How do I know whether to use + or − in a half-angle formula?

Determine which quadrant the half-angle (not the full angle) falls in. If the trig function is positive in that quadrant, use +; if negative, use −. For example, if θ/2 is in the second quadrant, sine is positive but cosine is negative.

Half-Angle Formulas — Definition, Formula & Examples

Half-angle formulas are trigonometric identities that let you find the sine, cosine, or tangent of half an angle when you know the trigonometric values of the full angle. They are derived from the double-angle identities by solving for the half-angle expression.

Given an angle $\theta$ , the half-angle identities state: $\sin\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$ , $\cos\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$ , and $\tan\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$ , where the sign of each expression is determined by the quadrant in which $\frac{\theta}{2}$ lies.

Key Formula

\sin\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}, \quad \cos\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}, \quad \tan\!\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta}

Where:

$\theta$ = The full angle whose half-angle trig value you want to find
$\pm$ = Sign chosen based on the quadrant of θ/2

How It Works

You use half-angle formulas when you need the exact value of a trig function at an angle that is half of a known reference angle. For instance, since you know the exact cosine of

60°

, you can find the exact sine or cosine of

30°

—or more usefully, angles like

15°

from the known value of

\cos 30°

. To apply the formula, substitute the full angle

\theta

into the right-hand side, simplify the expression under the radical, and then choose the correct sign (

+

-

) based on which quadrant

\frac{\theta}{2}

falls in. An equivalent and often more convenient form for tangent is

\tan\!\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}

, which avoids the

\pm

ambiguity entirely.

Worked Example

Problem: Find the exact value of cos 15° using a half-angle formula.

Identify the full angle: Since 15° is half of 30°, set θ = 30°.

\cos 15° = \cos\!\left(\frac{30°}{2}\right)

Apply the half-angle formula: Use the cosine half-angle identity. Because 15° is in the first quadrant, cosine is positive, so choose the + sign.

\cos 15° = +\sqrt{\frac{1 + \cos 30°}{2}}

Substitute the known value: Recall that cos 30° = √3/2.

\cos 15° = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}}

Simplify: Take the square root of the denominator.

\cos 15° = \frac{\sqrt{2 + \sqrt{3}}}{2}

Answer:

\cos 15° = \dfrac{\sqrt{2 + \sqrt{3}}}{2} \approx 0.9659

Another Example

Problem: Find the exact value of sin 105° using a half-angle formula.

Identify the full angle: Since 105° is half of 210°, set θ = 210°.

\sin 105° = \sin\!\left(\frac{210°}{2}\right)

Choose the sign: 105° is in the second quadrant, where sine is positive. Choose the + sign.

\sin 105° = +\sqrt{\frac{1 - \cos 210°}{2}}

Substitute and simplify: cos 210° = −√3/2.

\sin 105° = \sqrt{\frac{1 - \left(-\frac{\sqrt{3}}{2}\right)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2+\sqrt{3}}}{2}

Answer:

\sin 105° = \dfrac{\sqrt{2+\sqrt{3}}}{2} \approx 0.9659

Why It Matters

Half-angle formulas appear throughout precalculus and AP Calculus when you need exact trigonometric values or must simplify integrals involving even powers of sine and cosine. In calculus, the identity

\cos^2 x = \frac{1+\cos 2x}{2}

(a rearranged half-angle formula) is essential for evaluating integrals like

\int \cos^2 x\, dx

. Engineers also rely on these identities in signal processing when decomposing waveforms.

Common Mistakes

Mistake: Forgetting the ± sign or always choosing +

Correction: The sign depends on the quadrant of θ/2, not θ. Determine where the half-angle lies and use the sign rules for that quadrant.

Mistake: Confusing the half-angle and double-angle formulas

Correction: The half-angle formula for cosine has 1 + cos θ under the radical, while the double-angle formula states cos 2θ = 2cos²θ − 1. They are algebraically related but serve opposite purposes—one halves the angle, the other doubles it.