Question 1

Convert the expression. ​​ $( 2 \sin x + 2 \cos x ) ^ { 2 }$ ​

Accepted Answer

A) ​ $4 - 4 \sin 2 x$ 
B) ​ $2 + 4 \sin x$ 
C) ​ $4 + 4 \sin x$ 
D) ​ $4 - 4 \sin x$ 
E) ​ $4 + 4 \sin 2 x$ 
A) ​ $4 - 4 \sin 2 x$ 
B) ​ $2 + 4 \sin x$ 
C) ​ $4 + 4 \sin x$ 
D) ​ $4 - 4 \sin x$ 
E) ​ $4 + 4 \sin 2 x$

Question 2

Use the figure to find the exact value of the trigonometric function. ​
Csc 2θ​   ​
A = 1,b = 6
​

Accepted Answer

A) ​ $\frac { 37 } { 12 }$ 
B) ​ $\frac { 13 } { 37 }$ 
C)  $\frac { 37 } { 13 }$ 
D) ​ $\frac { 12 } { 13 }$ 
E)  $\frac { 12 } { 37 }$ 
A) ​ $\frac { 37 } { 12 }$ 
B) ​ $\frac { 13 } { 37 }$ 
C)  $\frac { 37 } { 13 }$ 
D) ​ $\frac { 12 } { 13 }$ 
E)  $\frac { 12 } { 37 }$

Question 3

Use the figure to find the exact value of the trigonometric function. ​
Tan 2θ​   ​
A = 1,b = 6
​

Accepted Answer

A)  $\frac { 35 } { 12 }$ 
B)  $\frac { 12 } { 35 }$ 
C) ​ $\frac { 12 } { 37 }$ 
D) ​ $\frac { 35 } { 37 }$ 
E) ​ $\frac { 37 } { 35 }$ 
A)  $\frac { 35 } { 12 }$ 
B)  $\frac { 12 } { 35 }$ 
C) ​ $\frac { 12 } { 37 }$ 
D) ​ $\frac { 35 } { 37 }$ 
E) ​ $\frac { 37 } { 35 }$

Question 4

Use the sum-to-product formulas to write the given expression as a product. ​​ $$\sin 5 \theta - \sin 3 \theta$$ ​

Accepted Answer

A) ​​2 sin 4θ cos θ 
B) ​-2 sin 4θ sin θ 
C) -​2 cos 4θ cos θ 
D) ​2 cos 4θ sin θ 
E) ​​2 cos 4θ cos θ 
A) ​​2 sin 4θ cos θ 
B) ​-2 sin 4θ sin θ 
C) -​2 cos 4θ cos θ 
D) ​2 cos 4θ sin θ 
E) ​​2 cos 4θ cos θ

Question 5

Use the figure to find the exact value of the trigonometric function.​   $$\begin{array} { l } 
a = 12 , b = 3 \
c = 5 , d = 4
\end{array}$$ ​ cos 2β
​

Accepted Answer

A) ​ $\frac { 41 } { 5 }$ 
B) ​ $\frac { 3 } { 10 }$ 
C)  $\frac { 9 } { 41 }$ 
D) ​ $\frac { 12 } { 13 }$ 
E) ​ $\frac { 41 } { 9 }$ 
A) ​ $\frac { 41 } { 5 }$ 
B) ​ $\frac { 3 } { 10 }$ 
C)  $\frac { 9 } { 41 }$ 
D) ​ $\frac { 12 } { 13 }$ 
E) ​ $\frac { 41 } { 9 }$

Question 6

Use the half-angle formulas to simplify the expression. ​​ $$\sqrt { \frac { 1 + \cos 8 x } { 2 } }$$ ​

Accepted Answer

A) ​- ​|cos x| 
B) ​|cos x| 
C) ​​|cos 4x| 
D) ​​- |cos 4x| 
E) ​- ​|cos 8x| 
A) ​- ​|cos x| 
B) ​|cos x| 
C) ​​|cos 4x| 
D) ​​- |cos 4x| 
E) ​- ​|cos 8x|

Question 7

Use the half-angle formulas to simplify the expression.​ $$- \sqrt { \frac { 1 + \cos 6 x } { 1 - \cos 6 x } }$$ ​

Accepted Answer

A) ​- |cot x| 
B) ​- |3 tan x| 
C) ​- |3 tan 6x| 
D) ​- |3 cot 3x| 
E) ​- |cot 3x| 
A) ​- |cot x| 
B) ​- |3 tan x| 
C) ​- |3 tan 6x| 
D) ​- |3 cot 3x| 
E) ​- |cot 3x|

Question 8

Use a double-angle formula to rewrite the expression. ​​ $2 \sin x \cos x$ ​

Accepted Answer

A) ​sin x 
B) ​​2 sin x 
C) ​​​2 sin 2x 
D) ​ sin x 
E) ​ sin 2x 
A) ​sin x 
B) ​​2 sin x 
C) ​​​2 sin 2x 
D) ​ sin x 
E) ​ sin 2x

