Question 1

If x =10sin?, use trigonometric substitution to write $\sqrt{100-x^{2}}$ as a trigonometric function of ?, where $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$

Accepted Answer

A)  10sin? 
B)  10cos? 
C)  10 tan? 
D)  10csc? 
E)  10sec? 
A)  10sin? 
B)  10cos? 
C)  10 tan? 
D)  10csc? 
E)  10sec? B

Question 2

Use a graphing utility to approximate the solutions (to three decimal places) of the given equation in the interval $[ 0,2 \pi )$.
$$( \cos x ) ( 15 \cos x + 4 ) - 3 = 0$$

Accepted Answer

A)  $x = 0.484,1.799,3.626,4.490$ 
B)  $x = 0.624,1.932,3.776,4.435$ 
C)  $x = 1.055,2.078,3.896,4.721$ 
D)  $x = 1.101,2.118,3.982,5.104$ 
E)  $x = 1.231,2.214,4.069,5.052$ 
A)  $x = 0.484,1.799,3.626,4.490$ 
B)  $x = 0.624,1.932,3.776,4.435$ 
C)  $x = 1.055,2.078,3.896,4.721$ 
D)  $x = 1.101,2.118,3.982,5.104$ 
E)  $x = 1.231,2.214,4.069,5.052$ E

Question 3

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

Question 4

Solve the multi-angle equation below.
$$\sin ( 2 x ) = \frac { \sqrt { 3 } } { 2 }$$

Accepted Answer

A)  $\frac { \pi } { 8 } + n \pi , \frac { \pi } { 4 } + n \pi$, where $n$ is an integer 
B)  $\frac { \pi } { 6 } + n \pi , \frac { \pi } { 4 } + n \pi$, where $n$ is an integer 
C)  $\frac { \pi } { 6 } + 2 n \pi , \frac { \pi } { 3 } + 2 n \pi$, where $n$ is an integer 
D)  $\frac { \pi } { 8 } + 2 n \pi , \frac { \pi } { 4 } + 2 n \pi$, where $n$ is an integer 
E)  $\frac { \pi } { 6 } + n \pi , \frac { \pi } { 3 } + n \pi$, where $n$ is an integer 
A)  $\frac { \pi } { 8 } + n \pi , \frac { \pi } { 4 } + n \pi$, where $n$ is an integer 
B)  $\frac { \pi } { 6 } + n \pi , \frac { \pi } { 4 } + n \pi$, where $n$ is an integer 
C)  $\frac { \pi } { 6 } + 2 n \pi , \frac { \pi } { 3 } + 2 n \pi$, where $n$ is an integer 
D)  $\frac { \pi } { 8 } + 2 n \pi , \frac { \pi } { 4 } + 2 n \pi$, where $n$ is an integer 
E)  $\frac { \pi } { 6 } + n \pi , \frac { \pi } { 3 } + n \pi$, where $n$ is an integer

Question 5

Use the cofunction identities to evaluate the expression below without the aid of a calculator. $\cos ^ { 2 } 50 ^ { \circ } + \cos ^ { 2 } 58 ^ { \circ } + \cos ^ { 2 } 40 ^ { \circ } + \cos ^ { 2 } 32 ^ { \circ }$

Accepted Answer

A)  1 
B)  2 
C)  $- 1$ 
D)  0 
E)  $\frac { 1 } { 2 }$ 
A)  1 
B)  2 
C)  $- 1$ 
D)  0 
E)  $\frac { 1 } { 2 }$

Question 6

Solve the following equation.
$\tan x - \sqrt { 3 } = 0$

Accepted Answer

A)  $x = \frac { \pi } { 6 } + n \pi$ and $x = \frac { 5 \pi } { 6 } + n \pi$, where $n$ is an integer 
B)  $x = \frac { \pi } { 3 } + n \pi$, where $n$ is an integer 
C)  $x = \frac { \pi } { 4 } + n \pi$, where $n$ is an integer 
D)  $x = \frac { \pi } { 6 } + n \pi$, where $n$ is an integer 
E)  $x = \frac { 2 \pi } { 3 } + n \pi$ and $x = \frac { 4 \pi } { 3 } + n \pi$, where $n$ is an integer 
A)  $x = \frac { \pi } { 6 } + n \pi$ and $x = \frac { 5 \pi } { 6 } + n \pi$, where $n$ is an integer 
B)  $x = \frac { \pi } { 3 } + n \pi$, where $n$ is an integer 
C)  $x = \frac { \pi } { 4 } + n \pi$, where $n$ is an integer 
D)  $x = \frac { \pi } { 6 } + n \pi$, where $n$ is an integer 
E)  $x = \frac { 2 \pi } { 3 } + n \pi$ and $x = \frac { 4 \pi } { 3 } + n \pi$, where $n$ is an integer

