The half angle formulas are not new. They are the double angle formula for cosine, solved for the thing inside the square.
Step 1: Let's Learn
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The derivation
From cos 2θ = 1 − 2sin²θ, rearranging gives sin²θ = (1 − cos 2θ) ÷ 2.
Renaming
Writing A for 2θ makes θ equal A ÷ 2, so sin²(A/2) = (1 − cos A) ÷ 2.
The sign is yours to choose
Taking a square root gives plus or minus. The quadrant of the half angle, not the original angle, decides which one is right.
What they are for
They produce exact values for angles like 22.5° that no special triangle reaches, and they turn squared terms into first powers for integration later.
Run the double angle formula backwards
From cos 2θ = 1 − 2sin²θ, solving for sin θ gives the half angle formula. Deriving it this way means you need only remember the double angle version, which is used far more often.
The sign must be chosen
Half angle formulas carry a ± because taking a square root loses the sign. Which sign to use is decided by the quadrant of the half angle, not by the original angle. Working that out is part of the answer.
Where they matter
The power-reducing forms — sin²θ = (1 − cos 2θ)/2 — are essential in integration, where an even power of sine or cosine is otherwise intractable. This is one of the places trigonometry pays off directly in calculus.
More exact values
Since 15° is half of 30°, the half angle formula gives sin 15° exactly. Combined with the compound angle formulas, the set of angles with exact known values becomes surprisingly rich.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Amara Find sin 15°
Amara wants an exact value for sin 15°, using cos 30° ≈ 0.866.
- Step 1
She notices 15° is half of 30°, so the half angle formula applies.
Step 4: Your Turn
Practice makes it stick.
The Half
Problem 1 of 2
cos A = 0.5. What is sin²(A/2)?
The Sign
Problem 2 of 2
A = 300°, so A/2 = 150°. Is sin(A/2) positive or negative?
Halve the Angle
1 of 8
cos A = 0. What is sin²(A/2)?
2 of 8
cos A = 0.8. What is sin²(A/2)?
3 of 8
cos A = 0.8. What is cos²(A/2)?
4 of 8
sin²(A/2) = 0.1 and cos²(A/2) = 0.9. What do they total?
5 of 8
cos A = −1. What is sin²(A/2)?
6 of 8
A = 90°. What is cos²(A/2), given cos 90° = 0?
7 of 8
Put the derivation of the half angle formula in order.
- 1Rearrange to sin²θ = (1 − cos 2θ) ÷ 2.
- 2Write A in place of 2θ, so θ becomes A ÷ 2.
- 3Take the square root and choose the sign from the quadrant.
- 4Begin with cos 2θ = 1 − 2sin²θ.
8 of 8
cos A = 0.5, and A/2 lies in quadrant I. What is sin(A/2)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
cos A = 0.6. What is sin²(A/2)?
Question 2 of 2
Which angle decides the sign of sin(A/2)?
What You Learned
- The half angle formulas come from rearranging cos 2θ.
- sin²(A/2) = (1 − cos A) ÷ 2, and cos²(A/2) = (1 + cos A) ÷ 2.
- The quadrant of the half angle chooses the sign.