Question 1

Complete the identity
-$\tan ( \pi - \theta ) =$ ?

Accepted Answer

A)  $- \tan \theta$ 
B)  $\tan \theta$ 
C)  $- \cot \theta$ 
D)  $\cot \theta$ 
A)  $- \tan \theta$ 
B)  $\tan \theta$ 
C)  $- \cot \theta$ 
D)  $\cot \theta$

Question 2

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places.
-$\cos x = - 0.55$

Accepted Answer

A)  $2.15,4.13$ 
B)  $2.15,5.29$ 
C)  $0.99,4.13$ 
D)  $0.99,2.15$ Use a calculator to solve the equation on the interval [0, 2π). Round to the nearest hundredth of a radian. 
A)  $2.15,4.13$ 
B)  $2.15,5.29$ 
C)  $0.99,4.13$ 
D)  $0.99,2.15$ Use a calculator to solve the equation on the interval [0, 2π). Round to the nearest hundredth of a radian.

Question 3

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places.
-$\cos x = 0.76$

Accepted Answer

A)  $0.71,5.57$ 
B)  $0.71,2.43$ 
C)  $0.71,3.85$ 
D)  $0.71,2.28$ 
A)  $0.71,5.57$ 
B)  $0.71,2.43$ 
C)  $0.71,3.85$ 
D)  $0.71,2.28$

Question 4

Use the given information to find the exact value of the expression.
-$\tan \alpha = \frac { 3 } { 4 } , \alpha$ lies in quadrant III, and $\cos \beta = - \frac { 12 } { 13 } , \beta$ lies in quadrant II $\quad$ Find $\sin ( \alpha + \beta )$.

Accepted Answer

A)  $\frac { 16 } { 65 }$ 
B)  $\frac { 63 } { 65 }$ 
C)  $\frac { 33 } { 65 }$ 
D)  $\frac { 56 } { 65 }$ 
A)  $\frac { 16 } { 65 }$ 
B)  $\frac { 63 } { 65 }$ 
C)  $\frac { 33 } { 65 }$ 
D)  $\frac { 56 } { 65 }$

Question 5

Verify the identity.
-$$\tan \theta \cdot \csc \theta = \sec \theta$$

Accepted Answer

The answer of Verify the identity.
-\[\tan \theta \cdot \csc \theta...

Question 6

Use the Power-Reducing Formulas
-$\cos ^ { 4 } x$

Accepted Answer

A)  $\frac { 3 + 4 \cos 2 x + \cos 4 x } { 8 }$ 
B)  $\frac { 3 + 2 \cos 2 x + \cos 4 x } { 8 }$ 
C)  $\frac { 3 - 4 \cos 2 x + \cos 4 x } { 8 }$ 
D)  $\frac { 2 - 4 \cos 2 x + \cos 4 x } { 8 }$ 
A)  $\frac { 3 + 4 \cos 2 x + \cos 4 x } { 8 }$ 
B)  $\frac { 3 + 2 \cos 2 x + \cos 4 x } { 8 }$ 
C)  $\frac { 3 - 4 \cos 2 x + \cos 4 x } { 8 }$ 
D)  $\frac { 2 - 4 \cos 2 x + \cos 4 x } { 8 }$

Question 7

Express the product as a sum or difference.
-$\sin 5 x \cos 6 x$

Accepted Answer

A)  $\frac { 1 } { 2 } ( \sin 11 x - \sin x )$ 
B)  $\frac { 1 } { 2 } ( \cos 11 x + \sin x )$ 
C)  $\frac { 1 } { 2 } ( \cos 11 x - \cos x )$ 
D)  $\sin \left( \cos 30 x ^ { 2 } \right)$ 
A)  $\frac { 1 } { 2 } ( \sin 11 x - \sin x )$ 
B)  $\frac { 1 } { 2 } ( \cos 11 x + \sin x )$ 
C)  $\frac { 1 } { 2 } ( \cos 11 x - \cos x )$ 
D)  $\sin \left( \cos 30 x ^ { 2 } \right)$

Question 8

Use Sum and Difference Formulas for Tangents
Find the exact value by using a difference identity.
-$$\frac { \tan 55 ^ { \circ } + \tan 95 ^ { \circ } } { 1 - \tan 55 ^ { \circ } \tan 95 ^ { \circ } }$$

