Question 1

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$.
$$2 \sin ^ { 2 } x + \sin x = 1$$

Accepted Answer

A)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
B) $x = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
C) $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
D) $x = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
E)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
A)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
B) $x = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
C) $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
D) $x = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
E)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ D

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

Question 3

If $x = 2 \cot \theta$, use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$, where $0 < \theta < \pi$.

Accepted Answer

A)  $2 \cos \theta$ 
B)  $2 \csc \theta$ 
C)  $2 \cot \theta$ 
D)  $2 \sec \theta$ 
E)  $2 \sin \theta$ 
A)  $2 \cos \theta$ 
B)  $2 \csc \theta$ 
C)  $2 \cot \theta$ 
D)  $2 \sec \theta$ 
E)  $2 \sin \theta$ B

Question 4

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

Accepted Answer

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

Question 5

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$. (Both $u$ and $v$ are in Quadrant II.)

Accepted Answer

A)  $\cos ( u + v ) = \frac { 28 } { 65 }$ 
B)  $\cos ( u + v ) = - \frac { 16 } { 65 }$ 
C)  $\quad \cos ( u + v ) = \frac { 33 } { 65 }$ 
D)  $\cos ( u + v ) = - \frac { 28 } { 65 }$ 
E) 
$\cos ( u + v ) = \frac { 56 } { 65 }$ 
A)  $\cos ( u + v ) = \frac { 28 } { 65 }$ 
B)  $\cos ( u + v ) = - \frac { 16 } { 65 }$ 
C)  $\quad \cos ( u + v ) = \frac { 33 } { 65 }$ 
D)  $\cos ( u + v ) = - \frac { 28 } { 65 }$ 
E) 
$\cos ( u + v ) = \frac { 56 } { 65 }$

Question 6

Which of the following is a solution to the given equation? $\sec x - 2 = 0$

Accepted Answer

A)  $x = \frac { 7 \pi } { 6 }$ 
B)  $x = \frac { \pi } { 4 }$ 
C)  $x = \frac { 5 \pi } { 6 }$ 
D)  $x = \frac { 5 \pi } { 3 }$ 
E)  $x = \frac { 7 \pi } { 4 }$ 
A)  $x = \frac { 7 \pi } { 6 }$ 
B)  $x = \frac { \pi } { 4 }$ 
C)  $x = \frac { 5 \pi } { 6 }$ 
D)  $x = \frac { 5 \pi } { 3 }$ 
E)  $x = \frac { 7 \pi } { 4 }$

Question 7

Solve the following equation. $$\csc ^ { 2 } ( x ) - 4 = 0$$

Accepted Answer

A)  $\frac { \pi } { 3 } + \pi n , \frac { 2 \pi } { 3 } + \pi n$, where $n$ is an integer 
B)  $\frac { \pi } { 3 } + 2 \pi n , \frac { 2 \pi } { 3 } + 2 \pi n$, where $n$ is an integer 
C)  $\frac { \pi } { 6 } + \pi n , \frac { 5 \pi } { 6 } + \pi n$, where $n$ is an integer 
D)  $\frac { \pi } { 6 } + 2 \pi n , \frac { 2 \pi } { 3 } + 2 \pi n$, where $n$ is an integer 
E)  $\frac { \pi } { 6 } + 2 \pi n , \frac { 5 \pi } { 6 } + 2 \pi n$, where $n$ is an integer 
A)  $\frac { \pi } { 3 } + \pi n , \frac { 2 \pi } { 3 } + \pi n$, where $n$ is an integer 
B)  $\frac { \pi } { 3 } + 2 \pi n , \frac { 2 \pi } { 3 } + 2 \pi n$, where $n$ is an integer 
C)  $\frac { \pi } { 6 } + \pi n , \frac { 5 \pi } { 6 } + \pi n$, where $n$ is an integer 
D)  $\frac { \pi } { 6 } + 2 \pi n , \frac { 2 \pi } { 3 } + 2 \pi n$, where $n$ is an integer 
E)  $\frac { \pi } { 6 } + 2 \pi n , \frac { 5 \pi } { 6 } + 2 \pi n$, where $n$ is an integer

Question 8

Verify the identity shown below. $$\frac { 1 - \sin \theta } { 1 + \sin \theta } = 2 \sec ^ { 2 } \theta - 2 \sec \theta \tan \theta - 1$$

Accepted Answer

\[\begin{aligned}
\frac { 1 - \sin \thet

Question 9

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 10

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.
$$\sin x + 1 + \frac { 1 } { \sin x - 1 }$$

