Question 1

Solve the multi-angle equation below.
$\cos \left( \frac { x } { 2 } \right) = \frac { \sqrt { 2 } } { 2 }$

Accepted Answer

A)  $\frac { \pi } { 3 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
B)  $\frac { \pi } { 3 } + 2 n \pi , \frac { 7 \pi } { 3 } + 2 n \pi$, where $n$ is an integer 
C)  $\frac { \pi } { 2 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
D)  $\frac { \pi } { 2 } + n \pi , \frac { 7 \pi } { 2 } + n \pi$, where $n$ is an integer 
E)  $\frac { \pi } { 2 } + 4 n \pi , \frac { 7 \pi } { 2 } + 4 n \pi$, where $n$ is an integer 
A)  $\frac { \pi } { 3 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
B)  $\frac { \pi } { 3 } + 2 n \pi , \frac { 7 \pi } { 3 } + 2 n \pi$, where $n$ is an integer 
C)  $\frac { \pi } { 2 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
D)  $\frac { \pi } { 2 } + n \pi , \frac { 7 \pi } { 2 } + n \pi$, where $n$ is an integer 
E)  $\frac { \pi } { 2 } + 4 n \pi , \frac { 7 \pi } { 2 } + 4 n \pi$, where $n$ is an integer

Question 2

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 11 } { 61 }$ and $\cos v = - \frac { 40 } { 41 }$. (Both $u$ and $v$ are in Quadrant II.)

Accepted Answer

A)  $\cos ( u + v ) = \frac { 160 } { 2501 }$ 
B)  $\cos ( u + v ) = - \frac { 100 } { 2501 }$ 
C)  $\cos ( u + v ) = \frac { 2301 } { 2501 }$ 
D)  $\cos ( u + v ) = - \frac { 2290 } { 2501 }$ 
E)  $\cos ( u + v ) = \frac { 980 } { 2501 }$ 
A)  $\cos ( u + v ) = \frac { 160 } { 2501 }$ 
B)  $\cos ( u + v ) = - \frac { 100 } { 2501 }$ 
C)  $\cos ( u + v ) = \frac { 2301 } { 2501 }$ 
D)  $\cos ( u + v ) = - \frac { 2290 } { 2501 }$ 
E)  $\cos ( u + v ) = \frac { 980 } { 2501 }$

Question 3

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.
$$( \sin x + \cos x ) ( \sin x - \cos x )$$

Accepted Answer

A)  $2 \sin ^ { 2 } x - \sec ^ { 2 } x - \tan ^ { 2 } x$ 
B)  $\sin ^ { 2 } x - \cos ^ { 2 } x$ 
C)  $1 - 2 \cos ^ { 2 } x$ 
D)  $\csc ^ { 2 } x - \cot ^ { 2 } x - 2 \cos ^ { 2 } x$ 
E)  $1 - 2 \sin \left( \frac { \pi } { 2 } - x \right) \cos x$ 
A)  $2 \sin ^ { 2 } x - \sec ^ { 2 } x - \tan ^ { 2 } x$ 
B)  $\sin ^ { 2 } x - \cos ^ { 2 } x$ 
C)  $1 - 2 \cos ^ { 2 } x$ 
D)  $\csc ^ { 2 } x - \cot ^ { 2 } x - 2 \cos ^ { 2 } x$ 
E)  $1 - 2 \sin \left( \frac { \pi } { 2 } - x \right) \cos x$

Question 4

Verify the identity shown below.
$\frac { 1 + \tan \theta } { 1 + \cot \theta } = \tan \theta$

Accepted Answer

None

Question 5

Determine which of the following are trigonometric identities.
I. $\frac { \cos ( \mathrm { t } ) - \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) - \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } = 0$
II. $\frac { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } = 1$
III. $\frac { \cos ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) \sin ( \mathrm { s } ) } = \cos ( \mathrm { s } ) + \sin ( \mathrm { t } )$

Accepted Answer

A)  I is the only identity. 
B)  I, II, and III are identities. 
C)  II is the only identity. 
D)  II and II are the only identities. 
E)  I and II are the only identities. 
A)  I is the only identity. 
B)  I, II, and III are identities. 
C)  II is the only identity. 
D)  II and II are the only identities. 
E)  I and II are the only identities.