Question 9

Find the exact solutions of the given equation in the interval [0,2π). ​​ $$2 \sin ^ { 2 } x + 3 \sin x = - 1$$ ​

Accepted Answer

A) ​ $x = \pi , \frac { 7 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
B) ​ $x = \frac { 7 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 4 }$ 
C) ​ $x = \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
D) ​ $x = \frac { \pi } { 4 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 2 }$ 
E) ​ $x = 0 , \frac { 7 \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 2 }$ 
A) ​ $x = \pi , \frac { 7 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
B) ​ $x = \frac { 7 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 4 }$ 
C) ​ $x = \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
D) ​ $x = \frac { \pi } { 4 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 2 }$ 
E) ​ $x = 0 , \frac { 7 \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 2 }$

Question 10

Use the half-angle formulas to determine the exact value of the given trigonometric expression. ​​ $$\tan \frac { 3 \pi } { 8 }$$ ​

Accepted Answer

A) ​ $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } - 1$ 
B) ​ $\tan \frac { 3 \pi } { 8 } = \frac { \sqrt { 2 - \sqrt { 2 } } } { 4 }$ 
C) ​ $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 + \sqrt { 2 } }$ 
D) ​ $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } + 1$ ​ 
E) ​ $\tan \frac { 3 \pi } { 8 } = - \frac { \sqrt { 2 + \sqrt { 2 } } } { 4 }$ 
A) ​ $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } - 1$ 
B) ​ $\tan \frac { 3 \pi } { 8 } = \frac { \sqrt { 2 - \sqrt { 2 } } } { 4 }$ 
C) ​ $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 + \sqrt { 2 } }$ 
D) ​ $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } + 1$ ​ 
E) ​ $\tan \frac { 3 \pi } { 8 } = - \frac { \sqrt { 2 + \sqrt { 2 } } } { 4 }$

Exam 35: Multiple Angle and Product to Sum Formulas

Convert the expression. ​​ (2sin⁡x+2cos⁡x)2( 2 \sin x + 2 \cos x ) ^ { 2 }(2sinx+2cosx)2 ​

Use the figure to find the exact value of the trigonometric function. ​ Csc 2θ​ ​ A = 1,b = 6 ​

Use the figure to find the exact value of the trigonometric function. ​ Tan 2θ​ ​ A = 1,b = 6 ​

Use the sum-to-product formulas to write the given expression as a product. ​​ sin⁡5θ−sin⁡3θ\sin 5 \theta - \sin 3 \thetasin5θ−sin3θ ​

Use the figure to find the exact value of the trigonometric function.​ a=12,b=3 c=5,d=4 ​ cos 2β ​

Use the half-angle formulas to simplify the expression. ​​ 1+cos⁡8x2\sqrt { \frac { 1 + \cos 8 x } { 2 } }21+cos8x​​ ​

Use the half-angle formulas to simplify the expression.​ −1+cos⁡6x1−cos⁡6x- \sqrt { \frac { 1 + \cos 6 x } { 1 - \cos 6 x } }−1−cos6x1+cos6x​​ ​

Use a double-angle formula to rewrite the expression. ​​ 2sin⁡xcos⁡x2 \sin x \cos x2sinxcosx ​

Find the exact solutions of the given equation in the interval [0,2π). ​​ 2sin⁡2x+3sin⁡x=−12 \sin ^ { 2 } x + 3 \sin x = - 12sin2x+3sinx=−1 ​

Use the half-angle formulas to determine the exact value of the given trigonometric expression. ​​ tan⁡3π8\tan \frac { 3 \pi } { 8 }tan83π​ ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Hyperbolas

Rotation of Conics

Parametric Equations

Polar Coordinates

Polar Equations of Conics

Graphs of Polar Equations

Filters

Convert the expression. $( 2 \sin x + 2 \cos x ) ^ { 2 }$

Use the figure to find the exact value of the trigonometric function. Csc 2θ A = 1,b = 6

Use the figure to find the exact value of the trigonometric function. Tan 2θ A = 1,b = 6

Use the sum-to-product formulas to write the given expression as a product. $\sin 5 \theta - \sin 3 \theta$

Use the figure to find the exact value of the trigonometric function. $Use the figure to find the exact value of the trigonometric function. \begin{array} { l } a = 12 , b = 3 \\ c = 5 , d = 4 \end{array} cos 2β $ a=12,b=3 c=5,d=4 cos 2β

Use the half-angle formulas to simplify the expression. $\sqrt { \frac { 1 + \cos 8 x } { 2 } }$

Use the half-angle formulas to simplify the expression. $- \sqrt { \frac { 1 + \cos 6 x } { 1 - \cos 6 x } }$

Use a double-angle formula to rewrite the expression. $2 \sin x \cos x$

Find the exact solutions of the given equation in the interval [0,2π). $2 \sin ^ { 2 } x + 3 \sin x = - 1$

Use the half-angle formulas to determine the exact value of the given trigonometric expression. $\tan \frac { 3 \pi } { 8 }$