Question 1

Use the half-angle formulas to simplify the expression. ​​ $$\sqrt { \frac { 1 - \cos 10 x } { 2 } }$$ ​

Accepted Answer

A) |sin 5x| 
B) - |sin x| 
C) |sin 10x| 
D) - |sin 5x| 
E) |sin x| 
A) |sin 5x| 
B) - |sin x| 
C) |sin 10x| 
D) - |sin 5x| 
E) |sin x|

Question 2

Convert the expression.​ $$\cos ^ { 4 } b - \sin ^ { 4 } b$$ ​

Accepted Answer

A) 2 cos b 
B) ​cos 2b 
C) ​​cos b 
D) 2 ​cos 2b 
E) ​4 cos b 
A) 2 cos b 
B) ​cos 2b 
C) ​​cos b 
D) 2 ​cos 2b 
E) ​4 cos b

Question 3

Use a double angle formula to rewrite the given expression. $$10 \cos ^ { 2 } x - 5$$

Accepted Answer

A)  $5 \cos 5 x$ 
B)  $5 \cos 2 x$ ​ 
C)  $10 \cos 2 x$ ​ 
D)  $2 \cos 10 x$ ​ 
E)  $2 \cos 5 x$ ​ 
A)  $5 \cos 5 x$ 
B)  $5 \cos 2 x$ ​ 
C)  $10 \cos 2 x$ ​ 
D)  $2 \cos 10 x$ ​ 
E)  $2 \cos 5 x$ ​

Question 4

Convert the expression.​ $$\tan \frac { a } { 2 }$$ ​ ​

Accepted Answer

A) ​ $\csc a - \tan a$ 
B)  $\csc a + \cot a$ 
C) ​ $\cos a - \cot a$ 
D)  $\cos a + \cot a$ 
E)  $\csc a - \cot a$ 
A) ​ $\csc a - \tan a$ 
B)  $\csc a + \cot a$ 
C) ​ $\cos a - \cot a$ 
D)  $\cos a + \cot a$ 
E)  $\csc a - \cot a$

Question 5

Use the product-to-sum formula to write the given product as a sum or difference.​ $$12 \sin \frac { \pi } { 6 } \cos \frac { \pi } { 6 }$$ ​

Accepted Answer

A) ​ $6 - 6 \cos \frac { \pi } { 12 }$ 
B)  $6 \sin \frac { \pi } { 6 } + 6 \cos \frac { \pi } { 6 }$ ​ 
C)  $6 \sin \frac { \pi } { 3 } + 6 \sin 0$ ​ 
D)  $6 + 6 \cos \frac { \pi } { 12 }$ ​ 
E)  $- 6 \sin \frac { \pi } { 12 }$ ​ 
A) ​ $6 - 6 \cos \frac { \pi } { 12 }$ 
B)  $6 \sin \frac { \pi } { 6 } + 6 \cos \frac { \pi } { 6 }$ ​ 
C)  $6 \sin \frac { \pi } { 3 } + 6 \sin 0$ ​ 
D)  $6 + 6 \cos \frac { \pi } { 12 }$ ​ 
E)  $- 6 \sin \frac { \pi } { 12 }$ ​

Question 6

Use the sum-to-product formulas to rewrite the sum or difference as a product.​ $$\sin 3 \theta - \sin \theta$$ ​

Accepted Answer

A) ​ $2 \sin 3 \theta \cos \theta$ 
B) ​ $2 \sin 2 \theta \cos \theta$ 
C) ​ $2 \cos 2 \theta \cos \theta$ 
D) ​ $2 \sin 2 \theta \sin \theta$ 
E) ​ $2 \cos 2 \theta \sin \theta$ 
A) ​ $2 \sin 3 \theta \cos \theta$ 
B) ​ $2 \sin 2 \theta \cos \theta$ 
C) ​ $2 \cos 2 \theta \cos \theta$ 
D) ​ $2 \sin 2 \theta \sin \theta$ 
E) ​ $2 \cos 2 \theta \sin \theta$

