Question 1

Complete the identity.
-$$\frac { \sin 5 x + \sin 7 x } { \cos 2 x + \cos 4 x } = ?$$

Accepted Answer

A)  $\frac { \sin 6 x } { \cos 3 x }$ 
B)  $\tan \frac { 5 } { 2 } x + \tan \frac { 7 } { 4 } x$ 
C)  $\frac { \sin 12 x } { \cos 6 x }$ 
D)  $\frac { \cos 12 x } { \sin 6 x }$ 
A)  $\frac { \sin 6 x } { \cos 3 x }$ 
B)  $\tan \frac { 5 } { 2 } x + \tan \frac { 7 } { 4 } x$ 
C)  $\frac { \sin 12 x } { \cos 6 x }$ 
D)  $\frac { \cos 12 x } { \sin 6 x }$

Question 2

Verify the identity.
-$$\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$$

Accepted Answer

The answer of Verify the identity.
-\[\tan \left( \frac { \pi...

Question 3

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

Accepted Answer

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

Question 4

Complete the identity
-$$\tan \left( x + \frac { \pi } { 4 } \right) = ?$$

Accepted Answer

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

Question 5

Use Sum and Difference Formulas for Cosines and Sines
-$\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 5 } { 12 }$ 
D)  $\frac { \sqrt { 3 } } { 3 }$ 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 5 } { 12 }$ 
D)  $\frac { \sqrt { 3 } } { 3 }$

Question 6

Use the graph to complete the identity.
-$\cos ^ { 2 } x + \cos x - 1 + \sin ^ { 2 } x ; \cos x$

Accepted Answer

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

Question 7

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
-$\sin ( x + \pi ) = \sin x$

Accepted Answer

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

Question 8

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

Accepted Answer

A)  $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
B)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
C)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  no solution 
A)  $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
B)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
C)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  no solution

Question 9

Use Identities to Solve Trigonometric Equations
Solve the equation on the interval [0, 2π).
-$\sin 3 x = 0$

Accepted Answer

A)  $0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 } , 2 \pi$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  $\frac { \pi } { 8 } , \frac { 9 \pi } { 8 }$ 
D)  no solution 
A)  $0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 } , 2 \pi$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  $\frac { \pi } { 8 } , \frac { 9 \pi } { 8 }$ 
D)  no solution

Question 10

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

Accepted Answer

\[\begin{array} { l } 
\sin ( \alpha - \

Question 11

Verify the identity.
-$$\csc ^ { 2 } u - \cos u \sec u = \cot ^ { 2 } u$$

Accepted Answer

The answer of Verify the identity.
-\[\csc ^ { 2 }...

Question 12

Use the Formula for the Cosine of the Difference of Two Angles
-$\cos \left( \frac { 5 \pi } { 18 } - \frac { \pi } { 9 } \right)$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  1 Identify α and β in the following expression which is the right side of the formula for cos (α - β). 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  1 Identify α and β in the following expression which is the right side of the formula for cos (α - β).

Question 13

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

Accepted Answer

The answer of Verify the identity.
-\[\cos \left( x + \frac...

Question 14

Solve Apps: Trigonometric Equations
-The range $\mathrm { r }$ of a projectile is given by $\mathrm { r } = \frac { 1 } { 32 } \mathrm { v } ^ { 2 } \sin 2 \theta$, where $\mathrm { v }$ is the initial velocity and $\theta$ is the angle of elevation. If $r$ is to be $9000 \mathrm { ft }$ and $\mathrm { v } = 5000 \mathrm { ft } / \mathrm { sec }$, what must the angle of elevation be? Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $0.33 ^ { \circ }$ 
B)  $89.67 ^ { \circ }$ 
C)  $0.66 ^ { \circ }$ 
D)  $0.46 ^ { \circ }$ 
A)  $0.33 ^ { \circ }$ 
B)  $89.67 ^ { \circ }$ 
C)  $0.66 ^ { \circ }$ 
D)  $0.46 ^ { \circ }$

Question 15

Complete the identity.
-$\cos ( \alpha + \beta ) + \cos ( \alpha - \beta ) =$ ?

