Question 1

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 2

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 3

Verify the given identity.
$$\frac { \sin u + \sin v } { \cos u + \cos v } = \tan \frac { 1 } { 2 } ( u + v )$$

Accepted Answer

\[\begin{aligned}
\frac { \sin u + \sin

Question 4

Which of the following is a solution to the given equation?
$2 \sin x - 1 = 0$

Accepted Answer

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

Question 5

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

Question 6

Use a double angle formula to rewrite the following expression.
$$- 16 \sin ( x ) \cos ( x )$$

Accepted Answer

A)  $\sin ( - 16 x )$ 
B)  $2 \sin ( - 8 x )$ 
C)  $- 8 \cos ( 2 x )$ 
D)  $- 8 \sin ( 2 x )$ 
E)  $\cos ( - 16 x )$ 
A)  $\sin ( - 16 x )$ 
B)  $2 \sin ( - 8 x )$ 
C)  $- 8 \cos ( 2 x )$ 
D)  $- 8 \sin ( 2 x )$ 
E)  $\cos ( - 16 x )$

Question 7

Find the exact value of $\cos ( u - v )$ given that $\sin u = - \frac { 9 } { 41 }$ and $\cos v = \frac { 15 } { 17 }$. (Both $u$ and $v$ are in Quadrant IV.)

Accepted Answer

A)  $\cos ( u - v ) = - \frac { 194 } { 697 }$ 
B)  $\cos ( u - v ) = - \frac { 39 } { 697 }$ 
C)  $\cos ( u - v ) = \frac { 568 } { 697 }$ 
D)  $\cos ( u - v ) = - \frac { 528 } { 697 }$ 
E)  $\cos ( u - v ) = \frac { 672 } { 697 }$ 
A)  $\cos ( u - v ) = - \frac { 194 } { 697 }$ 
B)  $\cos ( u - v ) = - \frac { 39 } { 697 }$ 
C)  $\cos ( u - v ) = \frac { 568 } { 697 }$ 
D)  $\cos ( u - v ) = - \frac { 528 } { 697 }$ 
E)  $\cos ( u - v ) = \frac { 672 } { 697 }$

Question 8

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 9

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 10

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 11

Use the graph of the function $f ( x ) = - 2 \cos ( x ) + \sin ( x )$ to approximate the maximum points of the graph in the interval $[ 0,2 \pi ]$. Round your answer to one decimal.

Accepted Answer

A)  $( 2.7,2.2 ) , ( 6.3 , - 1.8 )$ 
B)  $( - 1.8,6.3 ) , ( 2.7,2.2 )$ 
C)  $( 2.7 , - 1.8 ) , ( 6.3,2.2 )$ 
D)  $( - 1.8,6.3 ) , ( 2.2,2.7 )$ 
E)  $( 2.2,2.7 ) , ( 6.3 , - 1.8 )$ 
A)  $( 2.7,2.2 ) , ( 6.3 , - 1.8 )$ 
B)  $( - 1.8,6.3 ) , ( 2.7,2.2 )$ 
C)  $( 2.7 , - 1.8 ) , ( 6.3,2.2 )$ 
D)  $( - 1.8,6.3 ) , ( 2.2,2.7 )$ 
E)  $( 2.2,2.7 ) , ( 6.3 , - 1.8 )$

Question 12

Use the half-angle formulas to determine the exact value of the following.
$$\cos \left( 22.5 ^ { \circ } \right)$$

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

Simplify the given expression algebraically. $\cos ( \pi + x )$

Accepted Answer

A)  $\sin x$ 
B)  $- \sin x$ 
C)  $\cos x$ 
D)  $- \cos x$ 
E)  1 
A)  $\sin x$ 
B)  $- \sin x$ 
C)  $\cos x$ 
D)  $- \cos x$ 
E)  1

Question 16

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 17

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 18

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 19

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 20

Use a double-angle formula to find the exact value of $\cos 2 u$ when $\sin u = \frac { 8 } { 17 }$, where $\frac { \pi } { 2 } < u < \pi$.

