Question 1

Find the exact value by using a half-angle identity.
-Find $\cos \frac { x } { 2 }$, given that $\sec x = 4$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$

Question 2

Use an identity to write the expression as a single trigonometric function or as a single number.
-$\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } x$

Accepted Answer

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

Question 3

Use an identity to write the expression as a single trigonometric function or as a single number.
-$\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$

Accepted Answer

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

Question 4

Find the exact value by using a half-angle identity.
-Write $\cos x$ in terms of $\cot x$.

Accepted Answer

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

Question 5

Use an identity to write the expression as a single trigonometric function or as a single number.
-$1 - 2 \sin ^ { 2 } 2 x$

Accepted Answer

A)  $\cos 4 \mathrm { x }$ 
B)  $\frac { 1 } { 2 } \sin 16 x$ 
C)  $\cos 8 x$ 
D)  $2 \sin 4 x$ 
A)  $\cos 4 \mathrm { x }$ 
B)  $\frac { 1 } { 2 } \sin 16 x$ 
C)  $\cos 8 x$ 
D)  $2 \sin 4 x$

Question 6

Provide an appropriate response.
-Graph the expression cos x + sin x tan 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 cos...

Question 7

Use the fundamental identities to simplify the expression.
-sin 2x = 2sin x cos x

Accepted Answer

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

Question 8

Use a sum or difference identity to find the exact value.
-$\tan x ( \cot x - \cos x )$

Accepted Answer

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

Question 9

Find the exact value of the expression using the provided information.
-$4 \sin 2 x \cos 2 x$

Accepted Answer

A)  $\frac { 1 } { 2 } \sin 16 x$ 
B)  $\cos 8 x$ 
C)  $\cos 4 x$ 
D)  $2 \sin 4 x$ 
A)  $\frac { 1 } { 2 } \sin 16 x$ 
B)  $\cos 8 x$ 
C)  $\cos 4 x$ 
D)  $2 \sin 4 x$

Question 10

Perform the indicated operations and simplify the result so there are no quotients.
-$- 1 = \tan ^ { 2 } x$

Accepted Answer

A)  $\cos ^ { 2 } x$ 
B)  $\sin ^ { 2 } x$ 
C)  $\cot ^ { 2 } x$ 
D)  $\sec ^ { 2 } x$ 
A)  $\cos ^ { 2 } x$ 
B)  $\sin ^ { 2 } x$ 
C)  $\cot ^ { 2 } x$ 
D)  $\sec ^ { 2 } x$

Question 11

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan ( \mathrm { s } - \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { 3 \sqrt { 13 } } { 13 }$, with $\mathrm { s }$ in quadrant $\mathrm { IV }$, and $\sin \mathrm { t } = - \frac { \sqrt { 10 } } { 10 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $- \frac { 11 } { 3 }$ 
B)  $- \frac { 11 } { 9 }$ 
C)  $\frac { 11 } { 3 }$ 
D)  $- \frac { 7 } { 9 }$ 
A)  $- \frac { 11 } { 3 }$ 
B)  $- \frac { 11 } { 9 }$ 
C)  $\frac { 11 } { 3 }$ 
D)  $- \frac { 7 } { 9 }$

Question 12

Solve the problem.
-$\cos \left( - 75 ^ { \circ } \right)$

Accepted Answer

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

Question 13

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-The power dissipated in an electric circuit is given by the expression $\mathrm { P } = \mathrm { RI } ^ { 2 }$, where $\mathrm { R }$ is the resistance of the circuit and $I$ is the current through the circuit. For a sinusoidal alternating current, the current might be represented by the relation $\mathrm { I } = \mathrm { A } \sin ( 2 \pi \mathrm { ft } )$, where $\mathrm { A }$ is the amplitude, $\mathrm { f }$ is the frequency, and $t$ is time. Write an expression for $P$ involving the sine function, and use a fundamental identity to write $P$ in terms of the cosine function.

Accepted Answer

A)  $P = R A ^ { 2 } \sin ^ { 2 } ( 2 \pi f t ) ; P = - R A ^ { 2 } \cos ^ { 2 } ( 2 \pi f t )$ 
B)  $P = R A \sin ^ { 2 } ( 2 \pi f t ) ; P = R A - \cos ^ { 2 } ( 2 \pi f t )$ 
C)  $P = R A ^ { 2 } \sin ^ { 2 } ( 2 \pi f \mathrm { ft } ) ; P = R A ^ { 2 } - R A ^ { 2 } \cos ^ { 2 } ( 2 \pi f t )$ 
D)  $P = R A \sin ^ { 2 } ( 2 \pi f \mathrm { ft } ) ; P = R A - R A \cos ^ { 2 } ( 2 \pi f t )$ 
A)  $P = R A ^ { 2 } \sin ^ { 2 } ( 2 \pi f t ) ; P = - R A ^ { 2 } \cos ^ { 2 } ( 2 \pi f t )$ 
B)  $P = R A \sin ^ { 2 } ( 2 \pi f t ) ; P = R A - \cos ^ { 2 } ( 2 \pi f t )$ 
C)  $P = R A ^ { 2 } \sin ^ { 2 } ( 2 \pi f \mathrm { ft } ) ; P = R A ^ { 2 } - R A ^ { 2 } \cos ^ { 2 } ( 2 \pi f t )$ 
D)  $P = R A \sin ^ { 2 } ( 2 \pi f \mathrm { ft } ) ; P = R A - R A \cos ^ { 2 } ( 2 \pi f t )$