Question 7

Convert the expression.​ $$3 \csc 2 \theta$$ ​

Accepted Answer

A) ​ $\frac { 3 \csc \theta } { 2 \sin \theta }$ 
B) ​ $\frac { 3 \csc \theta } { 2 \sec \theta }$ 
C) ​ $\frac { 3 \csc \theta } { \cos \theta }$ 
D)  $\frac { 3 \csc \theta } { 2 \cos \theta }$ 
E) ​ $\frac { 3 \sec \theta } { 2 \cos \theta }$ 
A) ​ $\frac { 3 \csc \theta } { 2 \sin \theta }$ 
B) ​ $\frac { 3 \csc \theta } { 2 \sec \theta }$ 
C) ​ $\frac { 3 \csc \theta } { \cos \theta }$ 
D)  $\frac { 3 \csc \theta } { 2 \cos \theta }$ 
E) ​ $\frac { 3 \sec \theta } { 2 \cos \theta }$

Question 8

Find all solutions of the given equation in the interval [0,2π). ​​ $$\frac { 1 } { 2 } \sin 2 x = \cos x$$ ​

Accepted Answer

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

Question 9

Use the sum-to-product formulas to rewrite the sum or difference as a product. ​
Cos 3θ + cos 8θ
​

Accepted Answer

A) ​ $\cos \frac { 11 \theta } { 2 } \cos \frac { 5 \theta } { 2 }$ 
B) ​ $2 \cos \frac { 11 \theta } { 2 } \sin - \frac { 5 \theta } { 2 }$ 
C) ​ $2 \sin \frac { 11 \theta } { 2 } \sin \frac { 5 \theta } { 2 }$ 
D) ​ $\cos - \frac { 11 \theta } { 2 } \cos \frac { 5 \theta } { 2 }$ 
E)  $2 \cos \frac { 11 \theta } { 2 } \cos - \frac { 5 \theta } { 2 }$ 
A) ​ $\cos \frac { 11 \theta } { 2 } \cos \frac { 5 \theta } { 2 }$ 
B) ​ $2 \cos \frac { 11 \theta } { 2 } \sin - \frac { 5 \theta } { 2 }$ 
C) ​ $2 \sin \frac { 11 \theta } { 2 } \sin \frac { 5 \theta } { 2 }$ 
D) ​ $\cos - \frac { 11 \theta } { 2 } \cos \frac { 5 \theta } { 2 }$ 
E)  $2 \cos \frac { 11 \theta } { 2 } \cos - \frac { 5 \theta } { 2 }$

Question 10

Use the product-to-sum formulas to rewrite the product as a sum or difference.​ $$10 \cos 45 ^ { \circ } \cos 20 ^ { \circ }$$ ​

Accepted Answer

A) ​ $10 \left( \cos 25 ^ { \circ } + \cos 65 ^ { \circ } \right)$ 
B) ​ $\cos 25 ^ { \circ } + \cos 65 ^ { \circ }$ 
C) ​ $5 \left( \cos 65 ^ { \circ } - \cos 25 ^ { \circ } \right)$ 
D)  $5 \left( \cos 25 ^ { \circ } + \cos 65 ^ { \circ } \right)$ 
E) ​ $5 \left( \cos 65 ^ { \circ } + \sin 25 ^ { \circ } \right)$ 
A) ​ $10 \left( \cos 25 ^ { \circ } + \cos 65 ^ { \circ } \right)$ 
B) ​ $\cos 25 ^ { \circ } + \cos 65 ^ { \circ }$ 
C) ​ $5 \left( \cos 65 ^ { \circ } - \cos 25 ^ { \circ } \right)$ 
D)  $5 \left( \cos 25 ^ { \circ } + \cos 65 ^ { \circ } \right)$ 
E) ​ $5 \left( \cos 65 ^ { \circ } + \sin 25 ^ { \circ } \right)$

Question 11

Use the sum-to-product formulas to rewrite the sum or difference as a product.​ $$\cos 12 \theta + \cos 8 \theta$$ ​