Question 6

Use the figure below to determine the exact value of the given function.

Accepted Answer

A)  $\sin 2 \theta = \frac { 5 } { 13 }$ 
B)  $\sin 2 \theta = \frac { 7 } { 13 }$ 
C)  $\sin 2 \theta = \frac { 12 } { 5 }$ 
D)  $\sin 2 \theta = \frac { 9 } { 13 }$ 
E)  $\sin 2 \theta = \frac { 12 } { 13 }$ 
A)  $\sin 2 \theta = \frac { 5 } { 13 }$ 
B)  $\sin 2 \theta = \frac { 7 } { 13 }$ 
C)  $\sin 2 \theta = \frac { 12 } { 5 }$ 
D)  $\sin 2 \theta = \frac { 9 } { 13 }$ 
E)  $\sin 2 \theta = \frac { 12 } { 13 }$

Question 7

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 3 } { 5 }$ and $\cos v = \frac { 24 } { 25 }$. (Both $u$ and $v$ are in Quadrant IV.)

Accepted Answer

A)  $\tan ( u + v ) = \frac { 41 } { 75 }$ 
B)  $\quad \tan ( u + v ) = \frac { 38 } { 75 }$ 
C)  $\tan ( u + v ) = - \frac { 4 } { 3 }$ 
D)  $\tan ( u + v ) = \frac { 89 } { 75 }$ 
E)  $\tan ( u + v ) = - \frac { 39 } { 25 }$ 
A)  $\tan ( u + v ) = \frac { 41 } { 75 }$ 
B)  $\quad \tan ( u + v ) = \frac { 38 } { 75 }$ 
C)  $\tan ( u + v ) = - \frac { 4 } { 3 }$ 
D)  $\tan ( u + v ) = \frac { 89 } { 75 }$ 
E)  $\tan ( u + v ) = - \frac { 39 } { 25 }$

Question 8

Use the product-to-sum formula to write the given product as a sum or difference.
$$10 \sin \frac { \pi } { 8 } \cos \frac { \pi } { 8 }$$

Accepted Answer

A)  $5 \sin \frac { \pi } { 8 } + 5 \cos \frac { \pi } { 8 }$ 
B)  $5 \sin \frac { \pi } { 4 }$ 
C)  $5 + 5 \cos \frac { \pi } { 16 }$ 
D)  $5 - 5 \cos \frac { \pi } { 16 }$ 
E)  $- 5 \sin \frac { \pi } { 16 }$ 
A)  $5 \sin \frac { \pi } { 8 } + 5 \cos \frac { \pi } { 8 }$ 
B)  $5 \sin \frac { \pi } { 4 }$ 
C)  $5 + 5 \cos \frac { \pi } { 16 }$ 
D)  $5 - 5 \cos \frac { \pi } { 16 }$ 
E)  $- 5 \sin \frac { \pi } { 16 }$

Question 9

Use the trigonometric substitution $x = 8 \sec ( \theta )$ to write the expression $\sqrt { x ^ { 2 } - 64 }$ as a trigonometric function of $\theta$, where $0 < \theta < \frac { \pi } { 2 }$.

Accepted Answer

A)  $8 \tan ( \theta )$ 
B)  $64 \tan ( \theta )$ 
C)  $64 \sec ( \theta )$ 
D)  $8 \sec ( \theta )$ 
E)  $8 \sec ( \theta ) - 1$ 
A)  $8 \tan ( \theta )$ 
B)  $64 \tan ( \theta )$ 
C)  $64 \sec ( \theta )$ 
D)  $8 \sec ( \theta )$ 
E)  $8 \sec ( \theta ) - 1$

Question 10

Determine which of the following are trigonometric identities.
I. $\frac { \sin ( x ) - \sin ( y ) } { \cos ( x ) + \cos ( y ) } + \frac { \cos ( x ) - \cos ( y ) } { \sin ( x ) + \sin ( y ) } = 0$
II. $\frac { \sin ( \mathrm { x } ) + \sin ( \mathrm { y } ) } { \cos ( \mathrm { x } ) + \cos ( \mathrm { y } ) } + \frac { \cos ( \mathrm { x } ) + \cos ( \mathrm { y } ) } { \sin ( \mathrm { x } ) + \sin ( \mathrm { y } ) } = 1$
III. $\frac { \sin ( \mathrm { x } ) + \cos ( \mathrm { y } ) } { \sin ( \mathrm { x } ) \cos ( \mathrm { y } ) } = \sin ( \mathrm { y } ) + \cos ( \mathrm { x } )$