Accepted Answer

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

Question 9

Express the product as a sum or difference.
-$\cos 5 x \cos 8 x$

Accepted Answer

A)  $\frac { 1 } { 2 } ( \cos 3 x + \cos 13 x )$ 
B)  $\frac { 1 } { 2 } ( \cos 13 x - \sin 3 x )$ 
C)  $\frac { 1 } { 2 } ( \cos 13 x - \cos 3 x )$ 
D)  $\cos ^ { 2 } 28 x ^ { 2 }$ 
A)  $\frac { 1 } { 2 } ( \cos 3 x + \cos 13 x )$ 
B)  $\frac { 1 } { 2 } ( \cos 13 x - \sin 3 x )$ 
C)  $\frac { 1 } { 2 } ( \cos 13 x - \cos 3 x )$ 
D)  $\cos ^ { 2 } 28 x ^ { 2 }$

Question 10

Use the graph to complete the identity.
-$\sin 3 x \cos \frac { 3 } { 2 } x - \cos 3 x \sin \frac { 3 } { 2 } x = ?$

Accepted Answer

A)  $y = \sin \frac { 3 } { 2 } x$ 
B)  $y = \cos \frac { 3 } { 2 } x$ 
C)  $y = \cos \frac { 9 } { 2 } x$ 
D)  $y = \sin \frac { 9 } { 2 } x$ 
A)  $y = \sin \frac { 3 } { 2 } x$ 
B)  $y = \cos \frac { 3 } { 2 } x$ 
C)  $y = \cos \frac { 9 } { 2 } x$ 
D)  $y = \sin \frac { 9 } { 2 } x$

Question 11

Solve Apps: Trigonometric Equations
-A generator produces an alternating current according to the equation $\mathrm { I } = 40 \sin 144 \pi \mathrm { t }$, where $t$ is time in seconds and $\mathrm { I }$ is the current in amperes. What is the smallest time $t$ such that $\mathrm { I } = 20$ ?

Accepted Answer

A)  $\frac { 1 } { 864 }$ second 
B)  $\frac { 1 } { 432 }$ second 
C)  $\frac { 1 } { 576 }$ second 
D)  $\frac { 1 } { 288 }$ second 
A)  $\frac { 1 } { 864 }$ second 
B)  $\frac { 1 } { 432 }$ second 
C)  $\frac { 1 } { 576 }$ second 
D)  $\frac { 1 } { 288 }$ second

Question 12

Use Sum and Difference Formulas for Cosines and Sines
-$\cos 15 ^ { \circ } \cos 45 ^ { \circ } - \sin 15 ^ { \circ } \sin 45 ^ { \circ }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\sqrt { 3 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\sqrt { 3 }$

Question 13

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

Accepted Answer

A)  $- \frac { 3 \sqrt { 10 } } { 10 }$ 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 3 \sqrt { 10 } } { 10 }$ 
D)  $- \frac { 2 } { 5 }$ 
A)  $- \frac { 3 \sqrt { 10 } } { 10 }$ 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 3 \sqrt { 10 } } { 10 }$ 
D)  $- \frac { 2 } { 5 }$

Question 14

Solve Apps: Trigonometric Equations
-A coil of wire rotating in a magnetic field induces a voltage given by $\mathrm { e } = 20 \sin \left( \frac { \pi \mathrm { t } } { 4 } - \frac { \pi } { 2 } \right)$, where $\mathrm { t }$ is time in seconds. Find the smallest positive time to produce a voltage of $10 \sqrt { 3 }$.

Accepted Answer

A)  $3.33$ seconds 
B)  3 seconds 
C)  $3.33 \pi$ seconds 
D)  $3 \pi$ seconds 
A)  $3.33$ seconds 
B)  3 seconds 
C)  $3.33 \pi$ seconds 
D)  $3 \pi$ seconds

Question 15

Complete the identity.
-$\tan x \cdot \cot x =$ ?

Accepted Answer

A)  1 
B)  $- 1$ 
C)  0 
D)  $\sin x$ 
A)  1 
B)  $- 1$ 
C)  0 
D)  $\sin x$

Question 16

Verify the identity.
-$$\sin ( \alpha + \beta ) - \sin ( \alpha - \beta ) = 2 \cos \alpha \sin \beta$$

Accepted Answer

The answer of Verify the identity.
-\[\sin ( \alpha + \beta...

