Question 1

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 2

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.

Accepted Answer

A) $\csc ^ { 3 } x - \csc ^ { 2 } x - \csc x + 1$
$\csc x \cot ^ { 2 } x - \cot ^ { 2 } x$ 
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) $\csc ^ { 3 } x - \csc ^ { 2 } x - \csc x + 1$
$\csc x \cot ^ { 2 } x - \cot ^ { 2 } x$ 
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 3

Which of the following is a solution to the given equation?
$\cot x + 1 = 0$

Accepted Answer

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

Question 4

Use the product-to-sum formulas to write the expression below as a sum or difference
$$\sin ( 8 \theta ) \cos ( 6 \theta )$$

Accepted Answer

A)  $\frac { 1 } { 2 } ( \cos ( 2 \theta ) + \cos ( 14 \theta ) )$ 
B)  $\frac { 1 } { 2 } ( \sin ( 2 \theta ) + \cos ( 14 \theta ) )$ 
C)  $\frac { 1 } { 2 } ( \sin ( 14 \theta ) + \sin ( 2 \theta ) )$ 
D)  $\frac { 1 } { 2 } ( \cos ( 2 \theta ) - \cos ( 14 \theta ) )$ 
E)  $\frac { 1 } { 2 } ( \sin ( 14 \theta ) - \sin ( 2 \theta ) )$ 
A)  $\frac { 1 } { 2 } ( \cos ( 2 \theta ) + \cos ( 14 \theta ) )$ 
B)  $\frac { 1 } { 2 } ( \sin ( 2 \theta ) + \cos ( 14 \theta ) )$ 
C)  $\frac { 1 } { 2 } ( \sin ( 14 \theta ) + \sin ( 2 \theta ) )$ 
D)  $\frac { 1 } { 2 } ( \cos ( 2 \theta ) - \cos ( 14 \theta ) )$ 
E)  $\frac { 1 } { 2 } ( \sin ( 14 \theta ) - \sin ( 2 \theta ) )$

Question 5

Use the graph below of the function to approximate the solutions to $3 \cos ( 2 x ) - \cos ( x ) = 0$ in the interval $[ 0,2 \pi )$. Round your answers to one decimal.

Accepted Answer

A)  0.7, 2.0, 1.5, 4.8 
B)  2.0, 1.5, 4.8, 5.6 
C)  2.0, 2.2, 4.0, 5.6 
D)  0.7, 2.2, 4.0, 5.6 
E)  0.7, 1.5, 4.0, 6.3 
A)  0.7, 2.0, 1.5, 4.8 
B)  2.0, 1.5, 4.8, 5.6 
C)  2.0, 2.2, 4.0, 5.6 
D)  0.7, 2.2, 4.0, 5.6 
E)  0.7, 1.5, 4.0, 6.3

Question 6

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 7

Use the graph below of the function to approximate the solutions to $4 \cos ( 2 x ) - \cos ( x ) = 0$ in the interval $[ 0,2 \pi )$. Round your answers to one decimal.

Accepted Answer

A)  0.7, 3.0, 1.5, 4.7 
B)  3.0, 1.5, 4.7, 5.6 
C)  3.0, 2.3, 4.0, 5.6 
D)  0.7, 2.3, 4.0, 5.6 
E)  0.7, 1.5, 4.0, 6.3 
A)  0.7, 3.0, 1.5, 4.7 
B)  3.0, 1.5, 4.7, 5.6 
C)  3.0, 2.3, 4.0, 5.6 
D)  0.7, 2.3, 4.0, 5.6 
E)  0.7, 1.5, 4.0, 6.3

Question 8

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 9

Use the figure below to determine the exact value of the given function. $\sin 2 \theta$

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 10

Use a double-angle formula to find the exact value of $\cos 2 u$ when $\sin u = \frac { 5 } { 13 }$, where $\frac { \pi } { 2 } < u < \pi$.

Accepted Answer

A)  $\cos 2 u = - \frac { 94 } { 169 }$ 
B)  $\quad \cos 2 u = \frac { 60 } { 169 }$ 
C)  $\cos 2 u = \frac { 120 } { 169 }$ 
D)  $\cos 2 u = \frac { 119 } { 169 }$ 
E)  $\cos 2 u = - \frac { 288 } { 169 }$ 
A)  $\cos 2 u = - \frac { 94 } { 169 }$ 
B)  $\quad \cos 2 u = \frac { 60 } { 169 }$ 
C)  $\cos 2 u = \frac { 120 } { 169 }$ 
D)  $\cos 2 u = \frac { 119 } { 169 }$ 
E)  $\cos 2 u = - \frac { 288 } { 169 }$

Question 11

Determine which of the following are trigonometric identities. 6 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 12

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 13

Use the figure below to determine the exact value of the given function.
$\sin 2 \theta$

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 14

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

Accepted Answer

\[\begin{aligned}
\sqrt { \frac { 1 + \s

Question 15

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)  $x = 1.231,2.214,4.069,5.052$

Question 16

Which of the following is a solution to the given equation?
$2 \cos x + \sqrt { 3 } = 0$

Accepted Answer

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

Question 17

Solve the following equation. $$3 \csc ^ { 2 } ( x ) - 4 = 0$$

Accepted Answer

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

Question 18

Solve the following trigonometric equation on the interval $[ 0,2 \pi )$.
$$\sin ( x ) + \cos ( x ) = 0$$

Accepted Answer

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

Question 19

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

Accepted Answer

\[\begin{aligned}
\tan ^ { 2 } \theta \s

Question 20

Which of the following is equivalent to the given expression?
$\frac { \cot ^ { 2 } x } { \csc x + 1 }$

Accepted Answer

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

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

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.