Accepted Answer

A)  $\cos 2 u = - \frac { 97 } { 289 }$ 
B)  $\cos 2 u = \frac { 120 } { 289 }$ 
C)  $\cos 2 u = \frac { 240 } { 289 }$ 
D)  $\cos 2 u = \frac { 161 } { 289 }$ 
E)  $\cos 2 u = - \frac { 450 } { 289 }$ 
A)  $\cos 2 u = - \frac { 97 } { 289 }$ 
B)  $\cos 2 u = \frac { 120 } { 289 }$ 
C)  $\cos 2 u = \frac { 240 } { 289 }$ 
D)  $\cos 2 u = \frac { 161 } { 289 }$ 
E)  $\cos 2 u = - \frac { 450 } { 289 }$

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

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

Verify the given identity. $\frac { \sin u + \sin v } { \cos u + \cos v } = \tan \frac { 1 } { 2 } ( u + v )$

Which of the following is a solution to the given equation? $2 \sin x - 1 = 0$

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 a double angle formula to rewrite the following expression. $- 16 \sin ( x ) \cos ( x )$

Find the exact value of $\cos ( u - v )$ given that $\sin u = - \frac { 9 } { 41 }$ and $\cos v = \frac { 15 } { 17 }$ . (Both $u$ and $v$ are in Quadrant IV.)

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

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

Use the half-angle formulas to determine the exact value of the following. $\cos \left( 22.5 ^ { \circ } \right)$

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

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

Simplify the given expression algebraically. $\cos ( \pi + x )$

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

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

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

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

Use a double-angle formula to find the exact value of $\cos 2 u$ when $\sin u = \frac { 8 } { 17 }$ , where $\frac { \pi } { 2 } < u < \pi$ .

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

Calculus Practice Problems

Filters

Exam 5: Analytic Trigonometry

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

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​

Verify the given identity. sin⁡u+sin⁡vcos⁡u+cos⁡v=tan⁡12(u+v)\frac { \sin u + \sin v } { \cos u + \cos v } = \tan \frac { 1 } { 2 } ( u + v )cosu+cosvsinu+sinv​=tan21​(u+v)

Which of the following is a solution to the given equation? 2sin⁡x−1=02 \sin x - 1 = 02sinx−1=0

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 a double angle formula to rewrite the following expression. −16sin⁡(x)cos⁡(x)- 16 \sin ( x ) \cos ( x )−16sin(x)cos(x)

Find the exact value of cos⁡(u−v)\cos ( u - v )cos(u−v) given that sin⁡u=−941\sin u = - \frac { 9 } { 41 }sinu=−419​ and cos⁡v=1517\cos v = \frac { 15 } { 17 }cosv=1715​ . (Both uuu and vvv are in Quadrant IV.)

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.

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

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

Use the graph of the function f(x)=−2cos⁡(x)+sin⁡(x)f ( x ) = - 2 \cos ( x ) + \sin ( x )f(x)=−2cos(x)+sin(x) to approximate the maximum points of the graph in the interval [0,2π][ 0,2 \pi ][0,2π] . Round your answer to one decimal.

Use the half-angle formulas to determine the exact value of the following. cos⁡(22.5∘)\cos \left( 22.5 ^ { \circ } \right)cos(22.5∘)

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. tan⁡3x−tan⁡2x+tan⁡x−1\tan ^ { 3 } x - \tan ^ { 2 } x + \tan x - 1tan3x−tan2x+tanx−1

Simplify the given expression algebraically. cos⁡(π+x)\cos ( \pi + x )cos(π+x)

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

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

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

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

Use a double-angle formula to find the exact value of cos⁡2u\cos 2 ucos2u when sin⁡u=817\sin u = \frac { 8 } { 17 }sinu=178​ , where π2<u<π\frac { \pi } { 2 } < u < \pi2π​<u<π .

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

Calculus Practice Problems

Filters

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

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

Verify the given identity. $\frac { \sin u + \sin v } { \cos u + \cos v } = \tan \frac { 1 } { 2 } ( u + v )$

Which of the following is a solution to the given equation? $2 \sin x - 1 = 0$

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 a double angle formula to rewrite the following expression. $- 16 \sin ( x ) \cos ( x )$

Find the exact value of $\cos ( u - v )$ given that $\sin u = - \frac { 9 } { 41 }$ and $\cos v = \frac { 15 } { 17 }$ . (Both $u$ and $v$ are in Quadrant IV.)

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

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

Use the half-angle formulas to determine the exact value of the following. $\cos \left( 22.5 ^ { \circ } \right)$

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

Simplify the given expression algebraically. $\cos ( \pi + x )$

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

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

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

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

Use a double-angle formula to find the exact value of $\cos 2 u$ when $\sin u = \frac { 8 } { 17 }$ , where $\frac { \pi } { 2 } < u < \pi$ .