Question 14

Factor the trigonometric expression and simplify.
-$1 - \sin ^ { 3 } x$

Accepted Answer

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

Question 15

Complete the sentence so the result is an identity. Let x be any real number.
-$\cot \frac { 17 \pi } { 18 }$

Accepted Answer

A)  $\tan \frac { \pi } { 18 }$ 
B)  $\cot \frac { \pi } { 18 }$ 
C)  $\cot \frac { - 4 \pi } { 9 }$ 
D)  $\tan \frac { - 4 \pi } { 9 }$ 
A)  $\tan \frac { \pi } { 18 }$ 
B)  $\cot \frac { \pi } { 18 }$ 
C)  $\cot \frac { - 4 \pi } { 9 }$ 
D)  $\tan \frac { - 4 \pi } { 9 }$

Question 16

Perform the transformation.
-Write $\sec x$ in terms of $\tan x$.

Accepted Answer

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

Question 17

Verify that each equation is an identity.
-$$\sec \beta + \tan \beta = \frac { \cos \beta } { 1 - \sin \beta }$$

Accepted Answer

\[\begin{array} { l } 
\sec \beta + \tan

Question 18

Use an identity to write the expression as a single trigonometric function or as a single number.
-$1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x$

Accepted Answer

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

Question 19

Find the exact value of the expression using the provided information.
-sin 87°

Accepted Answer

A) csc 93° 
B) cos 93° 
C) csc 3° 
D) cos 3° 
A) csc 93° 
B) cos 93° 
C) csc 3° 
D) cos 3°

Question 20

Use the fundamental identities to simplify the expression.
-$\sin 260 ^ { \circ } \cos 20 ^ { \circ } - \cos 260 ^ { \circ } \sin 20 ^ { \circ }$

Accepted Answer

A)  $- \frac { 13 } { 3 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $- \frac { 1 } { 2 }$ 
A)  $- \frac { 13 } { 3 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $- \frac { 1 } { 2 }$

Find the exact value by using a half-angle identity. -Find $\cos \frac { x } { 2 }$ , given that $\sec x = 4$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$ .

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } x$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$

Find the exact value by using a half-angle identity. -Write $\cos x$ in terms of $\cot x$ .

Use an identity to write the expression as a single trigonometric function or as a single number. - $1 - 2 \sin ^ { 2 } 2 x$

Provide an appropriate response. -Graph the expression cos x + sin x tan x on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Use the fundamental identities to simplify the expression. -sin 2x = 2sin x cos x

Use a sum or difference identity to find the exact value. - $\tan x ( \cot x - \cos x )$

Find the exact value of the expression using the provided information. - $4 \sin 2 x \cos 2 x$

Perform the indicated operations and simplify the result so there are no quotients. - $- 1 = \tan ^ { 2 } x$

Solve the problem. - $\cos \left( - 75 ^ { \circ } \right)$

Factor the trigonometric expression and simplify. - $1 - \sin ^ { 3 } x$

Complete the sentence so the result is an identity. Let x be any real number. - $\cot \frac { 17 \pi } { 18 }$

Perform the transformation. -Write $\sec x$ in terms of $\tan x$ .

Verify that each equation is an identity. - $\sec \beta + \tan \beta = \frac { \cos \beta } { 1 - \sin \beta }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x$

Find the exact value of the expression using the provided information. -sin 87°

Use the fundamental identities to simplify the expression. - $\sin 260 ^ { \circ } \cos 20 ^ { \circ } - \cos 260 ^ { \circ } \sin 20 ^ { \circ }$

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

Find the exact value by using a half-angle identity. -Find cos⁡x2\cos \frac { x } { 2 }cos2x​ , given that sec⁡x=4\sec x = 4secx=4 and xxx terminates in 0<x<π20 < x < \frac { \pi } { 2 }0<x<2π​ .

Use an identity to write the expression as a single trigonometric function or as a single number. - sec⁡4x+sec⁡2xtan⁡2x−2tan⁡4x\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } xsec4x+sec2xtan2x−2tan4x

Use an identity to write the expression as a single trigonometric function or as a single number. - sin⁡2x+sin⁡2xcot⁡2x\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } xsin2x+sin2xcot2x

Find the exact value by using a half-angle identity. -Write cos⁡x\cos xcosx in terms of cot⁡x\cot xcotx .

Use an identity to write the expression as a single trigonometric function or as a single number. - 1−2sin⁡22x1 - 2 \sin ^ { 2 } 2 x1−2sin22x