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta }$

Verify that each equation is an identity. - $\cos ( 4 u ) = 2 \cos ^ { 2 } ( 2 u ) - 1$

Write the product as a sum or difference of trigonometric functions. - $\sin 15 ^ { \circ } \cos 105 ^ { \circ } + \cos 15 ^ { \circ } \sin 105 ^ { \circ }$

Use a sum or difference identity to find the exact value. - $\tan \left( - \frac { 5 \pi } { 12 } \right)$

Complete the sentence so the result is an identity. Let x be any real number. -Write $\cot x$ in terms of $\sin x$ .

Find the exact value of the expression using the provided information. - $\frac { 2 \tan 30 ^ { \circ } } { 1 - \tan ^ { 2 } 30 ^ { \circ } }$

Factor the trigonometric expression and simplify. -Find $\sin \theta$ if $\cos \theta = \frac { 5 } { 6 }$ and $\theta$ is in quadrant IV.

Using a sum or difference identity, write the following as an expression involving functions of x. - $\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)$

Decide whether the expression is or is not an identity. - $\frac { \csc x \cot x } { \sec x }$

Express the function as a trigonometric function of x. - $\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$

Use the fundamental identities to find the value of the trigonometric function. -

Express the function as a trigonometric function of x. -Find $\sin \theta$ if $\cos \theta = - \frac { \sqrt { 11 } } { 6 }$ and $\theta$ is in quadrant III.

Complete the sentence so the result is an identity. Let x be any real number. - $\sqrt { 1 - \cos ^ { 2 } x } = 1 - \cos x$

Use a sum or difference identity to find the exact value. - $\sin \frac { 7 \pi } { 24 } \cos \frac { \pi } { 8 } - \cos \frac { 7 \pi } { 24 } \sin \frac { \pi } { 8 }$

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Inverse Circular Functions and Trigonometric Equations

Applications of Trigonometry and Vectors

Complex Numbers, Polar Equations, and Parametric Equations

Filters

Exam 5: Trigonometric Identities

Use the fundamental identities to simplify the expression. - sin⁡11π12\sin \frac { 11 \pi } { 12 }sin1211π​

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Use identities to write each expression as a function of . - cos⁡(θ+270∘)\cos \left( \theta + 270 ^ { \circ } \right)cos(θ+270∘)

Provide an appropriate response. -Graph the expression sec⁡2x−sec⁡2xsin⁡2x\sec ^ { 2 } x - \sec ^ { 2 } x \sin ^ { 2 } xsec2x−sec2xsin2x on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Solve the problem. - (sin⁡θ+cos⁡θ)21+2sin⁡θcos⁡θ\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta }1+2sinθcosθ(sinθ+cosθ)2​

Verify that each equation is an identity. - cos⁡(4u)=2cos⁡2(2u)−1\cos ( 4 u ) = 2 \cos ^ { 2 } ( 2 u ) - 1cos(4u)=2cos2(2u)−1

Write the product as a sum or difference of trigonometric functions. - sin⁡15∘cos⁡105∘+cos⁡15∘sin⁡105∘\sin 15 ^ { \circ } \cos 105 ^ { \circ } + \cos 15 ^ { \circ } \sin 105 ^ { \circ }sin15∘cos105∘+cos15∘sin105∘

Use a sum or difference identity to find the exact value. - tan⁡(−5π12)\tan \left( - \frac { 5 \pi } { 12 } \right)tan(−125π​)

Complete the sentence so the result is an identity. Let x be any real number. -Write cot⁡x\cot xcotx in terms of sin⁡x\sin xsinx .

Find the exact value of the expression using the provided information. - 2tan⁡30∘1−tan⁡230∘\frac { 2 \tan 30 ^ { \circ } } { 1 - \tan ^ { 2 } 30 ^ { \circ } }1−tan230∘2tan30∘​

Factor the trigonometric expression and simplify. -Find sin⁡θ\sin \thetasinθ if cos⁡θ=56\cos \theta = \frac { 5 } { 6 }cosθ=65​ and θ\thetaθ is in quadrant IV.

Using a sum or difference identity, write the following as an expression involving functions of x. - sin⁡2x(cot⁡2x+1)\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)sin2x(cot2x+1)

Decide whether the expression is or is not an identity. - csc⁡xcot⁡xsec⁡x\frac { \csc x \cot x } { \sec x }secxcscxcotx​

Express the function as a trigonometric function of x. - sin⁡2x+sin⁡2xcot⁡2x\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } xsin2x+sin2xcot2x