Question 1

Use Identities to find the exact value.
Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$\sin 4 x$

Accepted Answer

A)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
B)  $1 - 8 \sin ^ { 2 } x \cos ^ { 2 } x$ 
C)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
D)  $3 \sin x - 4 \sin ^ { 3 } x$ 
A)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
B)  $1 - 8 \sin ^ { 2 } x \cos ^ { 2 } x$ 
C)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
D)  $3 \sin x - 4 \sin ^ { 3 } x$

Question 2

Determine all solutions of the equation in radians.
-Find $\cos ( s + t )$ given that $\cos s = \frac { 1 } { 3 }$, with $s$ in quadrant $I$, and $\sin t = - \frac { 1 } { 2 }$, with $t$ in quadrant IV.

Accepted Answer

A)  $\frac { 2 \sqrt { 6 } + 1 } { 6 }$ 
B)  $\frac { \sqrt { 3 } - 2 \sqrt { 2 } } { 6 }$ 
C)  $\frac { 2 \sqrt { 6 } - 1 } { 6 }$ 
D)  $\frac { \sqrt { 3 } + 2 \sqrt { 2 } } { 6 }$ 
A)  $\frac { 2 \sqrt { 6 } + 1 } { 6 }$ 
B)  $\frac { \sqrt { 3 } - 2 \sqrt { 2 } } { 6 }$ 
C)  $\frac { 2 \sqrt { 6 } - 1 } { 6 }$ 
D)  $\frac { \sqrt { 3 } + 2 \sqrt { 2 } } { 6 }$

Question 3

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$\sqrt { \sin ^ { 2 } x } = \sin x$

Accepted Answer

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

Question 4

Verify that each equation is an identity.
-$$\sin ( 4 u ) = 2 \sin ( 2 u ) \cos ( 2 u )$$

Accepted Answer

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

Question 5

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 6

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)$

Accepted Answer

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

Question 7

Use a sum or difference identity to find the exact value.
-$\cos 36 ^ { \circ } \cos 24 ^ { \circ } - \sin 36 ^ { \circ } \sin 24 ^ { \circ }$

Accepted Answer

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

Question 8

Factor the trigonometric expression and simplify.
-$$\sec \theta - \frac { 1 } { \sec \theta }$$

Accepted Answer

A)  $\sec \theta \csc \theta$ 
B)  $1 + \cot \theta$ 
C)  $- 2 \tan ^ { 2 } \theta$ 
D)  $\sin \theta \tan \theta$ 
A)  $\sec \theta \csc \theta$ 
B)  $1 + \cot \theta$ 
C)  $- 2 \tan ^ { 2 } \theta$ 
D)  $\sin \theta \tan \theta$

Question 9

Use identities to find the indicated value for each angle measure.
-$\cos 2 \theta = - \frac { 3 } { 8 }$ and $\frac { \pi } { 2 } < \theta < \pi \quad$ Find $\sin \theta$.

Accepted Answer

A)  $\sin \theta = - \frac { \sqrt { 11 } } { 4 }$ 
B)  $\sin \theta = \frac { \sqrt { 11 } } { 4 }$ 
C)  $\sin \theta = 0$ 
D)  $\sin \theta =$ 
A)  $\sin \theta = - \frac { \sqrt { 11 } } { 4 }$ 
B)  $\sin \theta = \frac { \sqrt { 11 } } { 4 }$ 
C)  $\sin \theta = 0$ 
D)  $\sin \theta =$

Question 10

Verify that the equation is an identity.
-$$\tan ^ { 2 } \mathrm { u } ( 1 + \cos ( 2 \mathrm { u } ) ) = 1 - \cos ( 2 \mathrm { u } )$$

Accepted Answer

The answer of Verify that the equation is an identity.
-\[\tan...

Question 11

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$2 \tan \theta - ( 1 + \tan \theta ) ^ { 2 }$

Accepted Answer

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

Question 12

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sec \theta$ if $\tan \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant $\mathrm { I }$.