Accepted Answer

A)  I is the only identity. 
B)  I and III are the only identities. 
C)  II and II are the only identities. 
D)  I and II are the only identities. 
E)  I, II, and III are identities. 
A)  I is the only identity. 
B)  I and III are the only identities. 
C)  II and II are the only identities. 
D)  I and II are the only identities. 
E)  I, II, and III are identities.

Question 11

Find all solutions of the following equation in the interval $[ 0,2 \pi )$.
$$\csc ^ { 2 } x = \cot x + 1$$

Accepted Answer

A)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 4 \pi } { 3 }$ 
B)  $x = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
C)  $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
D)  $x = \frac { \pi } { 4 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 4 } , \frac { 3 \pi } { 2 }$ 
E)  $x = \frac { \pi } { 4 } , \frac { 4 \pi } { 3 } , \frac { 11 \pi } { 6 }$ 
A)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 4 \pi } { 3 }$ 
B)  $x = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
C)  $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
D)  $x = \frac { \pi } { 4 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 4 } , \frac { 3 \pi } { 2 }$ 
E)  $x = \frac { \pi } { 4 } , \frac { 4 \pi } { 3 } , \frac { 11 \pi } { 6 }$

Question 12

Find all solutions of the following equation in the interval $[ 0,2 \pi )$.
$2 \cos ^ { 2 } x = 2 + \sin x$

Accepted Answer

A)  $x = 0 , \frac { \pi } { 4 } , \frac { 3 \pi } { 4 }$ 
B)  $x = 0 , \frac { 3 \pi } { 4 } , \frac { 4 \pi } { 3 } , \frac { 11 \pi } { 6 }$ 
C)  $x = \frac { \pi } { 2 } , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , \frac { 3 \pi } { 2 }$ 
D)  $x = 0 , \pi , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
E)  $x = \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 6 }$ 
A)  $x = 0 , \frac { \pi } { 4 } , \frac { 3 \pi } { 4 }$ 
B)  $x = 0 , \frac { 3 \pi } { 4 } , \frac { 4 \pi } { 3 } , \frac { 11 \pi } { 6 }$ 
C)  $x = \frac { \pi } { 2 } , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , \frac { 3 \pi } { 2 }$ 
D)  $x = 0 , \pi , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
E)  $x = \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 6 }$

Question 13

Write the given expression as the cosine of an angle. $\cos 30 ^ { \circ } \cos 50 ^ { \circ } + \sin 30 ^ { \circ } \sin 50 ^ { \circ }$

Accepted Answer

A)  $\cos \left( 50 ^ { \circ } \right)$ 
B)  $\cos \left( - 20 ^ { \circ } \right)$ 
C)  $\cos \left( 80 ^ { \circ } \right)$ 
D)  $\cos \left( 30 ^ { \circ } \right)$ 
E)  $\cos \left( - 100 ^ { \circ } \right)$ 
A)  $\cos \left( 50 ^ { \circ } \right)$ 
B)  $\cos \left( - 20 ^ { \circ } \right)$ 
C)  $\cos \left( 80 ^ { \circ } \right)$ 
D)  $\cos \left( 30 ^ { \circ } \right)$ 
E)  $\cos \left( - 100 ^ { \circ } \right)$

Question 14

Expand the expression below and use fundamental trigonometric identities to simplify.
$( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Accepted Answer

A)  $\sin ^ { 2 } ( \omega ) + \cos ^ { 2 } ( \omega )$ 
B)  $2 \tan ( \omega ) + 1$ 
C)  $2 \sin ( \omega ) \cos ( \omega ) + 1$ 
D)  1 
E)  $2 \cot ( \omega ) + 1$ 
A)  $\sin ^ { 2 } ( \omega ) + \cos ^ { 2 } ( \omega )$ 
B)  $2 \tan ( \omega ) + 1$ 
C)  $2 \sin ( \omega ) \cos ( \omega ) + 1$ 
D)  1 
E)  $2 \cot ( \omega ) + 1$

Question 15

Rewrite the expression $\frac { \sin ( y ) } { 1 - \cos ( y ) }$ so that it is not in fractional form.

