Half Angle Identities

Show your work for each problem.

1.
Which formula correctly represents ?
1. a)
2. b)
3. c)
4. d)
Answer:
The sine half-angle formula is . The second option is for cosine, the third is for tangent, and the fourth is simply wrong.
2.
To find using a half-angle formula, what angle should you use for ?
1. a)
2. b)
3. c)
4. d)
Answer:
Since , we need . We use the half-angle formula with because we know from the unit circle.
3.
The half-angle formula for sine has a sign. If , which sign should you use?
1. a) (positive)
2. b)Neither
3. c)Both are correct
4. d) (negative)
Answer: (positive)
is in Quadrant I (between and ), where sine is positive. Therefore, we use the positive sign.
4.
If , what is ?
Answer: 1/2
. This value appears in the numerator of the sine half-angle formula.
5.
What is calculated using the half-angle formula with ?
1. a)
2. b)
3. c)
4. d)
Answer:
. This verifies our known value!
6.
Using , calculate . Give your answer as a fraction with in the numerator.
Answer: (2+sqrt(2))/2
. This appears in the cosine half-angle formula.
7.
Find the exact value of using the half-angle formula.
Answer: sqrt(2+sqrt(2))/2
1. What angle should you use? (22.5° is half of which angle?) 45
2. What is ? sqrt(2)/2
3. Calculate (2+sqrt(2))/2
4. Divide by 2: (2+sqrt(2))/4
5. Take the square root (positive, since 22.5° is in QI) sqrt(2+sqrt(2))/2
8.
Which of these is a valid formula for ?
1. a)
2. b)
3. c)
4. d)
Answer:
The tangent half-angle formula is (or equivalently ). The other options are incorrect variations.
9.
Calculate using the half-angle formula. Express as where is an integer.
Answer: 1
. So .
10.
Find the exact value of .
Answer: sqrt(2-sqrt(3))/2
1. 15° is half of which angle? 30
2. What is ? sqrt(3)/2
3. Calculate (2-sqrt(3))/2
4. Divide by 2 to get the radicand (2-sqrt(3))/4
5. Take the square root (positive for QI) sqrt(2-sqrt(3))/2
11.
If , what is the sign of ?
1. a)Zero
2. b)Positive
3. c)Negative
4. d)Undefined
Answer: Negative
. The angle 120° is in Quadrant II, where cosine is negative. Therefore, we use the negative sign.
12.
Using half-angle formulas, find . Express your answer as and give the value of .
Answer: 2
. So , , .
13.
Find the exact value of .
Answer: sqrt(2)+1
1. 67.5° is half of which angle? 135
2. What is ? sqrt(2)/2
3. What is ? -sqrt(2)/2
4. Calculate 1+sqrt(2)/2
5. Apply and simplify sqrt(2)+1
14.
If and is in Quadrant I, find . Express as a fraction.
Answer: 7/25
From , we get . Using : , so , thus .
15.
Find using the half-angle formula.
Answer: sqrt(2+sqrt(3))/2
1. Convert to degrees 75
2. 75° is half of which angle? 150
3. What is ? -sqrt(3)/2
4. Calculate (2+sqrt(3))/2
5. Apply the formula and simplify (positive for QI) sqrt(2+sqrt(3))/2