Question 1

Determine all solutions of the equation in radians.
-$$( \sin x + \cos x ) ^ { 2 } = 1$$

Accepted Answer

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

Question 2

Use identities to find the indicated value for each angle measure.
-$\cos 2 \theta = \frac { 4 } { 5 }$ and $\theta$ terminates in quadrant III $\quad$ Find $\cos \theta$.

Accepted Answer

A)  $\cos \theta = \frac { 3 \sqrt { 10 } } { 10 }$ 
B)  $\cos \theta = 0$ 
C)  $\cos \theta = - \frac { \sqrt { 10 } } { 3 }$ 
D)  $\cos \theta = - \frac { 3 \sqrt { } } { 10 }$ 
A)  $\cos \theta = \frac { 3 \sqrt { 10 } } { 10 }$ 
B)  $\cos \theta = 0$ 
C)  $\cos \theta = - \frac { \sqrt { 10 } } { 3 }$ 
D)  $\cos \theta = - \frac { 3 \sqrt { } } { 10 }$

Question 3

Complete the sentence so the result is an identity. Let x be any real number.
-Find $\cos \theta$ if $\csc \theta = - \frac { 5 } { 3 }$ and $\theta$ is in quadrant IV.

Accepted Answer

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

Question 4

Use the fundamental identities to find the value of the trigonometric function.
-$$\cos \frac { 17 \pi } { 72 } = \cos \frac { \pi } { 9 } \cos \frac { \pi } { 8 } - \sin \frac { \pi } { 9 } \sin \frac { \pi } { 8 }$$

Accepted Answer

A) True 
 B)False

Question 5

Tell whether the statement is true or false.
-$\cos \left( - 165 ^ { \circ } \right)$

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 6

Solve the problem.
-When a person bends at the waist with a straight back, the force, F, exerted by the person's lower back muscles can be estimated with the formula F = 2.89w sin (ϴ + 90°), where w is the person's
Weight, and ϴ is the angle made by the person's torso relative to the horizontal. Suppose that a
120-pound person bends at an angle of 30°. Determine the force exerted by the person's back
Muscles, and estimate the angle that requires the back muscles to exert a force of 208 pounds.
Round the force to the nearest pound, and the angle to the nearest tenth, if necessary.

Accepted Answer

A) 300 pounds; 30° 
B) 300 pounds; 53.1° 
C) 26 pounds; 53.1° 
D) 173 pounds; 47.3° 
A) 300 pounds; 30° 
B) 300 pounds; 53.1° 
C) 26 pounds; 53.1° 
D) 173 pounds; 47.3°

Question 7

Use the fundamental identities to simplify the expression.
-$$\frac { \sin x \cos x } { \tan x }$$

Accepted Answer

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

Question 8

Use the fundamental identities to find the value of the trigonometric function.
-$2 \cos ^ { 2 } 75 ^ { \circ } - 1$

Accepted Answer

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

Question 9

Solve the equation for exact solutions.
-$$\sec ^ { 4 } x - \tan ^ { 4 } x = \sec ^ { 2 } x + \tan ^ { 2 } x$$

Accepted Answer

\[\begin{array} { l } 
\sec ^ { 4 } x -

Question 10

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta$

Accepted Answer

A)  $\sin \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sec ^ { 2 } \theta$ 
D)  $\cos ^ { 3 } \theta$ 
A)  $\sin \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sec ^ { 2 } \theta$ 
D)  $\cos ^ { 3 } \theta$

Question 11

Perform the indicated operations and simplify the result so there are no quotients.
Solve the problem.
-$\cos 4 x$

Accepted Answer

A)  $3 \sin x - 4 \sin ^ { 3 } x$ 
B)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
C)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
D)  $\cos ^ { 4 } x - 6 \sin ^ { 2 } x \cos ^ { 2 } x + \sin ^ { 4 } x$ 
A)  $3 \sin x - 4 \sin ^ { 3 } x$ 
B)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
C)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
D)  $\cos ^ { 4 } x - 6 \sin ^ { 2 } x \cos ^ { 2 } x + \sin ^ { 4 } x$

Question 12

Verify that the equation is an identity.
-$$\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$$

Accepted Answer

\[\begin{array} { l } 
\tan \left( \frac

Question 13

Provide an appropriate response.
-Graph the expression $\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }$ 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 \(\frac...

Question 14

Use the fundamental identities to find the value of the trigonometric function.
-Two sound waves, producing the pressures $\mathrm { P } _ { 1 } = 8 \sin ( 460 \pi \mathrm { t } )$ and $\mathrm { P } _ { 2 } = 6 \cos ( 460 \pi \mathrm { t } )$, combine on a person's ear drum to produce the pressure $\mathrm { P } = \mathrm { P } _ { 1 } + \mathrm { P } _ { 2 }$. Graph $\mathrm { P }$ in $[ - 0.01,0.01,0.001 ]$ by $[ - 20,20,2 ]$ and use the graph to estimate values for $a$ and $k$ so that $P = a \sin ( 460 \pi t + k )$. Round $k$ to the nearest ten-thousandth, if necessary.