Question 7

Find all solutions of the following equation on the interval $[ 0,2 \pi )$. $\cot ( x ) + \sqrt { 3 } = 0$

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

Determine which of the following are trigonometric identities.
I. $\frac { \cos ( 4 x ) + \cos ( 2 x ) } { 2 \cot ( 3 x ) } = \cot ( x )$
II. $\frac { \cos ( 4 x ) + \cos ( x ) } { \sin ( 3 x ) - \sin ( x ) } = \cot ( 2 x )$
III. $\frac { \cos ( 6 x ) + \cos ( 2 x ) } { \sin ( 4 x ) + \sin ( 2 x ) } = \cot ( 3 x )$

Accepted Answer

A)  III is the only identity. 
B)  I and III are the only identities. 
C)  None are identities. 
D)  II is the only identity. 
E)  I and II are the only identities. 
A)  III is the only identity. 
B)  I and III are the only identities. 
C)  None are identities. 
D)  II is the only identity. 
E)  I and II are the only identities.

Question 10

Use the figure below to find the exact value of the given trigonometric expression.
$\cot \frac { x } { 2 }$

Accepted Answer

A)  $\cot \frac { x } { 2 } = \frac { \sqrt { 2 } } { 10 }$ 
B)  $\cot \frac { x } { 2 } = \frac { 1 } { 7 }$ 
C)  $\cot \frac { x } { 2 } = 7$ 
D)  $\cot \frac { x } { 2 } = \frac { 7 \sqrt { 2 } } { 10 }$ 
E)  $\cot \frac { x } { 2 } = \frac { 7 } { 12 }$ 
A)  $\cot \frac { x } { 2 } = \frac { \sqrt { 2 } } { 10 }$ 
B)  $\cot \frac { x } { 2 } = \frac { 1 } { 7 }$ 
C)  $\cot \frac { x } { 2 } = 7$ 
D)  $\cot \frac { x } { 2 } = \frac { 7 \sqrt { 2 } } { 10 }$ 
E)  $\cot \frac { x } { 2 } = \frac { 7 } { 12 }$

Question 11

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)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$

Question 12

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.
$$\csc ^ { 3 } x - \csc ^ { 2 } x - \csc x + 1$$

Accepted Answer

A)  $\csc x \cot ^ { 2 } x - \cot ^ { 2 } x$ 
B)  $\cos ^ { 2 } x \left( \frac { 1 - \sin x } { \sin ^ { 3 } x } \right)$ 
C)  $\frac { \cos ^ { 4 } x } { \sin ^ { 3 } x + \sin ^ { 4 } x }$ 
D)  $\frac { 1 - \sin x - \sin ^ { 2 } x + \sin ^ { 3 } x } { \sin ^ { 3 } x }$ 
E)  $\frac { \sec ^ { 2 } x + \tan ^ { 2 } x } { \csc x }$ 
A)  $\csc x \cot ^ { 2 } x - \cot ^ { 2 } x$ 
B)  $\cos ^ { 2 } x \left( \frac { 1 - \sin x } { \sin ^ { 3 } x } \right)$ 
C)  $\frac { \cos ^ { 4 } x } { \sin ^ { 3 } x + \sin ^ { 4 } x }$ 
D)  $\frac { 1 - \sin x - \sin ^ { 2 } x + \sin ^ { 3 } x } { \sin ^ { 3 } x }$ 
E)  $\frac { \sec ^ { 2 } x + \tan ^ { 2 } x } { \csc x }$

Question 13

Find all solutions of the following equation on the interval $[ 0,2 \pi )$.
$$\csc ^ { 2 } ( x ) - 2 = 0$$

Accepted Answer

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

Question 14

If $x = 6 \sin \theta$, use trigonometric substitution to write $\sqrt { 36 - x ^ { 2 } }$ as a trigonometric function of $\theta$, where $0 < \theta < \frac { \pi } { 2 }$.

