Math101 · Independent Ontario learning library
Math101Half Angle Identities
Half-angle identities express trig values at $\theta/2$ using functions of $\theta$: $\sin(\theta/2)=\pm\sqrt{(1-\cos\theta)/2}$, $\cos(\theta/2)=\pm\sqrt{(1+\cos\theta)/2}$, and $\tan(\theta/2)=\sin\theta/(1+\cos\theta)=(1-\cos\theta)/\sin\theta$ where denominators are nonzero.
Half-angle identities produce exact values, solve equations, and support integration and radical simplification. Quadrant reasoning prevents sign ambiguity.
Intuition and core definition
Half-angle identities express trig values at $\theta/2$ using functions of $\theta$: $\sin(\theta/2)=\pm\sqrt{(1-\cos\theta)/2}$, $\cos(\theta/2)=\pm\sqrt{(1+\cos\theta)/2}$, and $\tan(\theta/2)=\sin\theta/(1+\cos\theta)=(1-\cos\theta)/\sin\theta$ where denominators are nonzero.
Notation, language, and conditions
The square-root sign in sine and cosine formulas carries $\pm$ because the identity gives a magnitude; the quadrant of $\theta/2$ determines sign. The principal square root alone is nonnegative. Tangent also has $\pm\sqrt{(1-\cos\theta)/(1+\cos\theta)}$ with quadrant selection.
Why this idea matters
Half-angle identities recover trigonometric values at a halved angle, with quadrant information deciding the square-root sign.
A dependable method
- Determine a coterminal representative for $\theta/2$ and its quadrant.
- Select the identity involving the known trig value.
- Choose the correct sign from the half-angle quadrant.
- Substitute exactly and simplify radicals.
- Check with a known unit-circle value or Pythagorean identity.
