Question 1

Complete the identity.
-$\cos \left( x + \frac { \pi } { 2 } \right) =$ ?

Accepted Answer

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

Question 2

Verify the identity.
-$$\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \theta$$

Accepted Answer

The answer of Verify the identity.
-\[\sin \left( \frac { 3...

Question 3

Use the figure to find the exact value of the trigonometric function.
-$\cos \theta = \frac { 4 } { 5 } , \theta$ lies in quadrant IV Find $\sin 2 \theta$.

Accepted Answer

A)  $- \frac { 24 } { 25 }$ 
B)  $- \frac { 7 } { 25 }$ 
C)  $\frac { 7 } { 25 }$ 
D)  $\frac { 24 } { 25 }$ 
A)  $- \frac { 24 } { 25 }$ 
B)  $- \frac { 7 } { 25 }$ 
C)  $\frac { 7 } { 25 }$ 
D)  $\frac { 24 } { 25 }$

Question 4

Express the sum or difference as a product.
-$\sin 9 x - \sin 3 x$

Accepted Answer

A)  $2 \sin 3 x \cos 6 x$ 
B)  $2 \cos 3 x \cos 6 x$ 
C)  $2 \sin 6 x \cos 3 x$ 
D)  $2 \sin 3 x$ 
A)  $2 \sin 3 x \cos 6 x$ 
B)  $2 \cos 3 x \cos 6 x$ 
C)  $2 \sin 6 x \cos 3 x$ 
D)  $2 \sin 3 x$

Question 5

Use a graph in a [ $$\left[ - 2 \pi , 2 \pi , \frac { \pi } { 2 } \right] \text { by } [ - 3,3,1 ]$$ 3, 3, 1] viewing rectangle to complete the identity.
-$\frac { 1 - 2 \cos 2 x } { 2 \sin x - 1 } =$ ?

Accepted Answer

A)  $2 \sin x + 1$ 
B)  $2 \cos x + 1$ 
C)  $\sin x + 1$ 
D)  $\cos x - 1$ 
A)  $2 \sin x + 1$ 
B)  $2 \cos x + 1$ 
C)  $\sin x + 1$ 
D)  $\cos x - 1$

Question 6

Complete the identity.
-$\frac { ( \tan x + 1 ) ( \tan x + 1 ) - \sec ^ { 2 } x } { \tan x } =$ ?

Accepted Answer

A)  2 
B)  1 
C)  0 
D)  $\tan x$ 
A)  2 
B)  1 
C)  0 
D)  $\tan x$

Question 7

Use the given information to find the exact value of the trigonometric function.
-$\sec \theta = - \frac { 25 } { 24 } , \theta$ lies in quadrant II Find $\sin \frac { \theta } { 2 }$.

Accepted Answer

A)  $\frac { 7 \sqrt { 2 } } { 10 }$ 
B)  $- \frac { \sqrt { 2 } } { 10 }$ 
C)  $- \frac { 7 } { 50 }$ 
D)  $\frac { \sqrt { 2 } } { 10 }$ 
A)  $\frac { 7 \sqrt { 2 } } { 10 }$ 
B)  $- \frac { \sqrt { 2 } } { 10 }$ 
C)  $- \frac { 7 } { 50 }$ 
D)  $\frac { \sqrt { 2 } } { 10 }$

Question 8

Use The Half-Angle Formulas
-$\cos \frac { 3 \pi } { 8 }$

Accepted Answer

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

Question 9

Complete the identity.
-$$\frac { 2 \cot x } { 1 + \cot ^ { 2 } x } = ?$$

Accepted Answer

A)  $\sin 2 x$ 
B)  $\cos 2 x$ 
C)  $\cot 2 x$ 
D)  $\tan 2 x$ 
A)  $\sin 2 x$ 
B)  $\cos 2 x$ 
C)  $\cot 2 x$ 
D)  $\tan 2 x$

Question 10

Use The Half-Angle Formulas
-$\cos \frac { 5 \pi } { 12 }$

Accepted Answer

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

Question 11

Complete the identity.
-$\cot \frac { \theta } { 2 } = ?$

Accepted Answer

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

Question 12

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.
-$$\cos ^ { 2 } 60 ^ { \circ } - \sin ^ { 2 } 60 ^ { \circ }$$