Accepted Answer

A)  $6 \csc \theta$ 
B)  $6 \sec \theta$ 
C)  $6 \sin \theta$ 
D)  $6 \cos \theta$ 
E)  $6 \tan \theta$ 
A)  $6 \csc \theta$ 
B)  $6 \sec \theta$ 
C)  $6 \sin \theta$ 
D)  $6 \cos \theta$ 
E)  $6 \tan \theta$

Question 15

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 16

Verify the identity shown below.
$\sqrt { \frac { 1 + \sin \theta } { 1 - \sin \theta } } = \frac { 1 + \sin \theta } { | \cos \theta | }$

Accepted Answer

None

Question 17

Verify the identity shown below.
$$\tan ^ { 2 } \theta \sec ^ { 2 } \theta + \sec ^ { 2 } \theta = \sec ^ { 4 } \theta$$

Accepted Answer

None

Question 18

Determine which of the following are trigonometric identities.
I. $\cot ( \theta ) \csc ( \theta ) = \sec ( \theta )$
II. $\cot ( \theta ) \sec ( \theta ) = \csc ( \theta )$
III. $\sec ( \theta ) \csc ( \theta ) = \cot ( \theta )$
IV. $\cot ( \theta ) \sin ( \theta ) = 1$

Accepted Answer

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

Question 19

Use the sum-to-product formulas to find the exact value of the given expression. $\cos 150 ^ { \circ } - \cos 30 ^ { \circ }$

Accepted Answer

A)  1 
B)  $- 1$ 
C)  $- \sqrt { 3 }$ 
D)  $- \frac { \sqrt { 3 } } { 2 }$ 
E)  0 
A)  1 
B)  $- 1$ 
C)  $- \sqrt { 3 }$ 
D)  $- \frac { \sqrt { 3 } } { 2 }$ 
E)  0

Question 20

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

Accepted Answer

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

If x =10sin?, use trigonometric substitution to write $\sqrt{100-x^{2}}$ as a trigonometric function of ?, where $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$

Use a graphing utility to approximate the solutions (to three decimal places) of the given equation in the interval $[ 0,2 \pi )$ . $( \cos x ) ( 15 \cos x + 4 ) - 3 = 0$

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.)

Solve the multi-angle equation below. $\sin ( 2 x ) = \frac { \sqrt { 3 } } { 2 }$

Use the cofunction identities to evaluate the expression below without the aid of a calculator. $\cos ^ { 2 } 50 ^ { \circ } + \cos ^ { 2 } 58 ^ { \circ } + \cos ^ { 2 } 40 ^ { \circ } + \cos ^ { 2 } 32 ^ { \circ }$

Solve the following equation. $\tan x - \sqrt { 3 } = 0$

Find all solutions of the following equation on the interval $[ 0,2 \pi )$ . $\cot ( x ) + \sqrt { 3 } = 0$

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

Use the figure below to find the exact value of the given trigonometric expression. $\cot \frac { x } { 2 }$ $Use the figure below to find the exact value of the given trigonometric expression. \cot \frac { x } { 2 }$

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

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\csc ^ { 3 } x - \csc ^ { 2 } x - \csc x + 1$

Find all solutions of the following equation on the interval $[ 0,2 \pi )$ . $\csc ^ { 2 } ( x ) - 2 = 0$

If $x = 6 \sin \theta$ , use trigonometric substitution to write $\sqrt { 36 - x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ .

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

Verify the identity shown below. $\sqrt { \frac { 1 + \sin \theta } { 1 - \sin \theta } } = \frac { 1 + \sin \theta } { | \cos \theta | }$

Verify the identity shown below. $\tan ^ { 2 } \theta \sec ^ { 2 } \theta + \sec ^ { 2 } \theta = \sec ^ { 4 } \theta$

Determine which of the following are trigonometric identities. I. $\cot ( \theta ) \csc ( \theta ) = \sec ( \theta )$ II. $\cot ( \theta ) \sec ( \theta ) = \csc ( \theta )$ III. $\sec ( \theta ) \csc ( \theta ) = \cot ( \theta )$ IV. $\cot ( \theta ) \sin ( \theta ) = 1$

Use the sum-to-product formulas to find the exact value of the given expression. $\cos 150 ^ { \circ } - \cos 30 ^ { \circ }$

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

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

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Exam 5: Analytic Trigonometry

If x =10sin?, use trigonometric substitution to write 100−x2\sqrt{100-x^{2}}100−x2​ as a trigonometric function of ?, where −π2<θ<π2-\frac{\pi}{2}<\theta<\frac{\pi}{2}−2π​<θ<2π​

Use a graphing utility to approximate the solutions (to three decimal places) of the given equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . (cos⁡x)(15cos⁡x+4)−3=0( \cos x ) ( 15 \cos x + 4 ) - 3 = 0(cosx)(15cosx+4)−3=0

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.)