Accepted Answer

A) ​ $2 \sin 10 \theta \cos 2 \theta$ 
B) ​ $2 \cos 10 \theta \sin 2 \theta$ 
C) ​ $2 \sin 12 \theta \cos 8 \theta$ 
D) ​ $2 \cos 10 \theta \cos 2 \theta$ 
E) ​ $2 \sin 10 \theta \sin 2 \theta$ 
A) ​ $2 \sin 10 \theta \cos 2 \theta$ 
B) ​ $2 \cos 10 \theta \sin 2 \theta$ 
C) ​ $2 \sin 12 \theta \cos 8 \theta$ 
D) ​ $2 \cos 10 \theta \cos 2 \theta$ 
E) ​ $2 \sin 10 \theta \sin 2 \theta$

Question 12

Convert the expression.​ $$7 \sec 2 \theta$$ ​

Accepted Answer

A) ​ $\frac { 7 \sec ^ { 2 } \theta } { 2 + \sec ^ { 2 } \theta }$ 
B) ​ $\frac { \sec ^ { 2 } \theta } { 2 - \sec ^ { 2 } \theta }$ 
C) ​ $\frac { 7 \sec ^ { 2 } \theta } { 2 - \sec ^ { 2 } \theta }$ 
D) ​ $\frac { 7 \sec ^ { 2 } \theta } { 2 - \cos ^ { 2 } \theta }$ 
E) ​ $\frac { 7 \cos ^ { 2 } \theta } { 2 - \cos ^ { 2 } \theta }$ 
A) ​ $\frac { 7 \sec ^ { 2 } \theta } { 2 + \sec ^ { 2 } \theta }$ 
B) ​ $\frac { \sec ^ { 2 } \theta } { 2 - \sec ^ { 2 } \theta }$ 
C) ​ $\frac { 7 \sec ^ { 2 } \theta } { 2 - \sec ^ { 2 } \theta }$ 
D) ​ $\frac { 7 \sec ^ { 2 } \theta } { 2 - \cos ^ { 2 } \theta }$ 
E) ​ $\frac { 7 \cos ^ { 2 } \theta } { 2 - \cos ^ { 2 } \theta }$

Question 13

Use the sum-to-product formulas to select the sum or difference as a product.​ $$\cos ( 5 \phi + 2 \pi ) + \cos 5 \phi$$ ​

Accepted Answer

A) ​ $- 2 \cos 5 \phi \cos \pi$ 
B) ​ $- 2 \cos 5 \pi$ 
C)  $2 \cos ( 5 \phi + \pi ) \cos \pi$ 
D) ​ $2 \cos ( \phi + \pi ) \cos \pi$ 
E) ​ $2 \cos ( 5 \phi + \pi )$ 
A) ​ $- 2 \cos 5 \phi \cos \pi$ 
B) ​ $- 2 \cos 5 \pi$ 
C)  $2 \cos ( 5 \phi + \pi ) \cos \pi$ 
D) ​ $2 \cos ( \phi + \pi ) \cos \pi$ 
E) ​ $2 \cos ( 5 \phi + \pi )$

Question 14

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

Accepted Answer

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

Question 15

Use a double-angle formula to rewrite the expression. ​
3 - 6 sin2 x
​

Accepted Answer

A) 6 cos x 
B) ​3 sin 2x 
C) ​3 sin x 
D) ​3 cos 2x 
E) ​6 cos 2x 
A) 6 cos x 
B) ​3 sin 2x 
C) ​3 sin x 
D) ​3 cos 2x 
E) ​6 cos 2x

Question 16

Convert the expression.​ $$\cos 4 \alpha$$ ​

Accepted Answer

A) ​ $\cos ^ { 2 } 2 \alpha - \sin ^ { 2 } 2 \alpha$ 
B) ​ $\cos ^ { 2 } 2 \alpha + \sin ^ { 2 } 2 \alpha$ 
C) ​ $\cos ^ { 4 } 2 \alpha + \sin ^ { 4 } 2 \alpha$ 
D) ​ $\cos ^ { 4 } 2 \alpha - \sin ^ { 4 } 2 \alpha$ 
E) ​ $\cos ^ { 4 } 2 \alpha - \sin ^ { 2 } 2 \alpha$ 
A) ​ $\cos ^ { 2 } 2 \alpha - \sin ^ { 2 } 2 \alpha$ 
B) ​ $\cos ^ { 2 } 2 \alpha + \sin ^ { 2 } 2 \alpha$ 
C) ​ $\cos ^ { 4 } 2 \alpha + \sin ^ { 4 } 2 \alpha$ 
D) ​ $\cos ^ { 4 } 2 \alpha - \sin ^ { 4 } 2 \alpha$ 
E) ​ $\cos ^ { 4 } 2 \alpha - \sin ^ { 2 } 2 \alpha$