Question 17

Use the Formula for the Cosine of the Difference of Two Angles
-$\cos \left( \frac { 7 \pi } { 18 } \right) \cos \left( \frac { 2 \pi } { 9 } \right) + \sin \left( \frac { 7 \pi } { 18 } \right) \sin \left( \frac { 2 \pi } { 9 } \right)$

Accepted Answer

A)  $\alpha = \frac { 7 \pi } { 18 } , \beta = \frac { 2 \pi } { 9 }$ 
B)  $\alpha = \frac { 7 \pi } { 18 } , \beta = - \frac { 2 \pi } { 9 }$ 
C)  $\alpha = - \frac { 2 \pi } { 9 } , \beta = \frac { 7 \pi } { 18 }$ 
D)  $\alpha = \frac { 2 \pi } { 9 } , \beta = \frac { 7 \pi } { 18 }$ Write the expression as the cosine of an angle, knowing that the expression is the right side of the formula for $$\cos ( \alpha - \beta ) \text { with particular values for } \alpha \text { and } \beta .$$ 
A)  $\alpha = \frac { 7 \pi } { 18 } , \beta = \frac { 2 \pi } { 9 }$ 
B)  $\alpha = \frac { 7 \pi } { 18 } , \beta = - \frac { 2 \pi } { 9 }$ 
C)  $\alpha = - \frac { 2 \pi } { 9 } , \beta = \frac { 7 \pi } { 18 }$ 
D)  $\alpha = \frac { 2 \pi } { 9 } , \beta = \frac { 7 \pi } { 18 }$ Write the expression as the cosine of an angle, knowing that the expression is the right side of the formula for $$\cos ( \alpha - \beta ) \text { with particular values for } \alpha \text { and } \beta .$$

Question 18

Solve the equation on the interval [0, 2π).
-$- \tan ^ { 2 } x \sin x = - \tan ^ { 2 } x$

Accepted Answer

A)  $0 , \pi$ 
B)  $\pi , 2 \pi$ 
C)  $\pi , \frac { 3 \pi } { 2 }$ 
D)  $0 , \frac { 3 \pi } { 2 }$ 
A)  $0 , \pi$ 
B)  $\pi , 2 \pi$ 
C)  $\pi , \frac { 3 \pi } { 2 }$ 
D)  $0 , \frac { 3 \pi } { 2 }$

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.
-$$\frac { \sin 3 x + \sin 9 x } { \cos 9 x - \cos 3 x } = ?$$

Accepted Answer

A)  $- \cot 3 x$ 
B)  $\tan 3 x + \tan 9 x$ 
C)  $\cot 3 x$ 
D)  $\tan 6 x$ 
A)  $- \cot 3 x$ 
B)  $\tan 3 x + \tan 9 x$ 
C)  $\cot 3 x$ 
D)  $\tan 6 x$

Question 20

Use Identities to Solve Trigonometric Equations
Solve the equation on the interval [0, 2π).
-$2 \cos ^ { 2 } x + \sin x - 2 = 0$

Accepted Answer

A)  $0 , \pi , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
B)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
D)  $\frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
A)  $0 , \pi , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
B)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
D)  $\frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$

Complete the identity - $\tan ( \pi - \theta ) =$ ?

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - $\cos x = - 0.55$

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - $\cos x = 0.76$

Use the given information to find the exact value of the expression. - $\tan \alpha = \frac { 3 } { 4 } , \alpha$ lies in quadrant III, and $\cos \beta = - \frac { 12 } { 13 } , \beta$ lies in quadrant II $\quad$ Find $\sin ( \alpha + \beta )$ .

Verify the identity. - $\tan \theta \cdot \csc \theta = \sec \theta$

Use the Power-Reducing Formulas - $\cos ^ { 4 } x$

Express the product as a sum or difference. - $\sin 5 x \cos 6 x$

Use Sum and Difference Formulas for Tangents Find the exact value by using a difference identity. - $\frac { \tan 55 ^ { \circ } + \tan 95 ^ { \circ } } { 1 - \tan 55 ^ { \circ } \tan 95 ^ { \circ } }$

Express the product as a sum or difference. - $\cos 5 x \cos 8 x$

Use the graph to complete the identity. - $\sin 3 x \cos \frac { 3 } { 2 } x - \cos 3 x \sin \frac { 3 } { 2 } x = ?$ $Use the graph to complete the identity. - \sin 3 x \cos \frac { 3 } { 2 } x - \cos 3 x \sin \frac { 3 } { 2 } x = ?$

Solve Apps: Trigonometric Equations -A generator produces an alternating current according to the equation $\mathrm { I } = 40 \sin 144 \pi \mathrm { t }$ , where $t$ is time in seconds and $\mathrm { I }$ is the current in amperes. What is the smallest time $t$ such that $\mathrm { I } = 20$ ?