Solve the multi-angle equation below. sin⁡(2x)=32\sin ( 2 x ) = \frac { \sqrt { 3 } } { 2 }sin(2x)=23​​

Solve the following equation. tan⁡x−3=0\tan x - \sqrt { 3 } = 0tanx−3​=0

Find all solutions of the following equation on the interval [0,2π)[ 0,2 \pi )[0,2π) . cot⁡(x)+3=0\cot ( x ) + \sqrt { 3 } = 0cot(x)+3​=0

Find the exact solutions of the given equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . sin⁡xcos⁡2x+cos⁡xsin⁡2x=0\sin x \cos 2 x + \cos x \sin 2 x = 0sinxcos2x+cosxsin2x=0

Use the figure below to find the exact value of the given trigonometric expression. cot⁡x2\cot \frac { x } { 2 }cot2x​

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

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. csc⁡3x−csc⁡2x−csc⁡x+1\csc ^ { 3 } x - \csc ^ { 2 } x - \csc x + 1csc3x−csc2x−cscx+1

Find all solutions of the following equation on the interval [0,2π)[ 0,2 \pi )[0,2π) . csc⁡2(x)−2=0\csc ^ { 2 } ( x ) - 2 = 0csc2(x)−2=0

If x=6sin⁡θx = 6 \sin \thetax=6sinθ , use trigonometric substitution to write 36−x2\sqrt { 36 - x ^ { 2 } }36−x2​ as a trigonometric function of θ\thetaθ , where 0<θ<π20 < \theta < \frac { \pi } { 2 }0<θ<2π​ .

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=1+sin⁡θ∣cos⁡θ∣\sqrt { \frac { 1 + \sin \theta } { 1 - \sin \theta } } = \frac { 1 + \sin \theta } { | \cos \theta | }1−sinθ1+sinθ​​=∣cosθ∣1+sinθ​

Verify the identity shown below. tan⁡2θsec⁡2θ+sec⁡2θ=sec⁡4θ\tan ^ { 2 } \theta \sec ^ { 2 } \theta + \sec ^ { 2 } \theta = \sec ^ { 4 } \thetatan2θsec2θ+sec2θ=sec4θ

Use the sum-to-product formulas to find the exact value of the given expression. cos⁡150∘−cos⁡30∘\cos 150 ^ { \circ } - \cos 30 ^ { \circ }cos150∘−cos30∘

Use the product-to-sum formula to write the given product as a sum or difference. 8sin⁡π8sin⁡π88 \sin \frac { \pi } { 8 } \sin \frac { \pi } { 8 }8sin8π​sin8π​

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

If x =10sin?, use trigonometric substitution to write $\sqrt{100-x^{2}}$ as a trigonometric function of ?, where $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$

Use a graphing utility to approximate the solutions (to three decimal places) of the given equation in the interval $[ 0,2 \pi )$ . $( \cos x ) ( 15 \cos x + 4 ) - 3 = 0$

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.)

Solve the multi-angle equation below. $\sin ( 2 x ) = \frac { \sqrt { 3 } } { 2 }$

Solve the following equation. $\tan x - \sqrt { 3 } = 0$

Find all solutions of the following equation on the interval $[ 0,2 \pi )$ . $\cot ( x ) + \sqrt { 3 } = 0$

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

Use the figure below to find the exact value of the given trigonometric expression. $\cot \frac { x } { 2 }$ $Use the figure below to find the exact value of the given trigonometric expression. \cot \frac { x } { 2 }$

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

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\csc ^ { 3 } x - \csc ^ { 2 } x - \csc x + 1$

Find all solutions of the following equation on the interval $[ 0,2 \pi )$ . $\csc ^ { 2 } ( x ) - 2 = 0$

If $x = 6 \sin \theta$ , use trigonometric substitution to write $\sqrt { 36 - x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ .

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

Verify the identity shown below. $\sqrt { \frac { 1 + \sin \theta } { 1 - \sin \theta } } = \frac { 1 + \sin \theta } { | \cos \theta | }$

Verify the identity shown below. $\tan ^ { 2 } \theta \sec ^ { 2 } \theta + \sec ^ { 2 } \theta = \sec ^ { 4 } \theta$

Use the sum-to-product formulas to find the exact value of the given expression. $\cos 150 ^ { \circ } - \cos 30 ^ { \circ }$

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