Accepted Answer

A)  $2 \cos \alpha \cos \beta$ 
B)  $\cos \alpha \cos \beta$ 
C)  $2 \sin \alpha \cos \beta$ 
D)  $\sin \beta \cos \alpha$ 
A)  $2 \cos \alpha \cos \beta$ 
B)  $\cos \alpha \cos \beta$ 
C)  $2 \sin \alpha \cos \beta$ 
D)  $\sin \beta \cos \alpha$

Question 16

Use the graph to complete the identity.
-$\cos 2 x \cos 5 x + \sin 2 x \sin 5 x = ?$

Accepted Answer

A)  $y = \cos 3 x$ 
B)  $y = \sin 3 x$ 
C)  $y = \cos 7 x$ 
D)  $y = \sin 7 x$ 
A)  $y = \cos 3 x$ 
B)  $y = \sin 3 x$ 
C)  $y = \cos 7 x$ 
D)  $y = \sin 7 x$

Question 17

Complete the identity.
-$\frac { \sin ( \alpha + \beta ) } { \cos \alpha \cos \beta } = ?$

Accepted Answer

A)  $\tan \alpha + \tan \beta$ 
B)  $\tan \beta + \tan \alpha$ 
C)  $\cot \alpha + \cot \beta$ 
D)  $- \tan \alpha + \cot \beta$ 
A)  $\tan \alpha + \tan \beta$ 
B)  $\tan \beta + \tan \alpha$ 
C)  $\cot \alpha + \cot \beta$ 
D)  $- \tan \alpha + \cot \beta$

Question 18

Express the product as a sum or difference.
-$\cos 3 x \cos 2 x$

Accepted Answer

A)  $\frac { 1 } { 2 } ( \cos x + \cos 5 x )$ 
B)  $\frac { 1 } { 2 } ( \cos 5 x - \sin x )$ 
C)  $\frac { 1 } { 2 } ( \cos 5 x - \cos x )$ 
D)  $\cos ^ { 2 } 6 \mathrm { x } ^ { 2 }$ 
A)  $\frac { 1 } { 2 } ( \cos x + \cos 5 x )$ 
B)  $\frac { 1 } { 2 } ( \cos 5 x - \sin x )$ 
C)  $\frac { 1 } { 2 } ( \cos 5 x - \cos x )$ 
D)  $\cos ^ { 2 } 6 \mathrm { x } ^ { 2 }$

Question 19

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places.
-$\tan x = 5.1$

Accepted Answer

A)  $1.38,4.52$ 
B)  $1.38,4.90$ 
C)  $1.38,1.76$ 
D)  $1.38,2.95$ 
A)  $1.38,4.52$ 
B)  $1.38,4.90$ 
C)  $1.38,1.76$ 
D)  $1.38,2.95$

Question 20

Verify the identity.
-$$\left( 1 + \tan ^ { 2 } u \right) \left( 1 - \sin ^ { 2 } u \right) = 1$$

Accepted Answer

The answer of Verify the identity.
-\[\left( 1 + \tan ^...

Complete the identity. - $\frac { \sin 5 x + \sin 7 x } { \cos 2 x + \cos 4 x } = ?$

Verify the identity. - $\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$

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

Complete the identity - $\tan \left( x + \frac { \pi } { 4 } \right) = ?$

Use Sum and Difference Formulas for Cosines and Sines - $\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }$

Use the graph to complete the identity. - $\cos ^ { 2 } x + \cos x - 1 + \sin ^ { 2 } x ; \cos x$

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. - $\sin ( x + \pi ) = \sin x$

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

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - $\sin 3 x = 0$

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

Verify the identity. - $\csc ^ { 2 } u - \cos u \sec u = \cot ^ { 2 } u$

Use the Formula for the Cosine of the Difference of Two Angles - $\cos \left( \frac { 5 \pi } { 18 } - \frac { \pi } { 9 } \right)$

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

Complete the identity. - $\cos ( \alpha + \beta ) + \cos ( \alpha - \beta ) =$ ?

