Question 1

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 2

Find the exact value of the given expression using a sum or difference formula. $\sin 345 ^ { \circ }$

Accepted Answer

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

Question 3

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 4

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

Accepted Answer

\[\begin{array}{l}
\frac { 1 } { \sec \t

Question 5

Determine the exact value of the following expression.
$$\cos \left( 120 ^ { \circ } - 0 ^ { \circ } \right)$$

Accepted Answer

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

Question 6

If $x = 8 \cos \theta$, use trigonometric substitution to write $\sqrt { 64 - x ^ { 2 } }$ as a trigonometric function of $\theta$, where $0 < \theta < \pi$.

Accepted Answer

A)  $8 \sec \theta$ 
B)  $8 \sin \theta$ 
C)  $8 \cot \theta$ 
D)  $8 \cos \theta$ 
E)  $8 \csc \theta$ 
A)  $8 \sec \theta$ 
B)  $8 \sin \theta$ 
C)  $8 \cot \theta$ 
D)  $8 \cos \theta$ 
E)  $8 \csc \theta$

Question 7

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 8

If $\csc x = \frac { 4 \sqrt { 3 } } { 3 }$ and $\cos x < 0$, evaluate the function below. $\sec x$

Accepted Answer

A)  $\sec x = \frac { 4 } { \sqrt { 3 } }$ 
B)  $\sec x = - \frac { \sqrt { 3 } } { 4 }$ 
C)  $\sec x = - \frac { 4 \sqrt { 13 } } { 13 }$ 
D)  $\sec x = - \frac { \sqrt { 39 } } { 3 }$ 
E)  $\sec x = \frac { \sqrt { 39 } } { 3 }$ 
A)  $\sec x = \frac { 4 } { \sqrt { 3 } }$ 
B)  $\sec x = - \frac { \sqrt { 3 } } { 4 }$ 
C)  $\sec x = - \frac { 4 \sqrt { 13 } } { 13 }$ 
D)  $\sec x = - \frac { \sqrt { 39 } } { 3 }$ 
E)  $\sec x = \frac { \sqrt { 39 } } { 3 }$

Question 9

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 10

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 11

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 12

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 13

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 14

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)  $\quad \cos 2 u = \frac { 240 } { 289 }$ 
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)  $\quad \cos 2 u = \frac { 240 } { 289 }$ 
D)  $\cos 2 u = \frac { 161 } { 289 }$ 
E)  $\cos 2 u = - \frac { 450 } { 289 }$

Question 15

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)  $2 \sin \theta$

Question 16

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 17

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 18

Determine which of the following are trigonometric identities. I. $\sin ( \theta ) + \cot ( \theta ) \cos ( \theta ) = \csc ( \theta )$
II. $\cot ( \theta ) - \sin ( \theta ) \cos ( \theta ) = 0$
III. $\sin ( \theta ) + \sin ( \theta ) \cos ( \theta ) = \csc ( \theta )$

Accepted Answer

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

Question 19

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 is 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 is the only identity. 
E)  II, III, and IV are the only identities.

Question 20

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)  $x = \frac { 7 \pi } { 4 }$

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

Find the exact value of the given expression using a sum or difference formula. $\sin 345 ^ { \circ }$

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

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

Determine the exact value of the following expression. $\cos \left( 120 ^ { \circ } - 0 ^ { \circ } \right)$

If $x = 8 \cos \theta$ , use trigonometric substitution to write $\sqrt { 64 - x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \pi$ .

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

If $\csc x = \frac { 4 \sqrt { 3 } } { 3 }$ and $\cos x < 0$ , evaluate the function below. $\sec x$

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.)

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

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

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

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

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

Determine which of the following are trigonometric identities. I. $\sin ( \theta ) + \cot ( \theta ) \cos ( \theta ) = \csc ( \theta )$ II. $\cot ( \theta ) - \sin ( \theta ) \cos ( \theta ) = 0$ III. $\sin ( \theta ) + \sin ( \theta ) \cos ( \theta ) = \csc ( \theta )$

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. $\csc ( \theta ) \sin ( \theta ) = 1$

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

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

Filters

Exam 5: Analytic Trigonometry

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 the given expression using a sum or difference formula. sin⁡345∘\sin 345 ^ { \circ }sin345∘

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

Verify the identity shown below. 1sec⁡θ−tan⁡θ=sec⁡θ+tan⁡θ\frac { 1 } { \sec \theta - \tan \theta } = \sec \theta + \tan \thetasecθ−tanθ1​=secθ+tanθ

Determine the exact value of the following expression. cos⁡(120∘−0∘)\cos \left( 120 ^ { \circ } - 0 ^ { \circ } \right)cos(120∘−0∘)

If x=8cos⁡θx = 8 \cos \thetax=8cosθ , use trigonometric substitution to write 64−x2\sqrt { 64 - x ^ { 2 } }64−x2​ as a trigonometric function of θ\thetaθ , where 0<θ<π0 < \theta < \pi0<θ<π .

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

If csc⁡x=433\csc x = \frac { 4 \sqrt { 3 } } { 3 }cscx=343​​ and cos⁡x<0\cos x < 0cosx<0 , evaluate the function below. sec⁡x\sec xsecx

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.)

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 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 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 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 .

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 .

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

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

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

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

Filters

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

Find the exact value of the given expression using a sum or difference formula. $\sin 345 ^ { \circ }$

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

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

Determine the exact value of the following expression. $\cos \left( 120 ^ { \circ } - 0 ^ { \circ } \right)$

If $x = 8 \cos \theta$ , use trigonometric substitution to write $\sqrt { 64 - x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \pi$ .

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

If $\csc x = \frac { 4 \sqrt { 3 } } { 3 }$ and $\cos x < 0$ , evaluate the function below. $\sec x$

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.)

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

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

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

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

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

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