Step by Step
1
The half-angle formula for sine
sin(A/2) = ±√((1−cos A)/2). Notice cosine appears with a MINUS sign inside the square root for sine's half-angle formula.
2
The half-angle formula for cosine
cos(A/2) = ±√((1+cos A)/2). Notice cosine appears with a PLUS sign inside the square root for cosine's half-angle formula — the opposite of sine's.
3
Determining the correct sign
The ± sign in front must be resolved by checking which quadrant the HALF angle A/2 itself falls in — not the quadrant of the original angle A. This is a subtle but critical distinction.
4
The related power-reducing formulas
sin²(x) = (1−cos(2x))/2, and cos²(x) = (1+cos(2x))/2. These are essentially the half-angle formulas rearranged, and they're especially useful for integrating powers of trig functions in calculus.
Applied Walkthrough
1
Find sin(15°) using the half-angle formula, treating 15° as half of 30°.
2
Apply the formula: sin(15°) = sin(30°/2) = ±√((1−cos(30°))/2) = ±√((1−√3/2)/2).
3
To resolve the sign, check the quadrant of the HALF angle itself — 15° — not the original 30°. Since 15° is in Quadrant I, where sine is positive, the correct sign is positive.
4
So sin(15°) = +√((1−√3/2)/2) — a fully simplified exact value obtained without a calculator.
Exam Application
Exams test whether you can correctly recall which cosine sign (plus or minus) belongs inside sine's versus cosine's half-angle formula, and specifically whether you check the quadrant of the HALF angle (not the original angle) to resolve the ± sign.
⚠ Common Trap
The single most common and specifically tested trap is checking the quadrant of the original angle A instead of the half angle A/2 when resolving the sign — always base your sign decision on where A/2 actually falls.
✓ Quick Self-Check
1. What is the half-angle formula for sine?
sin(A/2) = ±√((1−cos A)/2).
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2. What is the half-angle formula for cosine?
cos(A/2) = ±√((1+cos A)/2).
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3. Which angle's quadrant determines the correct sign in the half-angle formulas — A or A/2?
A/2, the half angle itself — not the original angle A.
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4. Find sin(15°) using the half-angle formula.
√((1−√3/2)/2), with a positive sign since 15° is in Quadrant I.
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5. What are the power-reducing formulas, and where do they come from?
sin²(x) = (1−cos(2x))/2 and cos²(x) = (1+cos(2x))/2 — rearranged versions of the half-angle formulas, useful for integrating powers of trig functions.
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