Accepted Answer

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

Question 11

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

Accepted Answer

A)  $\frac { 5 \pi } { 3 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
B)  $\frac { 5 \pi } { 8 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
C)  $\frac { 5 \pi } { 8 } + 2 n \pi , \frac { 7 \pi } { 8 } + 2 n \pi$, where $n$ is an integer 
D)  $\frac { 5 \pi } { 3 } + 2 n \pi , \frac { 7 \pi } { 3 } + 2 n \pi$, where $n$ is an integer 
E)  $\frac { 5 \pi } { 8 } + n \pi , \frac { 7 \pi } { 8 } + n \pi$, where $n$ is an integer 
A)  $\frac { 5 \pi } { 3 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
B)  $\frac { 5 \pi } { 8 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
C)  $\frac { 5 \pi } { 8 } + 2 n \pi , \frac { 7 \pi } { 8 } + 2 n \pi$, where $n$ is an integer 
D)  $\frac { 5 \pi } { 3 } + 2 n \pi , \frac { 7 \pi } { 3 } + 2 n \pi$, where $n$ is an integer 
E)  $\frac { 5 \pi } { 8 } + n \pi , \frac { 7 \pi } { 8 } + n \pi$, where $n$ is an integer

Question 12

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 13

Verify the identity shown below. $$\sec ^ { 2 } \left( \frac { \pi } { 2 } - y \right) - 1 = \cot ^ { 2 } y$$

Accepted Answer

\[\begin{aligned}
\sec ^ { 2 }

Question 14

Write the given expression as the sine of an angle. $\sin 75 ^ { \circ } \cos 35 ^ { \circ } - \sin 35 ^ { \circ } \cos 75 ^ { \circ }$

Accepted Answer

A)  $\sin \left( - 70 ^ { \circ } \right)$ 
B)  $\sin \left( 40 ^ { \circ } \right)$ 
C)  $\sin \left( 110 ^ { \circ } \right)$ 
D)  $\sin \left( 75 ^ { \circ } \right)$ 
E)  $\sin \left( 35 ^ { \circ } \right)$ 
A)  $\sin \left( - 70 ^ { \circ } \right)$ 
B)  $\sin \left( 40 ^ { \circ } \right)$ 
C)  $\sin \left( 110 ^ { \circ } \right)$ 
D)  $\sin \left( 75 ^ { \circ } \right)$ 
E)  $\sin \left( 35 ^ { \circ } \right)$

Question 15

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Accepted Answer

A)  $\sin \left( - 100 ^ { \circ } \right)$ 
B)  $\sin \left( 135 ^ { \circ } \right)$ 
C)  $\sin \left( 35 ^ { \circ } \right)$ 
D)  $\sin \left( 85 ^ { \circ } \right)$ 
E)  $\sin \left( 50 ^ { \circ } \right)$ 
A)  $\sin \left( - 100 ^ { \circ } \right)$ 
B)  $\sin \left( 135 ^ { \circ } \right)$ 
C)  $\sin \left( 35 ^ { \circ } \right)$ 
D)  $\sin \left( 85 ^ { \circ } \right)$ 
E)  $\sin \left( 50 ^ { \circ } \right)$

Question 16

Verify the identity shown below. $$\left( 1 + \cot ^ { 2 } \theta \right) \tan ^ { 2 } \theta = \sec ^ { 2 } \theta$$

Accepted Answer

\[\begin{aligned}
\left( 1 + \cot ^ { 2

Question 17

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$.
$\sin 2 x = \sin x$

Accepted Answer

A)  $x = \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
B)  $x = 0 , \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$ 
C)  $x = \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
E)  $x = 0$ 
A)  $x = \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
B)  $x = 0 , \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$ 
C)  $x = \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
E)  $x = 0$

Question 18

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 19

Verify the identity shown below. $$\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alpha$$

Accepted Answer

\[\begin{aligned}
\frac { \tan \alpha +

Question 20

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)  $\cos \left( - 100 ^ { \circ } \right)$

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ . $2 \sin ^ { 2 } x + \sin x = 1$

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.)

If $x = 2 \cot \theta$ , use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \pi$ .

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ . (Both $u$ and $v$ are in Quadrant II.)

Which of the following is a solution to the given equation? $\sec x - 2 = 0$

Solve the following equation. $\csc ^ { 2 } ( x ) - 4 = 0$

Verify the identity shown below. $\frac { 1 - \sin \theta } { 1 + \sin \theta } = 2 \sec ^ { 2 } \theta - 2 \sec \theta \tan \theta - 1$

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\sin x + 1 + \frac { 1 } { \sin x - 1 }$

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

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

Verify the identity shown below. $\sec ^ { 2 } \left( \frac { \pi } { 2 } - y \right) - 1 = \cot ^ { 2 } y$

Write the given expression as the sine of an angle. $\sin 75 ^ { \circ } \cos 35 ^ { \circ } - \sin 35 ^ { \circ } \cos 75 ^ { \circ }$

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Verify the identity shown below. $\left( 1 + \cot ^ { 2 } \theta \right) \tan ^ { 2 } \theta = \sec ^ { 2 } \theta$

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ . $\sin 2 x = \sin x$

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

Verify the identity shown below. $\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alpha$

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

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Exam 5: Analytic Trigonometry

Find the exact solutions of the given equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . 2sin⁡2x+sin⁡x=12 \sin ^ { 2 } x + \sin x = 12sin2x+sinx=1

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.)