Accepted Answer

A)  $- \frac { 3 } { 2 }$ 
B)  $- \frac { \sqrt { 7 } } { 9 }$ 
C)  $\frac { 3 \sqrt { 7 } } { 7 }$ 
D)  $\frac { 5 } { 4 }$ 
A)  $- \frac { 3 } { 2 }$ 
B)  $- \frac { \sqrt { 7 } } { 9 }$ 
C)  $\frac { 3 \sqrt { 7 } } { 7 }$ 
D)  $\frac { 5 } { 4 }$

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

Factor the trigonometric expression and simplify.
-$\tan \frac { \pi } { 12 }$

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 16

Use a sum or difference identity to find the exact value.
-Two voltages, $\mathrm { V } _ { 1 } = 80 \sin ( 110 \pi \mathrm { t } )$ and $\mathrm { V } _ { 2 } = 150 \cos ( 110 \pi \mathrm { t } )$, are added in a circuit to produce the sum $V = V _ { 1 } + V _ { 2 }$. Graph $V$ in $[ - 0.02,0.02,0.002 ]$ by $[ - 250,250,20 ]$ and use the graph to estimate values for $\mathrm { a }$ and $\mathrm { k }$ so that $\mathrm { V } = \mathrm { a } \sin ( 110 \pi \mathrm { t } + \mathrm { k } )$. Round $\mathrm { k }$ to the nearest ten-thousandth, if necessary.

Accepted Answer

A)  $a = 17 ; k = 1.0808$ 
B)  $a = 170 ; k = 169.317$ 
C)  $a = 170 ; \mathrm { k } = 1.0808$ 
D)  $a = 170 ; k = 373.5108$ 
A)  $a = 17 ; k = 1.0808$ 
B)  $a = 170 ; k = 169.317$ 
C)  $a = 170 ; \mathrm { k } = 1.0808$ 
D)  $a = 170 ; k = 373.5108$

Question 17

Use the fundamental identities to simplify the expression.
-$\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $\frac { - 12 \sqrt { 10 } } { 49 }$ 
B)  $\frac { 31 } { 49 }$ 
C)  $- \frac { 31 } { 49 }$ 
D)  $\frac { 12 \sqrt { 10 } } { 49 }$ 
A)  $\frac { - 12 \sqrt { 10 } } { 49 }$ 
B)  $\frac { 31 } { 49 }$ 
C)  $- \frac { 31 } { 49 }$ 
D)  $\frac { 12 \sqrt { 10 } } { 49 }$

Question 18

Solve the equation for exact solutions.
-$$( \sec \alpha + \tan \alpha ) ^ { 2 } = \frac { 1 + \sin \alpha } { 1 - \sin \alpha }$$

Accepted Answer

\[\begin{array} { l } 
( \sec \alpha + \

Question 19

Tell whether the statement is true or false.
-$\sin \left( - 15 ^ { \circ } \right)$

Accepted Answer

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

Question 20

Use the fundamental identities to find the value of the trigonometric function.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\theta$ is the vertex angle of the cone and $\mathrm { m }$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { \mathrm {~m} }$ $( \mathrm { m } > 1 )$. Find $\theta$ to the nearest degree if $\mathrm { m } = 2.9$.

Accepted Answer

A)  $45 ^ { \circ }$ 
B)  $43 ^ { \circ }$ 
C)  $38 ^ { \circ }$ 
D)  $40 ^ { \circ }$ 
A)  $45 ^ { \circ }$ 
B)  $43 ^ { \circ }$ 
C)  $38 ^ { \circ }$ 
D)  $40 ^ { \circ }$

Use Identities to find the exact value. Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\sin 4 x$

Determine all solutions of the equation in radians. -Find $\cos ( s + t )$ given that $\cos s = \frac { 1 } { 3 }$ , with $s$ in quadrant $I$ , and $\sin t = - \frac { 1 } { 2 }$ , with $t$ in quadrant IV.

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\sqrt { \sin ^ { 2 } x } = \sin x$

Verify that each equation is an identity. - $\sin ( 4 u ) = 2 \sin ( 2 u ) \cos ( 2 u )$

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

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)$

Use a sum or difference identity to find the exact value. - $\cos 36 ^ { \circ } \cos 24 ^ { \circ } - \sin 36 ^ { \circ } \sin 24 ^ { \circ }$

Factor the trigonometric expression and simplify. - $\sec \theta - \frac { 1 } { \sec \theta }$

Use identities to find the indicated value for each angle measure. - $\cos 2 \theta = - \frac { 3 } { 8 }$ and $\frac { \pi } { 2 } < \theta < \pi \quad$ Find $\sin \theta$ .