Question 17

Use a double-angle formula to rewrite the expression. ​
2 sin2 x - 1
​

Accepted Answer

A) cos x 
B) cos 2x ​ 
C) -2 cos x ​ 
D) 2 cos 2x ​ 
E) - cos 2x ​ 
A) cos x 
B) cos 2x ​ 
C) -2 cos x ​ 
D) 2 cos 2x ​ 
E) - cos 2x ​

Question 18

​Use the figure below to find the exact value of the given trigonometric expression. $$\sin \frac { \theta } { 2 }$$ ​​   10 24
(figure not necessarily to scale)
​

Accepted Answer

A) ​ $\frac { 5 } { 12 }$ 
B) ​ $\frac { 5 } { 26 }$ 
C) ​ $\frac { 5 \sqrt { 26 } } { 26 }$ 
D) ​ $\frac { \sqrt { 26 } } { 26 }$ 
E) ​ $\frac { \sqrt { 26 } } { 12 }$ 
A) ​ $\frac { 5 } { 12 }$ 
B) ​ $\frac { 5 } { 26 }$ 
C) ​ $\frac { 5 \sqrt { 26 } } { 26 }$ 
D) ​ $\frac { \sqrt { 26 } } { 26 }$ 
E) ​ $\frac { \sqrt { 26 } } { 12 }$

Question 19

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

Accepted Answer

A) ​ $\cos \frac { \theta } { 2 } = \frac { 7 \sqrt { 2 } } { 10 }$ 
B) ​ $\cos \frac { \theta } { 2 } = \frac { 7 \sqrt { 2 } } { 7 }$ 
C) ​ $\cos \frac { \theta } { 2 } = \frac { \sqrt { 2 } } { 10 }$ 
D) ​ $\cos \frac { \theta } { 2 } = 7$ 
E) ​ $\cos \frac { x } { 2 } = \frac { 7 \sqrt { 2 } } { 12 }$ 
A) ​ $\cos \frac { \theta } { 2 } = \frac { 7 \sqrt { 2 } } { 10 }$ 
B) ​ $\cos \frac { \theta } { 2 } = \frac { 7 \sqrt { 2 } } { 7 }$ 
C) ​ $\cos \frac { \theta } { 2 } = \frac { \sqrt { 2 } } { 10 }$ 
D) ​ $\cos \frac { \theta } { 2 } = 7$ 
E) ​ $\cos \frac { x } { 2 } = \frac { 7 \sqrt { 2 } } { 12 }$

Question 20

When two railroad tracks merge,the overlapping portions of the tracks are in the shapes of circular arcs (see figure).The radius of each arc r (in feet)and the angle θ are related by​ $$\frac { x } { 2 } = 2 r \sin ^ { 2 } \frac { \theta } { 2 }$$ ​ Write a formula for x in terms of cos θ.​   ​

Accepted Answer

A) ​ $x = r \left( 1 - \cos \frac { \theta } { 2 } \right)$ 
B) ​ $x = r ( 1 - \cos \theta )$ 
C)  $x = 2 r ( 1 - \cos \theta )$ ​ 
D)  $x = 2 r ( 1 + \cos \theta )$ ​ 
E)  $x = 2 r \left( 1 - \cos \frac { \theta } { 2 } \right)$ ​ 
A) ​ $x = r \left( 1 - \cos \frac { \theta } { 2 } \right)$ 
B) ​ $x = r ( 1 - \cos \theta )$ 
C)  $x = 2 r ( 1 - \cos \theta )$ ​ 
D)  $x = 2 r ( 1 + \cos \theta )$ ​ 
E)  $x = 2 r \left( 1 - \cos \frac { \theta } { 2 } \right)$ ​

Exam 35: Multiple Angle and Product to Sum Formulas

Use the half-angle formulas to simplify the expression. ​​ 1−cos⁡10x2\sqrt { \frac { 1 - \cos 10 x } { 2 } }21−cos10x​​ ​