Accepted Answer

A)  $\mathrm { a } = 3.2 ; \mathrm { k } = 929.9437$ 
B)  $a = 10 ; k = 929.9437$ 
C)  $a = 10 ; k = 0.6435$ 
D)  $a = 3.2 ; \mathrm { k } = 0.9273$ 
A)  $\mathrm { a } = 3.2 ; \mathrm { k } = 929.9437$ 
B)  $a = 10 ; k = 929.9437$ 
C)  $a = 10 ; k = 0.6435$ 
D)  $a = 3.2 ; \mathrm { k } = 0.9273$

Question 15

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\frac { \sin 44 ^ { \circ } } { 1 - \cos 44 ^ { \circ } }$$

Accepted Answer

A)  $\tan 22 ^ { \circ }$ 
B)  $\cot 22 ^ { \circ }$ 
C)  $\cos 22 ^ { \circ }$ 
D)  $\sin 22 ^ { \circ }$ 
A)  $\tan 22 ^ { \circ }$ 
B)  $\cot 22 ^ { \circ }$ 
C)  $\cos 22 ^ { \circ }$ 
D)  $\sin 22 ^ { \circ }$

Question 16

Use the fundamental identities to find the value of the trigonometric function.
-$$---+ \tan ^ { 2 } x = \sec ^ { 2 } x$$

Accepted Answer

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

Question 17

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-$\sin \frac { 11 \pi } { 12 }$

Accepted Answer

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

Question 18

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-$\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 x$

Accepted Answer

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

Question 19

Verify that each equation is an identity.
-$$\left( 1 + \tan ^ { 2 } s \right) \left( 1 - \sin ^ { 2 } s \right) = 1$$

Accepted Answer

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

Question 20

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-tan 105°

Accepted Answer

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

Determine all solutions of the equation in radians. - $( \sin x + \cos x ) ^ { 2 } = 1$

Use identities to find the indicated value for each angle measure. - $\cos 2 \theta = \frac { 4 } { 5 }$ and $\theta$ terminates in quadrant III $\quad$ Find $\cos \theta$ .

Complete the sentence so the result is an identity. Let x be any real number. -Find $\cos \theta$ if $\csc \theta = - \frac { 5 } { 3 }$ and $\theta$ is in quadrant IV.

Use the fundamental identities to find the value of the trigonometric function. - $\cos \frac { 17 \pi } { 72 } = \cos \frac { \pi } { 9 } \cos \frac { \pi } { 8 } - \sin \frac { \pi } { 9 } \sin \frac { \pi } { 8 }$

Tell whether the statement is true or false. - $\cos \left( - 165 ^ { \circ } \right)$

Use the fundamental identities to simplify the expression. - $\frac { \sin x \cos x } { \tan x }$

Use the fundamental identities to find the value of the trigonometric function. - $2 \cos ^ { 2 } 75 ^ { \circ } - 1$

Solve the equation for exact solutions. - $\sec ^ { 4 } x - \tan ^ { 4 } x = \sec ^ { 2 } x + \tan ^ { 2 } x$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta$

Perform the indicated operations and simplify the result so there are no quotients. Solve the problem. - $\cos 4 x$

Verify that the equation is an identity. - $\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$

Provide an appropriate response. -Graph the expression $\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { \sin 44 ^ { \circ } } { 1 - \cos 44 ^ { \circ } }$

Use the fundamental identities to find the value of the trigonometric function. - $---+ \tan ^ { 2 } x = \sec ^ { 2 } x$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. - $\sin \frac { 11 \pi } { 12 }$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. - $\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 x$

Verify that each equation is an identity. - $\left( 1 + \tan ^ { 2 } s \right) \left( 1 - \sin ^ { 2 } s \right) = 1$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. -tan 105°

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

Determine all solutions of the equation in radians. - (sin⁡x+cos⁡x)2=1( \sin x + \cos x ) ^ { 2 } = 1(sinx+cosx)2=1

Use identities to find the indicated value for each angle measure. - cos⁡2θ=45\cos 2 \theta = \frac { 4 } { 5 }cos2θ=54​ and θ\thetaθ terminates in quadrant III \quad Find cos⁡θ\cos \thetacosθ .

Complete the sentence so the result is an identity. Let x be any real number. -Find cos⁡θ\cos \thetacosθ if csc⁡θ=−53\csc \theta = - \frac { 5 } { 3 }cscθ=−35​ and θ\thetaθ is in quadrant IV.

Tell whether the statement is true or false. - cos⁡(−165∘)\cos \left( - 165 ^ { \circ } \right)cos(−165∘)

Use the fundamental identities to simplify the expression. - sin⁡xcos⁡xtan⁡x\frac { \sin x \cos x } { \tan x }tanxsinxcosx​

Use the fundamental identities to find the value of the trigonometric function. - 2cos⁡275∘−12 \cos ^ { 2 } 75 ^ { \circ } - 12cos275∘−1

Solve the equation for exact solutions. - sec⁡4x−tan⁡4x=sec⁡2x+tan⁡2x\sec ^ { 4 } x - \tan ^ { 4 } x = \sec ^ { 2 } x + \tan ^ { 2 } xsec4x−tan4x=sec2x+tan2x

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - sin⁡2θ+tan⁡2θ+cos⁡2θ\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \thetasin2θ+tan2θ+cos2θ

Perform the indicated operations and simplify the result so there are no quotients. Solve the problem. - cos⁡4x\cos 4 xcos4x

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

Provide an appropriate response. -Graph the expression 1−sin⁡2xsin⁡x−csc⁡x\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }sinx−cscx1−sin2x​ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Use an identity to write the expression as a single trigonometric function or as a single number. - sin⁡44∘1−cos⁡44∘\frac { \sin 44 ^ { \circ } } { 1 - \cos 44 ^ { \circ } }1−cos44∘sin44∘​

Use the fundamental identities to find the value of the trigonometric function. - −−−+tan⁡2x=sec⁡2x---+ \tan ^ { 2 } x = \sec ^ { 2 } x−−−+tan2x=sec2x

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. - 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. - cos⁡24x−sin⁡24x\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 xcos24x−sin24x

Verify that each equation is an identity. - (1+tan⁡2s)(1−sin⁡2s)=1\left( 1 + \tan ^ { 2 } s \right) \left( 1 - \sin ^ { 2 } s \right) = 1(1+tan2s)(1−sin2s)=1