Verify that the equation is an identity. - $\tan ^ { 2 } \mathrm { u } ( 1 + \cos ( 2 \mathrm { u } ) ) = 1 - \cos ( 2 \mathrm { u } )$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $2 \tan \theta - ( 1 + \tan \theta ) ^ { 2 }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sec \theta$ if $\tan \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant $\mathrm { I }$ .

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

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

Factor the trigonometric expression and simplify. - $\tan \frac { \pi } { 12 }$

Use the fundamental identities to simplify the expression. - $\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Solve the equation for exact solutions. - $( \sec \alpha + \tan \alpha ) ^ { 2 } = \frac { 1 + \sin \alpha } { 1 - \sin \alpha }$

Tell whether the statement is true or false. - $\sin \left( - 15 ^ { \circ } \right)$

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 Identities to find the exact value. Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - sin⁡4x\sin 4 xsin4x

Determine all solutions of the equation in radians. -Find cos⁡(s+t)\cos ( s + t )cos(s+t) given that cos⁡s=13\cos s = \frac { 1 } { 3 }coss=31​ , with sss in quadrant III , and sin⁡t=−12\sin t = - \frac { 1 } { 2 }sint=−21​ , with ttt in quadrant IV.

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - sin⁡2x=sin⁡x\sqrt { \sin ^ { 2 } x } = \sin xsin2x​=sinx

Verify that each equation is an identity. - sin⁡(4u)=2sin⁡(2u)cos⁡(2u)\sin ( 4 u ) = 2 \sin ( 2 u ) \cos ( 2 u )sin(4u)=2sin(2u)cos(2u)

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

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - sin⁡2x(cot⁡2x+1)\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)sin2x(cot2x+1)

Use a sum or difference identity to find the exact value. - cos⁡36∘cos⁡24∘−sin⁡36∘sin⁡24∘\cos 36 ^ { \circ } \cos 24 ^ { \circ } - \sin 36 ^ { \circ } \sin 24 ^ { \circ }cos36∘cos24∘−sin36∘sin24∘

Factor the trigonometric expression and simplify. - sec⁡θ−1sec⁡θ\sec \theta - \frac { 1 } { \sec \theta }secθ−secθ1​

Use identities to find the indicated value for each angle measure. - cos⁡2θ=−38\cos 2 \theta = - \frac { 3 } { 8 }cos2θ=−83​ and π2<θ<π\frac { \pi } { 2 } < \theta < \pi \quad2π​<θ<π Find sin⁡θ\sin \thetasinθ .

Verify that the equation is an identity. - tan⁡2u(1+cos⁡(2u))=1−cos⁡(2u)\tan ^ { 2 } \mathrm { u } ( 1 + \cos ( 2 \mathrm { u } ) ) = 1 - \cos ( 2 \mathrm { u } )tan2u(1+cos(2u))=1−cos(2u)

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - 2tan⁡θ−(1+tan⁡θ)22 \tan \theta - ( 1 + \tan \theta ) ^ { 2 }2tanθ−(1+tanθ)2

Use the fundamental identities to find the value of the trigonometric function. -Find sec⁡θ\sec \thetasecθ if tan⁡θ=34\tan \theta = \frac { 3 } { 4 }tanθ=43​ and θ\thetaθ is in quadrant I\mathrm { I }I .

Verify that the equation is an 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

Find the exact value by using a half-angle identity. - cos⁡75∘\cos 75 ^ { \circ }cos75∘

Factor the trigonometric expression and simplify. - tan⁡π12\tan \frac { \pi } { 12 }tan12π​

Use the fundamental identities to simplify the expression. - sin⁡θ=2107,tan⁡θ<0\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quadsinθ=7210​​,tanθ<0 Find sin⁡(2θ)\sin ( 2 \theta )sin(2θ)

Solve the equation for exact solutions. - (sec⁡α+tan⁡α)2=1+sin⁡α1−sin⁡α( \sec \alpha + \tan \alpha ) ^ { 2 } = \frac { 1 + \sin \alpha } { 1 - \sin \alpha }(secα+tanα)2=1−sinα1+sinα​

Tell whether the statement is true or false. - sin⁡(−15∘)\sin \left( - 15 ^ { \circ } \right)sin(−15∘)