If x=2cot⁡θx = 2 \cot \thetax=2cotθ , use trigonometric substitution to write 4+x2\sqrt { 4 + x ^ { 2 } }4+x2​ as a trigonometric function of θ\thetaθ , where 0<θ<π0 < \theta < \pi0<θ<π .

Find the exact value of cos⁡(u+v)\cos ( u + v )cos(u+v) given that sin⁡u=513\sin u = \frac { 5 } { 13 }sinu=135​ and cos⁡v=−45\cos v = - \frac { 4 } { 5 }cosv=−54​ . (Both uuu and vvv are in Quadrant II.)

Which of the following is a solution to the given equation? sec⁡x−2=0\sec x - 2 = 0secx−2=0

Solve the following equation. csc⁡2(x)−4=0\csc ^ { 2 } ( x ) - 4 = 0csc2(x)−4=0

Verify the identity shown below. 1−sin⁡θ1+sin⁡θ=2sec⁡2θ−2sec⁡θtan⁡θ−1\frac { 1 - \sin \theta } { 1 + \sin \theta } = 2 \sec ^ { 2 } \theta - 2 \sec \theta \tan \theta - 11+sinθ1−sinθ​=2sec2θ−2secθtanθ−1

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. sin⁡x+1+1sin⁡x−1\sin x + 1 + \frac { 1 } { \sin x - 1 }sinx+1+sinx−11​

Solve the multi-angle equation below. sin⁡(2x)=−22\sin ( 2 x ) = - \frac { \sqrt { 2 } } { 2 }sin(2x)=−22​​

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​ .

Verify the identity shown below. sec⁡2(π2−y)−1=cot⁡2y\sec ^ { 2 } \left( \frac { \pi } { 2 } - y \right) - 1 = \cot ^ { 2 } ysec2(2π​−y)−1=cot2y

Write the given expression as the sine of an angle. sin⁡75∘cos⁡35∘−sin⁡35∘cos⁡75∘\sin 75 ^ { \circ } \cos 35 ^ { \circ } - \sin 35 ^ { \circ } \cos 75 ^ { \circ }sin75∘cos35∘−sin35∘cos75∘

Write the given expression as the sine of an angle. sin⁡85∘cos⁡50∘+sin⁡50∘cos⁡85∘\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }sin85∘cos50∘+sin50∘cos85∘

Verify the identity shown below. (1+cot⁡2θ)tan⁡2θ=sec⁡2θ\left( 1 + \cot ^ { 2 } \theta \right) \tan ^ { 2 } \theta = \sec ^ { 2 } \theta(1+cot2θ)tan2θ=sec2θ

Find the exact solutions of the given equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . sin⁡2x=sin⁡x\sin 2 x = \sin xsin2x=sinx

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

Verify the identity shown below. tan⁡α+cot⁡βtan⁡αcot⁡β=tan⁡β+cot⁡α\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alphatanαcotβtanα+cotβ​=tanβ+cotα

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∘

Functions and Their Graphs

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Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ . $2 \sin ^ { 2 } x + \sin x = 1$

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.)

If $x = 2 \cot \theta$ , use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \pi$ .

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ . (Both $u$ and $v$ are in Quadrant II.)

Which of the following is a solution to the given equation? $\sec x - 2 = 0$

Solve the following equation. $\csc ^ { 2 } ( x ) - 4 = 0$

Verify the identity shown below. $\frac { 1 - \sin \theta } { 1 + \sin \theta } = 2 \sec ^ { 2 } \theta - 2 \sec \theta \tan \theta - 1$

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\sin x + 1 + \frac { 1 } { \sin x - 1 }$

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

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

Verify the identity shown below. $\sec ^ { 2 } \left( \frac { \pi } { 2 } - y \right) - 1 = \cot ^ { 2 } y$

Write the given expression as the sine of an angle. $\sin 75 ^ { \circ } \cos 35 ^ { \circ } - \sin 35 ^ { \circ } \cos 75 ^ { \circ }$

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Verify the identity shown below. $\left( 1 + \cot ^ { 2 } \theta \right) \tan ^ { 2 } \theta = \sec ^ { 2 } \theta$

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ . $\sin 2 x = \sin x$

Verify the identity shown below. $\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alpha$

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