Accepted Answer

A)  $\sin ^ { 2 } - \sin ( y ) \tan ( y )$ 
B)  $1 - \sin ( y ) \tan ( y )$ 
C)  $\csc ( y ) + \cot ( y )$ 
D)  $\sin ^ { 2 } + \sin ( y ) \tan ( y )$ 
E)  $1 - \cos ( y )$ 
A)  $\sin ^ { 2 } - \sin ( y ) \tan ( y )$ 
B)  $1 - \sin ( y ) \tan ( y )$ 
C)  $\csc ( y ) + \cot ( y )$ 
D)  $\sin ^ { 2 } + \sin ( y ) \tan ( y )$ 
E)  $1 - \cos ( y )$

Question 16

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.
$$\cot ^ { 2 } \alpha - \cot ^ { 2 } \alpha \cos ^ { 2 } \alpha$$

Accepted Answer

A)  $\cos ^ { 2 } \alpha$ 
B)  $1 - \sin ^ { 2 } \alpha$ 
C)  $\tan ^ { 2 } \alpha$ 
D)  $\sin ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$ 
E)  $\frac { 1 } { \sec ^ { 2 } \alpha }$ 
A)  $\cos ^ { 2 } \alpha$ 
B)  $1 - \sin ^ { 2 } \alpha$ 
C)  $\tan ^ { 2 } \alpha$ 
D)  $\sin ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$ 
E)  $\frac { 1 } { \sec ^ { 2 } \alpha }$

Question 17

Use the graph below to approximate the solutions of the equation $- 2 \cos ( x ) - \sin ( x ) = 0$ on the interval $[ 0,2 \pi )$. Round your answer to one decimal.

Accepted Answer

A)  $- 2.0,5.9$ 
B)  $2.0,5.9$ 
C)  $5.2,5.9$ 
D)  $2,5.2$ 
E)  $- 2,5.2$ 
A)  $- 2.0,5.9$ 
B)  $2.0,5.9$ 
C)  $5.2,5.9$ 
D)  $2,5.2$ 
E)  $- 2,5.2$

Question 18

Determine which of the following are trigonometric identities.
I. $\cos ^ { 4 } ( \mathrm { t } ) + \sin ^ { 4 } ( \mathrm { t } ) = 1 - 2 \sin ^ { 2 } ( \mathrm { t } ) + 2 \sin ^ { 4 } ( \mathrm { t } )$
II. $\sin ^ { 5 } ( \mathrm { t } ) = \sin ^ { 3 } ( \mathrm { t } ) \cos ^ { 2 } ( \mathrm { t } ) - \sin ^ { 3 } ( \mathrm { t } )$
III. $\sin ^ { 3 } ( \mathrm { t } ) \cos ^ { 2 } ( \mathrm { t } ) = \left( \cos ^ { 2 } ( \mathrm { t } ) - \cos ^ { 4 } ( \mathrm { t } ) \right) \sin ( \mathrm { t } )$

Accepted Answer

A)  I is the only identity. 
B)  I and III are the only identities. 
C)  III is the only identity. 
D)  I, II, and III are identities. 
E)  I and II are the only identities. 
A)  I is the only identity. 
B)  I and III are the only identities. 
C)  III is the only identity. 
D)  I, II, and III are identities. 
E)  I and II are the only identities.

Question 19

Approximate the solutions of the equation $2 \sin ^ { 2 } ( x ) - 4 \sin ( x ) + 1 = 0$ by considering its graph below. Round your answer to one decimal.