Use Sum and Difference Formulas for Cosines and Sines - $\cos 15 ^ { \circ } \cos 45 ^ { \circ } - \sin 15 ^ { \circ } \sin 45 ^ { \circ }$

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

Complete the identity. - $\tan x \cdot \cot x =$ ?

Verify the identity. - $\sin ( \alpha + \beta ) - \sin ( \alpha - \beta ) = 2 \cos \alpha \sin \beta$

Use the Formula for the Cosine of the Difference of Two Angles - $\cos \left( \frac { 7 \pi } { 18 } \right) \cos \left( \frac { 2 \pi } { 9 } \right) + \sin \left( \frac { 7 \pi } { 18 } \right) \sin \left( \frac { 2 \pi } { 9 } \right)$

Solve the equation on the interval [0, 2π). - $- \tan ^ { 2 } x \sin x = - \tan ^ { 2 } x$

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - $2 \cos ^ { 2 } x + \sin x - 2 = 0$

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 - tan⁡(π−θ)=\tan ( \pi - \theta ) =tan(π−θ)= ?

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - cos⁡x=−0.55\cos x = - 0.55cosx=−0.55

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - cos⁡x=0.76\cos x = 0.76cosx=0.76

Verify the identity. - tan⁡θ⋅csc⁡θ=sec⁡θ\tan \theta \cdot \csc \theta = \sec \thetatanθ⋅cscθ=secθ

Use the Power-Reducing Formulas - cos⁡4x\cos ^ { 4 } xcos4x

Express the product as a sum or difference. - sin⁡5xcos⁡6x\sin 5 x \cos 6 xsin5xcos6x

Use Sum and Difference Formulas for Tangents Find the exact value by using a difference identity. - tan⁡55∘+tan⁡95∘1−tan⁡55∘tan⁡95∘\frac { \tan 55 ^ { \circ } + \tan 95 ^ { \circ } } { 1 - \tan 55 ^ { \circ } \tan 95 ^ { \circ } }1−tan55∘tan95∘tan55∘+tan95∘​

Express the product as a sum or difference. - cos⁡5xcos⁡8x\cos 5 x \cos 8 xcos5xcos8x

Use the graph to complete the identity. - sin⁡3xcos⁡32x−cos⁡3xsin⁡32x=?\sin 3 x \cos \frac { 3 } { 2 } x - \cos 3 x \sin \frac { 3 } { 2 } x = ?sin3xcos23​x−cos3xsin23​x=?

Use Sum and Difference Formulas for Cosines and Sines - cos⁡15∘cos⁡45∘−sin⁡15∘sin⁡45∘\cos 15 ^ { \circ } \cos 45 ^ { \circ } - \sin 15 ^ { \circ } \sin 45 ^ { \circ }cos15∘cos45∘−sin15∘sin45∘

Use the given information 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 cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ .

Complete the identity. - tan⁡x⋅cot⁡x=\tan x \cdot \cot x =tanx⋅cotx= ?

Verify the identity. - sin⁡(α+β)−sin⁡(α−β)=2cos⁡αsin⁡β\sin ( \alpha + \beta ) - \sin ( \alpha - \beta ) = 2 \cos \alpha \sin \betasin(α+β)−sin(α−β)=2cosαsinβ

Solve the equation on the interval [0, 2π). - −tan⁡2xsin⁡x=−tan⁡2x- \tan ^ { 2 } x \sin x = - \tan ^ { 2 } x−tan2xsinx=−tan2x

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. - sin⁡3x+sin⁡9xcos⁡9x−cos⁡3x=?\frac { \sin 3 x + \sin 9 x } { \cos 9 x - \cos 3 x } = ?cos9x−cos3xsin3x+sin9x​=?

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - 2cos⁡2x+sin⁡x−2=02 \cos ^ { 2 } x + \sin x - 2 = 02cos2x+sinx−2=0