Half Angle Identities: Step-by-Step Guide
Half angle identities let you find the trig functions of x/2 using cos(x). sin(x/2) = plus or minus the square root of ((1 - cos x) / 2). cos(x/2) = plus or minus the square root of ((1 + cos x) / 2). tan(x/2) = (1 - cos x) / sin x = sin x / (1 + cos x). The plus or minus depends on the quadrant that x/2 is in.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Find the quadrant of x/2. Cut the angle in half and decide which quadrant x/2 is in.\[x \in [0, 2\pi) \Rightarrow \tfrac{x}{2} \in [0, \pi)\]
- Find cos(x). Use the given information or a Pythagorean identity to find cos(x).\[\cos x = \pm\sqrt{1 - \sin^2 x}\]
- Use the half-angle identity. Put cos(x) into the sin, cos, or tan formula.\[\sin\tfrac{x}{2} = \pm\sqrt{\tfrac{1 - \cos x}{2}}\]
- Choose the sign. Use the quadrant of x/2 from step 1.\[\text{Quadrant of } \tfrac{x}{2}\Rightarrow\text{sign}\]
- Simplify. Simplify any roots and rationalize the bottom if needed.\[\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\]
Worked example
Find sin(pi/12) using the half-angle identity.
Step 1: pi / 12 is half of pi / 6, so let x = pi / 6. x/2 = pi / 12 is in Quadrant I, so sine is positive.
Step 2: cos(pi / 6) = root 3 / 2.
Step 3: sin(pi / 12) = plus or minus the square root of ((1 - root 3 / 2) / 2) = plus or minus the square root of ((2 - root 3) / 4) = plus or minus (the square root of (2 - root 3)) / 2.
Step 4: Quadrant I means positive: sin(pi / 12) = (the square root of (2 - root 3)) / 2.
Step 5: Check: the square root of (2 - 1.732) / 2 = the square root of 0.268 / 2 = 0.5176 / 2 = 0.2588, which is sin(15 degrees). It works.
The answer is sin(pi / 12) = (the square root of (2 - root 3)) / 2.
Tips and common mistakes
- Find the quadrant of x/2 BEFORE you choose the plus or minus.
- tan(x/2) has two forms with no plus or minus: (1 - cos x) / sin x and sin x / (1 + cos x). Use whichever works.
- Half-angle identities are useful for integrating the square root of (1 +/- cos x) later.
- Power reduction is a special case: sin^2(x) = (1 - cos 2x) / 2.
- If x is between 0 and 2 pi, then x/2 is between 0 and pi, so sin(x/2) is always positive or zero.
- Check with a calculator. Half-angle answers can look strange but still be right.
- Sometimes you get a root inside a root. These can sometimes be simplified.
- Half-angle is the reverse of double-angle: cos(x) = 1 - 2 sin^2(x/2).