Accepted Answer

A)  0.3,2.8 
B)  $0.3,1.0$ 
C)  $1.0,4.6$ 
D)  $0.3,4.6$ 
E)  $2.8,4.6$ 
A)  0.3,2.8 
B)  $0.3,1.0$ 
C)  $1.0,4.6$ 
D)  $0.3,4.6$ 
E)  $2.8,4.6$

Question 20

If $\sin x = \frac { 1 } { 2 }$ and $\cos x = \frac { \sqrt { 3 } } { 2 }$, evaluate the following function. $\csc x$

Accepted Answer

A)  $\csc x = \frac { \sqrt { 3 } } { 3 }$ 
B)  $\csc x = 2$ 
C)  $\csc x = \sqrt { 3 }$ 
D)  $\csc x = \frac { 1 } { 3 }$ 
E)  $\csc x = \frac { 2 \sqrt { 3 } } { 3 }$ 
A)  $\csc x = \frac { \sqrt { 3 } } { 3 }$ 
B)  $\csc x = 2$ 
C)  $\csc x = \sqrt { 3 }$ 
D)  $\csc x = \frac { 1 } { 3 }$ 
E)  $\csc x = \frac { 2 \sqrt { 3 } } { 3 }$

Solve the multi-angle equation below. $\cos \left( \frac { x } { 2 } \right) = \frac { \sqrt { 2 } } { 2 }$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 11 } { 61 }$ and $\cos v = - \frac { 40 } { 41 }$ . (Both $u$ and $v$ are in Quadrant II.)

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $( \sin x + \cos x ) ( \sin x - \cos x )$

Verify the identity shown below. $\frac { 1 + \tan \theta } { 1 + \cot \theta } = \tan \theta$

Use the figure below to determine the exact value of the given function.

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 3 } { 5 }$ and $\cos v = \frac { 24 } { 25 }$ . (Both $u$ and $v$ are in Quadrant IV.)

Use the product-to-sum formula to write the given product as a sum or difference. $10 \sin \frac { \pi } { 8 } \cos \frac { \pi } { 8 }$

Use the trigonometric substitution $x = 8 \sec ( \theta )$ to write the expression $\sqrt { x ^ { 2 } - 64 }$ as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ .

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ . $\csc ^ { 2 } x = \cot x + 1$

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ . $2 \cos ^ { 2 } x = 2 + \sin x$

Write the given expression as the cosine of an angle. $\cos 30 ^ { \circ } \cos 50 ^ { \circ } + \sin 30 ^ { \circ } \sin 50 ^ { \circ }$

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Rewrite the expression $\frac { \sin ( y ) } { 1 - \cos ( y ) }$ so that it is not in fractional form.

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\cot ^ { 2 } \alpha - \cot ^ { 2 } \alpha \cos ^ { 2 } \alpha$

If $\sin x = \frac { 1 } { 2 }$ and $\cos x = \frac { \sqrt { 3 } } { 2 }$ , evaluate the following function. $\csc x$

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Exam 5: Analytic Trigonometry

Solve the multi-angle equation below. cos⁡(x2)=22\cos \left( \frac { x } { 2 } \right) = \frac { \sqrt { 2 } } { 2 }cos(2x​)=22​​

Find the exact value of cos⁡(u+v)\cos ( u + v )cos(u+v) given that sin⁡u=1161\sin u = \frac { 11 } { 61 }sinu=6111​ and cos⁡v=−4041\cos v = - \frac { 40 } { 41 }cosv=−4140​ . (Both uuu and vvv are in Quadrant II.)

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. (sin⁡x+cos⁡x)(sin⁡x−cos⁡x)( \sin x + \cos x ) ( \sin x - \cos x )(sinx+cosx)(sinx−cosx)

Verify the identity shown below. 1+tan⁡θ1+cot⁡θ=tan⁡θ\frac { 1 + \tan \theta } { 1 + \cot \theta } = \tan \theta1+cotθ1+tanθ​=tanθ

Use the figure below to determine the exact value of the given function.

Find the exact value of tan⁡(u+v)\tan ( u + v )tan(u+v) given that sin⁡u=−35\sin u = - \frac { 3 } { 5 }sinu=−53​ and cos⁡v=2425\cos v = \frac { 24 } { 25 }cosv=2524​ . (Both uuu and vvv are in Quadrant IV.)