Convert the expression.​ cos⁡4b−sin⁡4b\cos ^ { 4 } b - \sin ^ { 4 } bcos4b−sin4b ​

Use a double angle formula to rewrite the given expression. 10cos⁡2x−510 \cos ^ { 2 } x - 510cos2x−5

Convert the expression.​ tan⁡a2\tan \frac { a } { 2 }tan2a​ ​ ​

Use the product-to-sum formula to write the given product as a sum or difference.​ 12sin⁡π6cos⁡π612 \sin \frac { \pi } { 6 } \cos \frac { \pi } { 6 }12sin6π​cos6π​ ​

Use the sum-to-product formulas to rewrite the sum or difference as a product.​ sin⁡3θ−sin⁡θ\sin 3 \theta - \sin \thetasin3θ−sinθ ​

Convert the expression.​ 3csc⁡2θ3 \csc 2 \theta3csc2θ ​

Find all solutions of the given equation in the interval [0,2π). ​​ 12sin⁡2x=cos⁡x\frac { 1 } { 2 } \sin 2 x = \cos x21​sin2x=cosx ​

Use the sum-to-product formulas to rewrite the sum or difference as a product. ​ Cos 3θ + cos 8θ ​

Use the product-to-sum formulas to rewrite the product as a sum or difference.​ 10cos⁡45∘cos⁡20∘10 \cos 45 ^ { \circ } \cos 20 ^ { \circ }10cos45∘cos20∘ ​

Use the sum-to-product formulas to rewrite the sum or difference as a product.​ cos⁡12θ+cos⁡8θ\cos 12 \theta + \cos 8 \thetacos12θ+cos8θ ​

Convert the expression.​ 7sec⁡2θ7 \sec 2 \theta7sec2θ ​

Use the sum-to-product formulas to select the sum or difference as a product.​ cos⁡(5ϕ+2π)+cos⁡5ϕ\cos ( 5 \phi + 2 \pi ) + \cos 5 \phicos(5ϕ+2π)+cos5ϕ ​

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

Use a double-angle formula to rewrite the expression. ​ 3 - 6 sin2 x ​

Convert the expression.​ cos⁡4α\cos 4 \alphacos4α ​

Use a double-angle formula to rewrite the expression. ​ 2 sin2 x - 1 ​

​Use the figure below to find the exact value of the given trigonometric expression. sin⁡θ2\sin \frac { \theta } { 2 }sin2θ​ ​​ 10 24 (figure not necessarily to scale) ​

Use the figure below to find the exact value of the given trigonometric expression. cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​

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

Use the half-angle formulas to simplify the expression. $\sqrt { \frac { 1 - \cos 10 x } { 2 } }$

Convert the expression. $\cos ^ { 4 } b - \sin ^ { 4 } b$

Use a double angle formula to rewrite the given expression. $10 \cos ^ { 2 } x - 5$

Convert the expression. $\tan \frac { a } { 2 }$

Use the product-to-sum formula to write the given product as a sum or difference. $12 \sin \frac { \pi } { 6 } \cos \frac { \pi } { 6 }$

Use the sum-to-product formulas to rewrite the sum or difference as a product. $\sin 3 \theta - \sin \theta$

Convert the expression. $3 \csc 2 \theta$

Find all solutions of the given equation in the interval [0,2π). $\frac { 1 } { 2 } \sin 2 x = \cos x$

Use the sum-to-product formulas to rewrite the sum or difference as a product. Cos 3θ + cos 8θ

Use the product-to-sum formulas to rewrite the product as a sum or difference. $10 \cos 45 ^ { \circ } \cos 20 ^ { \circ }$

Use the sum-to-product formulas to rewrite the sum or difference as a product. $\cos 12 \theta + \cos 8 \theta$

Convert the expression. $7 \sec 2 \theta$

Use the sum-to-product formulas to select the sum or difference as a product. $\cos ( 5 \phi + 2 \pi ) + \cos 5 \phi$

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

Use a double-angle formula to rewrite the expression. 3 - 6 sin²x

Convert the expression. $\cos 4 \alpha$

Use a double-angle formula to rewrite the expression. 2 sin²x - 1

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