Double Angle Identities

Show your work for each problem.

1.
What is the double angle formula for sine?
1. a)
2. b)
3. c)
4. d)
Answer:
The double angle formula for sine is . This comes from the sum formula with .
2.
Which of the following is NOT a valid form of ?
1. a)
2. b)
3. c)
4. d)
Answer:
is incorrect. The three valid forms are: , , and .
3.
Calculate using the double angle formula with . Give your answer as a decimal rounded to 3 decimal places.
Answer: 0.866
4.
Simplify . Give your exact answer as a decimal rounded to 3 decimal places.
Answer: 0.707
5.
If and is in Quadrant I, what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
(positive in Q1). Then .
6.
If and is in Quadrant IV, find . Give your answer as a fraction in the form a/b.
Answer: -7/25
7.
What is the double angle formula for tangent?
1. a)
2. b)
3. c)
4. d)
Answer:
The double angle formula for tangent is . Note the minus sign in the denominator!
8.
Find if and is in Quadrant I.
Answer: 24/25
1. From , if opp = 4 and adj = 3, what is the hypotenuse? 5
2. What is ? 4/5
3. What is ? 3/5
4. Calculate 24/25
9.
Derive the formula from .
Answer: 1 - 2sin^2(theta)
1. Start with: . What identity relates and ? sin^2 + cos^2 = 1
2. Solve for from the Pythagorean identity. cos^2 = 1 - sin^2
3. Substitute into the original formula. What do you get? (1 - sin^2) - sin^2
4. Simplify the expression. 1 - 2sin^2
10.
Calculate if . Give your answer as a fraction in the form a/b.
Answer: -4/3
11.
Solve for .
Answer: 0, pi/3, pi, 5pi/3
1. Apply the double angle formula to the left side. 2sin(x)cos(x) = sin(x)
2. Rearrange to factor. sin(x)(2cos(x) - 1) = 0
3. What are the two cases from the factored form? sin(x) = 0 or cos(x) = 1/2
4. Find all solutions for in . 0, pi
5. Find all solutions for in . pi/3, 5pi/3
12.
Find if and is in Quadrant III.
Answer: 161/289
1. Which form of is most convenient when you know only ? 1 - 2sin^2
2. What is ? 64/289
3. What is ? 128/289
4. Calculate . 161/289
13.
At what angle does the projectile range formula give maximum range?
1. a)
2. b)
3. c)
4. d)
Answer:
Maximum range occurs when , which means , so .
14.
Simplify using the double angle identity. Write your answer in terms of .
Answer: cos(2theta)
is one of the three equivalent forms of the cosine double angle identity.
15.
Use the power-reducing identity to express in terms of .
Answer: (1 - cos(2x))/2
1. Start with the double angle identity: . Solve for . 2sin^2(x) = 1 - cos(2x)
2. Divide both sides by 2 to get . sin^2(x) = (1 - cos(2x))/2