Accepted Answer

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

Question 13

Find the exact value under the given conditions.
-$\sin \left( \frac { - 5 \pi } { 6 } - \alpha \right) \cos \left( \frac { - 5 \pi } { 6 } + \alpha \right) + \cos \left( \frac { - 5 \pi } { 6 } - \alpha \right) \sin \left( \frac { - 5 \pi } { 6 } + \alpha \right)$

Accepted Answer

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

Question 14

Use the Formula for the Cosine of the Difference of Two Angles
-$\cos \left( 160 ^ { \circ } \right) \cos \left( 40 ^ { \circ } \right) + \sin \left( 160 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right)$

Accepted Answer

A)  $\cos \left( 120 ^ { \circ } \right)$ 
B)  $\cos \left( 210 ^ { \circ } \right)$ 
C)  $\cos \left( 190 ^ { \circ } \right)$ 
D)  $\cos \left( 220 ^ { \circ } \right)$ 
A)  $\cos \left( 120 ^ { \circ } \right)$ 
B)  $\cos \left( 210 ^ { \circ } \right)$ 
C)  $\cos \left( 190 ^ { \circ } \right)$ 
D)  $\cos \left( 220 ^ { \circ } \right)$

Question 15

Complete the identity.
-$\sin 2 x - \cot x =$ ?

Accepted Answer

A)  $- \cot x \cos 2 x$ 
B)  $\cot x \cos 2 x$ 
C)  $- \tan x \cos 2 x$ 
D)  $\cot x \cos x$ 
A)  $- \cot x \cos 2 x$ 
B)  $\cot x \cos 2 x$ 
C)  $- \tan x \cos 2 x$ 
D)  $\cot x \cos x$

Question 16

Verify the identity.
-$$\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta } = \cot \beta - \tan \alpha$$

Accepted Answer

The answer of Verify the identity.
-\[\frac { \cos ( \alpha...

Question 17

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places.
-$\sin x = - 0.37$

Accepted Answer

A)  $3.52,5.90$ 
B)  $0.38,5.90$ 
C)  $0.38,3.52$ 
D)  $0.38,1.95$ 
A)  $3.52,5.90$ 
B)  $0.38,5.90$ 
C)  $0.38,3.52$ 
D)  $0.38,1.95$

Question 18

Use The Half-Angle Formulas
-$$\sin \frac { 5 \pi } { 12 }$$

Accepted Answer

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

Question 19

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
-Graph in a $\left[ - \pi , \pi , \frac { \pi } { 2 } \right]$ by $[ - 3,3,1 ]$ viewing rectangle $y = 2 \left( \frac { \sin x } { 1 } - \frac { \sin 2 x } { 2 } + \frac { \sin 3 x } { 3 } - \frac { \sin 4 x } { 4 } + \frac { \sin 5 x } { 5 } - \frac { \sin 6 x } { 6 } + \frac { \sin 7 x } { 7 } \right)$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 20

Express the product as a sum or difference.
-$\sin 3 x \sin 7 x$

Accepted Answer

A)  $\frac { 1 } { 2 } ( \cos 4 x - \cos 10 x )$ 
B)  $\frac { 1 } { 2 } ( \cos 10 x - \sin 4 x )$ 
C)  $\frac { 1 } { 2 } ( - \cos 4 x - \cos 10 x )$ 
D)  $\sin ^ { 2 } 21 x ^ { 2 }$ 
A)  $\frac { 1 } { 2 } ( \cos 4 x - \cos 10 x )$ 
B)  $\frac { 1 } { 2 } ( \cos 10 x - \sin 4 x )$ 
C)  $\frac { 1 } { 2 } ( - \cos 4 x - \cos 10 x )$ 
D)  $\sin ^ { 2 } 21 x ^ { 2 }$

Complete the identity. - $\cos \left( x + \frac { \pi } { 2 } \right) =$ ?

Verify the identity. - $\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \theta$

Use the figure to find the exact value of the trigonometric function. - $\cos \theta = \frac { 4 } { 5 } , \theta$ lies in quadrant IV Find $\sin 2 \theta$ .

Express the sum or difference as a product. - $\sin 9 x - \sin 3 x$

Use a graph in a [ $\left[ - 2 \pi , 2 \pi , \frac { \pi } { 2 } \right] \text { by } [ - 3,3,1 ]$ 3, 3, 1] viewing rectangle to complete the identity. - $\frac { 1 - 2 \cos 2 x } { 2 \sin x - 1 } =$ ?