Use the graph to complete the identity. - $\cos 2 x \cos 5 x + \sin 2 x \sin 5 x = ?$ $Use the graph to complete the identity. - \cos 2 x \cos 5 x + \sin 2 x \sin 5 x = ?$

Complete the identity. - $\frac { \sin ( \alpha + \beta ) } { \cos \alpha \cos \beta } = ?$

Express the product as a sum or difference. - $\cos 3 x \cos 2 x$

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - $\tan x = 5.1$

Verify the identity. - $\left( 1 + \tan ^ { 2 } u \right) \left( 1 - \sin ^ { 2 } u \right) = 1$

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. - sin⁡5x+sin⁡7xcos⁡2x+cos⁡4x=?\frac { \sin 5 x + \sin 7 x } { \cos 2 x + \cos 4 x } = ?cos2x+cos4xsin5x+sin7x​=?

Verify the identity. - tan⁡(π2+x)=−cot⁡x\tan \left( \frac { \pi } { 2 } + x \right) = - \cot xtan(2π​+x)=−cotx

Use the graph to complete the identity. - cos⁡3xcos⁡32x+sin⁡3xsin⁡32x=\cos 3 x \cos \frac { 3 } { 2 } x + \sin 3 x \sin \frac { 3 } { 2 } x =cos3xcos23​x+sin3xsin23​x= ?

Complete the identity - tan⁡(x+π4)=?\tan \left( x + \frac { \pi } { 4 } \right) = ?tan(x+4π​)=?

Use Sum and Difference Formulas for Cosines and Sines - sin⁡25∘cos⁡35∘+cos⁡25∘sin⁡35∘\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }sin25∘cos35∘+cos25∘sin35∘

Use the graph to complete the identity. - cos⁡2x+cos⁡x−1+sin⁡2x;cos⁡x\cos ^ { 2 } x + \cos x - 1 + \sin ^ { 2 } x ; \cos xcos2x+cosx−1+sin2x;cosx

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. - sin⁡(x+π)=sin⁡x\sin ( x + \pi ) = \sin xsin(x+π)=sinx

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - cos⁡2x=2−cos⁡2x\cos 2 x = \sqrt { 2 } - \cos 2 xcos2x=2​−cos2x

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - sin⁡3x=0\sin 3 x = 0sin3x=0

Verify the identity. - sin⁡(α−β)cos⁡(α+β)=sin⁡αcos⁡α−sin⁡βcos⁡β\sin ( \alpha - \beta ) \cos ( \alpha + \beta ) = \sin \alpha \cos \alpha - \sin \beta \cos \betasin(α−β)cos(α+β)=sinαcosα−sinβcosβ

Verify the identity. - csc⁡2u−cos⁡usec⁡u=cot⁡2u\csc ^ { 2 } u - \cos u \sec u = \cot ^ { 2 } ucsc2u−cosusecu=cot2u

Use the Formula for the Cosine of the Difference of Two Angles - cos⁡(5π18−π9)\cos \left( \frac { 5 \pi } { 18 } - \frac { \pi } { 9 } \right)cos(185π​−9π​)

Verify the identity. - cos⁡(x+π2)=−sin⁡x\cos \left( x + \frac { \pi } { 2 } \right) = - \sin xcos(x+2π​)=−sinx

Complete the identity. - cos⁡(α+β)+cos⁡(α−β)=\cos ( \alpha + \beta ) + \cos ( \alpha - \beta ) =cos(α+β)+cos(α−β)= ?

Use the graph to complete the identity. - cos⁡2xcos⁡5x+sin⁡2xsin⁡5x=?\cos 2 x \cos 5 x + \sin 2 x \sin 5 x = ?cos2xcos5x+sin2xsin5x=?

Complete the identity. - sin⁡(α+β)cos⁡αcos⁡β=?\frac { \sin ( \alpha + \beta ) } { \cos \alpha \cos \beta } = ?cosαcosβsin(α+β)​=?

Express the product as a sum or difference. - cos⁡3xcos⁡2x\cos 3 x \cos 2 xcos3xcos2x

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - tan⁡x=5.1\tan x = 5.1tanx=5.1

Verify the identity. - (1+tan⁡2u)(1−sin⁡2u)=1\left( 1 + \tan ^ { 2 } u \right) \left( 1 - \sin ^ { 2 } u \right) = 1(1+tan2u)(1−sin2u)=1