Half Angle Identities

Core Idea

Half angle identities express sin(A/2), cos(A/2), and tan(A/2) in terms of cos(A). They are derived by solving the double angle formulas for the half-angle. For example, sin(A/2) = +/- sqrt((1 - cos(A))/2). The sign depends on the quadrant of A/2. These identities are useful for finding exact values at angles like 15 or 22.5 degrees and appear in certain integration techniques.

How It's Best Learned

Derive from the double angle identity for cosine by replacing A with A/2. Practice determining the correct sign based on the quadrant of the half angle. Use them to find exact values that cannot be obtained from sum/difference identities alone.

Common Misconceptions

• Forgetting the +/- sign and not checking the quadrant.
• Confusing half-angle identities with simply dividing the full-angle value by 2.
• Mixing up which double angle form to start from for each half-angle identity.

Explainer

The double-angle identity for cosine comes in three forms, and the most useful for deriving half-angle identities is cos(2A) = 1 - 2sin²(A). To get a half-angle identity, substitute θ = 2A, so A = θ/2: cos(θ) = 1 - 2sin²(θ/2). Solving for sin(θ/2): sin²(θ/2) = (1 - cos θ)/2, so sin(θ/2) = ±√((1 - cos θ)/2). The parallel derivation from the form cos(2A) = 2cos²(A) - 1 gives cos(θ/2) = ±√((1 + cos θ)/2). Both follow in one algebraic step from the double-angle formulas you already know — there is nothing to memorize separately beyond recognizing which double-angle form to start from.

The ± sign is not optional or cosmetic — it carries essential information. The half-angle θ/2 lies in a specific quadrant, and that quadrant determines the sign of sin(θ/2) and cos(θ/2) independently. For example, to find sin(15°), write it as sin(30°/2). Since 15° lies in the first quadrant, sin(15°) > 0, so the + sign applies: sin(15°) = +√((1 - cos 30°)/2) = √((1 - √3/2)/2) = √((2 - √3)/4) = √(2 - √3)/2. This exact value cannot be reached using sum-or-difference identities because 15° does not decompose as a sum of standard angles in a useful way. Half-angle identities open up a new class of exact values, including 22.5°, 67.5°, and others that are halves of familiar angles.

For tan(θ/2), divide the half-angle formulas: sin(θ/2)/cos(θ/2) = √((1 - cos θ)/(1 + cos θ)). But there are two cleaner sign-free forms: multiplying numerator and denominator strategically (and using sin θ = 2sin(θ/2)cos(θ/2)) yields tan(θ/2) = sin θ/(1 + cos θ) and equivalently tan(θ/2) = (1 - cos θ)/sin θ. These are sign-free because the sign of sin θ and the sign of tan(θ/2) already agree automatically. These forms matter in calculus: the Weierstrass substitution t = tan(θ/2) converts any rational expression in sin θ and cos θ into a rational expression in t, enabling integration by substitution. The half-angle identities are the algebraic foundation for that technique.

The key habit: never apply a half-angle identity without determining the quadrant of θ/2 first. A wrong sign produces an answer that is the exact negative of the correct value — numerically plausible but wrong. Sketch the angle θ/2 on the unit circle, confirm which quadrant it occupies, and assign the sign before computing. This takes seconds and eliminates a persistent class of errors.