Complete the identity. - $\frac { ( \tan x + 1 ) ( \tan x + 1 ) - \sec ^ { 2 } x } { \tan x } =$ ?

Use the given information to find the exact value of the trigonometric function. - $\sec \theta = - \frac { 25 } { 24 } , \theta$ lies in quadrant II Find $\sin \frac { \theta } { 2 }$ .

Use The Half-Angle Formulas - $\cos \frac { 3 \pi } { 8 }$

Complete the identity. - $\frac { 2 \cot x } { 1 + \cot ^ { 2 } x } = ?$

Use The Half-Angle Formulas - $\cos \frac { 5 \pi } { 12 }$

Complete the identity. - $\cot \frac { \theta } { 2 } = ?$

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. - $\cos ^ { 2 } 60 ^ { \circ } - \sin ^ { 2 } 60 ^ { \circ }$

Find the exact value under the given conditions. - $\sin \left( \frac { - 5 \pi } { 6 } - \alpha \right) \cos \left( \frac { - 5 \pi } { 6 } + \alpha \right) + \cos \left( \frac { - 5 \pi } { 6 } - \alpha \right) \sin \left( \frac { - 5 \pi } { 6 } + \alpha \right)$

Use the Formula for the Cosine of the Difference of Two Angles - $\cos \left( 160 ^ { \circ } \right) \cos \left( 40 ^ { \circ } \right) + \sin \left( 160 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right)$

Complete the identity. - $\sin 2 x - \cot x =$ ?

Verify the identity. - $\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta } = \cot \beta - \tan \alpha$

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - $\sin x = - 0.37$

Use The Half-Angle Formulas - $\sin \frac { 5 \pi } { 12 }$

Express the product as a sum or difference. - $\sin 3 x \sin 7 x$

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Exam 6: Analytic Trigonometry

Complete the identity. - cos⁡(x+π2)=\cos \left( x + \frac { \pi } { 2 } \right) =cos(x+2π​)= ?

Verify the identity. - sin⁡(3π2−θ)=−cos⁡θ\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \thetasin(23π​−θ)=−cosθ

Use the figure to find the exact value of the trigonometric function. - cos⁡θ=45,θ\cos \theta = \frac { 4 } { 5 } , \thetacosθ=54​,θ lies in quadrant IV Find sin⁡2θ\sin 2 \thetasin2θ .

Express the sum or difference as a product. - sin⁡9x−sin⁡3x\sin 9 x - \sin 3 xsin9x−sin3x

Complete the identity. - (tan⁡x+1)(tan⁡x+1)−sec⁡2xtan⁡x=\frac { ( \tan x + 1 ) ( \tan x + 1 ) - \sec ^ { 2 } x } { \tan x } =tanx(tanx+1)(tanx+1)−sec2x​= ?

Use the given information to find the exact value of the trigonometric function. - sec⁡θ=−2524,θ\sec \theta = - \frac { 25 } { 24 } , \thetasecθ=−2425​,θ lies in quadrant II Find sin⁡θ2\sin \frac { \theta } { 2 }sin2θ​ .

Use The Half-Angle Formulas - cos⁡3π8\cos \frac { 3 \pi } { 8 }cos83π​

Complete the identity. - 2cot⁡x1+cot⁡2x=?\frac { 2 \cot x } { 1 + \cot ^ { 2 } x } = ?1+cot2x2cotx​=?

Use The Half-Angle Formulas - cos⁡5π12\cos \frac { 5 \pi } { 12 }cos125π​

Complete the identity. - cot⁡θ2=?\cot \frac { \theta } { 2 } = ?cot2θ​=?

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. - cos⁡260∘−sin⁡260∘\cos ^ { 2 } 60 ^ { \circ } - \sin ^ { 2 } 60 ^ { \circ }cos260∘−sin260∘

Complete the identity. - sin⁡2x−cot⁡x=\sin 2 x - \cot x =sin2x−cotx= ?

Verify the identity. - cos⁡(α+β)cos⁡αsin⁡β=cot⁡β−tan⁡α\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta } = \cot \beta - \tan \alphacosαsinβcos(α+β)​=cotβ−tanα

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - sin⁡x=−0.37\sin x = - 0.37sinx=−0.37

Use The Half-Angle Formulas - sin⁡5π12\sin \frac { 5 \pi } { 12 }sin125π​

Express the product as a sum or difference. - sin⁡3xsin⁡7x\sin 3 x \sin 7 xsin3xsin7x