Use the product-to-sum formula to write the given product as a sum or difference. 10sin⁡π8cos⁡π810 \sin \frac { \pi } { 8 } \cos \frac { \pi } { 8 }10sin8π​cos8π​

Use the trigonometric substitution x=8sec⁡(θ)x = 8 \sec ( \theta )x=8sec(θ) to write the expression x2−64\sqrt { x ^ { 2 } - 64 }x2−64​ as a trigonometric function of θ\thetaθ , where 0<θ<π20 < \theta < \frac { \pi } { 2 }0<θ<2π​ .

Find all solutions of the following equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . csc⁡2x=cot⁡x+1\csc ^ { 2 } x = \cot x + 1csc2x=cotx+1

Find all solutions of the following equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . 2cos⁡2x=2+sin⁡x2 \cos ^ { 2 } x = 2 + \sin x2cos2x=2+sinx

Write the given expression as the cosine of an angle. cos⁡30∘cos⁡50∘+sin⁡30∘sin⁡50∘\cos 30 ^ { \circ } \cos 50 ^ { \circ } + \sin 30 ^ { \circ } \sin 50 ^ { \circ }cos30∘cos50∘+sin30∘sin50∘

Expand the expression below and use fundamental trigonometric identities to simplify. (sin⁡(ω)+cos⁡(ω))2( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }(sin(ω)+cos(ω))2

Rewrite the expression sin⁡(y)1−cos⁡(y)\frac { \sin ( y ) } { 1 - \cos ( y ) }1−cos(y)sin(y)​ so that it is not in fractional form.

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. cot⁡2α−cot⁡2αcos⁡2α\cot ^ { 2 } \alpha - \cot ^ { 2 } \alpha \cos ^ { 2 } \alphacot2α−cot2αcos2α

Use the graph below to approximate the solutions of the equation −2cos⁡(x)−sin⁡(x)=0- 2 \cos ( x ) - \sin ( x ) = 0−2cos(x)−sin(x)=0 on the interval [0,2π)[ 0,2 \pi )[0,2π) . Round your answer to one decimal.

Approximate the solutions of the equation 2sin⁡2(x)−4sin⁡(x)+1=02 \sin ^ { 2 } ( x ) - 4 \sin ( x ) + 1 = 02sin2(x)−4sin(x)+1=0 by considering its graph below. Round your answer to one decimal.

If sin⁡x=12\sin x = \frac { 1 } { 2 }sinx=21​ and cos⁡x=32\cos x = \frac { \sqrt { 3 } } { 2 }cosx=23​​ , evaluate the following function. csc⁡x\csc xcscx

Functions and Their Graphs

Polynomial and Rational Functions

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Solve the multi-angle equation below. $\cos \left( \frac { x } { 2 } \right) = \frac { \sqrt { 2 } } { 2 }$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 11 } { 61 }$ and $\cos v = - \frac { 40 } { 41 }$ . (Both $u$ and $v$ are in Quadrant II.)

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $( \sin x + \cos x ) ( \sin x - \cos x )$

Verify the identity shown below. $\frac { 1 + \tan \theta } { 1 + \cot \theta } = \tan \theta$

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 3 } { 5 }$ and $\cos v = \frac { 24 } { 25 }$ . (Both $u$ and $v$ are in Quadrant IV.)

Use the product-to-sum formula to write the given product as a sum or difference. $10 \sin \frac { \pi } { 8 } \cos \frac { \pi } { 8 }$

Use the trigonometric substitution $x = 8 \sec ( \theta )$ to write the expression $\sqrt { x ^ { 2 } - 64 }$ as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ .

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ . $\csc ^ { 2 } x = \cot x + 1$

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ . $2 \cos ^ { 2 } x = 2 + \sin x$

Write the given expression as the cosine of an angle. $\cos 30 ^ { \circ } \cos 50 ^ { \circ } + \sin 30 ^ { \circ } \sin 50 ^ { \circ }$

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Rewrite the expression $\frac { \sin ( y ) } { 1 - \cos ( y ) }$ so that it is not in fractional form.

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\cot ^ { 2 } \alpha - \cot ^ { 2 } \alpha \cos ^ { 2 } \alpha$

If $\sin x = \frac { 1 } { 2 }$ and $\cos x = \frac { \sqrt { 3 } } { 2 }$ , evaluate the following function. $\csc x$