Half Angle Formulas

Run the double angle formula backwards.

12 problems · about 21 minutes · F-TF.C.9

What this lesson teaches

The student derives and applies the half angle formulas for sine and cosine.

Straightforward practice. Get the method working first.

5 problems

cos A = 0.6. What is sin²(A/2)?
Answer 0.2
Why 0.2.
Which angle decides the sign of sin(A/2)?
Answer The half angle, A/2.
Why The half angle.
cos A = 0. What is sin²(A/2)?
Answer 0.5
Why (1 − 0) ÷ 2.
cos A = 0.8. What is sin²(A/2)?
Answer 0.1
Why (1 − 0.8) ÷ 2.
cos A = 0.8. What is cos²(A/2)?
Answer 0.9
Why (1 + 0.8) ÷ 2.

The same ideas with more to keep track of.

3 problems

sin²(A/2) = 0.1 and cos²(A/2) = 0.9. What do they total?
Answer 1
Why The Pythagorean identity has to hold.
cos A = −1. What is sin²(A/2)?
Answer 1
Why (1 + 1) ÷ 2.
A = 90°. What is cos²(A/2), given cos 90° = 0?
Answer 0.5
Why (1 + 0) ÷ 2.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the derivation of the half angle formula in order.
Answer 1. Begin with cos 2θ = 1 − 2sin²θ. 2. Rearrange to sin²θ = (1 − cos 2θ) ÷ 2. 3. Write A in place of 2θ, so θ becomes A ÷ 2. 4. Take the square root and choose the sign from the quadrant.
Why The renaming happens after the rearranging.
cos A = 0.5, and A/2 lies in quadrant I. What is sin(A/2)?
Answer 0.5
Why sin²(A/2) = 0.25, then take the positive root.
The Half: cos A = 0.5. What is sin²(A/2)?
Answer 0.25
Why 0.25.
The Sign: A = 300°, so A/2 = 150°. Is sin(A/2) positive or negative?
Answer Positive, because 150° is in quadrant II.
Why Positive. Sine is positive in quadrant II, where 150° lies.