Half Angle Identities

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

1. Determine a coterminal representative for $\theta/2$ and its quadrant.
2. Select the identity involving the known trig value.
3. Choose the correct sign from the half-angle quadrant.
4. Substitute exactly and simplify radicals.
5. Check with a known unit-circle value or Pythagorean identity.

Worked example

Representations and interpretation

The identities come from solving double-angle formulas such as $\cos(2u)=1-2\sin^2u$ for $\sin u$. Solving a square introduces two signs, and the unit circle selects the appropriate one.

Reasoning about variations

Knowing $\cos\theta$ alone may not determine the quadrant of $\theta/2$ unless the range of $\theta$ is specified. Equivalent tangent half-angle formulas have different excluded denominators but agree on common domains.

Common mistakes

How to check your work

• Approximate the exact radical and compare with calculator in correct mode.
• Square sine/cosine half-angle values and verify their sum is $1$.
• Double the half-angle and check the source cosine value.

Practice

1. Given $\theta/2$ is in Quadrant I and $\cos\theta=1/2$, find $\sin(\theta/2)$.
2. Which information determines the $\pm$ sign?
3. Find $\cos45^\circ$ using half of $90^\circ$.

Answers and brief solutions

Synthesis and transfer

To obtain an exact value at $15^\circ$, halve a familiar $30^\circ$ angle and choose the positive branch from Quadrant I before simplifying the nested radical.

Using $\cos30^\circ=\sqrt3/2$, the identity gives $\sin15^\circ=\sqrt{(1-\sqrt3/2)/2}=\frac{\sqrt{2-\sqrt3}}2$. The positive root is chosen because $15^\circ$ lies in Quadrant I. Squaring the result and substituting into $2\sin^2(15^\circ)=1-\cos30^\circ$ verifies the radical without relying on a decimal. If the half-angle lay in another quadrant, the same radicand would determine magnitude but not sign. Since halving an angle does not simply halve its sine or cosine, the identity encodes a nonlinear geometric relationship rather than an arithmetic shortcut.

Related topics

• Double Angle Identities
• Trigonometric Identities
• Exact Trigonometric Values

Teaching and accessibility note