Which of the following is a solution to the given equation? $\cot x + 1 = 0$

Use the product-to-sum formulas to write the expression below as a sum or difference $\sin ( 8 \theta ) \cos ( 6 \theta )$

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

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

Use the figure below to determine the exact value of the given function. $\sin 2 \theta$ $Use the figure below to determine the exact value of the given function. \sin 2 \theta$

Use a double-angle formula to find the exact value of $\cos 2 u$ when $\sin u = \frac { 5 } { 13 }$ , where $\frac { \pi } { 2 } < u < \pi$ .

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. $\sin 2 \theta$ $Use the figure below to determine the exact value of the given function. \sin 2 \theta$

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

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

Which of the following is a solution to the given equation? $2 \cos x + \sqrt { 3 } = 0$

Solve the following equation. $3 \csc ^ { 2 } ( x ) - 4 = 0$

Solve the following trigonometric equation on the interval $[ 0,2 \pi )$ . $\sin ( x ) + \cos ( x ) = 0$

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

Which of the following is equivalent to the given expression? $\frac { \cot ^ { 2 } x } { \csc x + 1 }$

Functions and Their Graphs

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Trigonometric Functions

Additional Topics in Trigonometry

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Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

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Exam 5: Analytic Trigonometry

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∘

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.

Which of the following is a solution to the given equation? cot⁡x+1=0\cot x + 1 = 0cotx+1=0

Use the product-to-sum formulas to write the expression below as a sum or difference sin⁡(8θ)cos⁡(6θ)\sin ( 8 \theta ) \cos ( 6 \theta )sin(8θ)cos(6θ)

Use the graph below of the function to approximate the solutions to 3cos⁡(2x)−cos⁡(x)=03 \cos ( 2 x ) - \cos ( x ) = 03cos(2x)−cos(x)=0 in the interval [0,2π)[ 0,2 \pi )[0,2π) . Round your answers to one decimal.

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

Use the graph below of the function to approximate the solutions to 4cos⁡(2x)−cos⁡(x)=04 \cos ( 2 x ) - \cos ( x ) = 04cos(2x)−cos(x)=0 in the interval [0,2π)[ 0,2 \pi )[0,2π) . Round your answers to one decimal.

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​ .

Use the figure below to determine the exact value of the given function. sin⁡2θ\sin 2 \thetasin2θ

Use a double-angle formula to find the exact value of cos⁡2u\cos 2 ucos2u when sin⁡u=513\sin u = \frac { 5 } { 13 }sinu=135​ , where π2<u<π\frac { \pi } { 2 } < u < \pi2π​<u<π .

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. sin⁡2θ\sin 2 \thetasin2θ

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​

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

Which of the following is a solution to the given equation? 2cos⁡x+3=02 \cos x + \sqrt { 3 } = 02cosx+3​=0

Solve the following equation. 3csc⁡2(x)−4=03 \csc ^ { 2 } ( x ) - 4 = 03csc2(x)−4=0

Solve the following trigonometric equation on the interval [0,2π)[ 0,2 \pi )[0,2π) . sin⁡(x)+cos⁡(x)=0\sin ( x ) + \cos ( x ) = 0sin(x)+cos(x)=0

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

Which of the following is equivalent to the given expression? cot⁡2xcsc⁡x+1\frac { \cot ^ { 2 } x } { \csc x + 1 }cscx+1cot2x​

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

Review of Graphs, Equations, and Inequalities

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

Which of the following is a solution to the given equation? $\cot x + 1 = 0$

Use the product-to-sum formulas to write the expression below as a sum or difference $\sin ( 8 \theta ) \cos ( 6 \theta )$

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

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

Use the figure below to determine the exact value of the given function. $\sin 2 \theta$ $Use the figure below to determine the exact value of the given function. \sin 2 \theta$

Use a double-angle formula to find the exact value of $\cos 2 u$ when $\sin u = \frac { 5 } { 13 }$ , where $\frac { \pi } { 2 } < u < \pi$ .

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. $\sin 2 \theta$ $Use the figure below to determine the exact value of the given function. \sin 2 \theta$

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

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

Which of the following is a solution to the given equation? $2 \cos x + \sqrt { 3 } = 0$

Solve the following equation. $3 \csc ^ { 2 } ( x ) - 4 = 0$

Solve the following trigonometric equation on the interval $[ 0,2 \pi )$ . $\sin ( x ) + \cos ( x ) = 0$

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

Which of the following is equivalent to the given expression? $\frac { \cot ^ { 2 } x } { \csc x + 1 }$