Half Angle Identities — Formulas, Table & Examples

Half Angle Identities
Half Number Identities

Trig identities that show how to find the sine, cosine, or tangent of half a given angle.

Half Angle Identities

cos(x/2) = ±√((1 + cos x) / 2) or

or or

See also

Key Formula

\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}}

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

Where:

$\theta$ = The full angle whose half you want to evaluate
$\pm$ = Choose + or − based on the quadrant where θ/2 lies

Worked Example

Problem: Find the exact value of sin(15°) using a half angle identity.

Step 1: Recognize that 15° is half of 30°, so set θ = 30°.

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

Step 2: Apply the half angle identity for sine.

\sin\!\left(\frac{30°}{2}\right) = \pm\sqrt{\frac{1 - \cos 30°}{2}}

Step 3: Substitute cos 30° = √3/2.

= \pm\sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \pm\sqrt{\frac{\frac{2 - \sqrt{3}}{2}}{2}} = \pm\sqrt{\frac{2 - \sqrt{3}}{4}}

Step 4: Since 15° is in the first quadrant, sine is positive. Choose the positive root.

\sin(15°) = \frac{\sqrt{2 - \sqrt{3}}}{2}

Step 5: This simplifies to the equivalent well-known form.

\sin(15°) = \frac{\sqrt{6} - \sqrt{2}}{4} \approx 0.2588

Answer: sin(15°) = (√6 − √2)/4 ≈ 0.2588

Another Example

This example shows how the ± sign matters: because 105° lies in the second quadrant where cosine is negative, we must choose the negative root, unlike the first example where the positive root was used.

Problem: Find the exact value of cos(105°) using a half angle identity.

Step 1: Recognize that 105° is half of 210°, so set θ = 210°.

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

Step 2: Apply the half angle identity for cosine.

\cos\!\left(\frac{210°}{2}\right) = \pm\sqrt{\frac{1 + \cos 210°}{2}}

Step 3: Substitute cos 210° = −√3/2.

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

Step 4: Since 105° is in the second quadrant, cosine is negative. Choose the negative root.

\cos(105°) = -\frac{\sqrt{2 - \sqrt{3}}}{2} = -\frac{\sqrt{6} - \sqrt{2}}{4} \approx -0.2588

Answer: cos(105°) = −(√6 − √2)/4 ≈ −0.2588

Frequently Asked Questions

How do you know whether to use + or − in the half angle identity?

The sign depends on the quadrant where the half angle θ/2 lies, not the full angle θ. If the function (sine, cosine, or tangent) 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.

What is the difference between half angle and double angle identities?

Double angle identities express functions of 2θ in terms of functions of θ (e.g., cos 2θ = 2cos²θ − 1). Half angle identities do the reverse: they express functions of θ/2 in terms of functions of θ. In fact, half angle identities are derived by solving the double angle formulas for the half angle.

Why does the tangent half angle identity have alternate forms without a ± sign?

The tangent half angle can be rewritten as tan(θ/2) = sin θ/(1 + cos θ) or tan(θ/2) = (1 − cos θ)/sin θ. These forms avoid the ± ambiguity because the signs of sin θ and the denominators automatically produce the correct sign for the result. Many students prefer these forms for that reason.

Half Angle Identities vs. Double Angle Identities

	Half Angle Identities	Double Angle Identities
Purpose	Find trig values of θ/2 from θ	Find trig values of 2θ from θ
Sine formula	sin(θ/2) = ±√((1 − cos θ)/2)	sin(2θ) = 2 sin θ cos θ
Cosine formula	cos(θ/2) = ±√((1 + cos θ)/2)	cos(2θ) = cos²θ − sin²θ
Contains ± sign?	Yes — depends on quadrant of θ/2	No
Typical use	Finding exact values like sin 15° or cos 22.5°	Simplifying expressions or solving equations involving 2θ

Why It Matters

Half angle identities appear frequently in precalculus and calculus courses, especially when you need exact values for angles that are not standard reference angles (like 15°, 22.5°, or π/8). In calculus, these identities are essential for integrating expressions like sin²x or cos²x, since they let you rewrite powers of trig functions in terms of first-power expressions. They also appear in physics and engineering when analyzing wave interference and signal processing.

Common Mistakes

Mistake: Choosing the wrong sign (+ or −) based on the quadrant of the full angle θ instead of the half angle θ/2.

Correction: Always determine which quadrant θ/2 falls in, then decide the sign based on whether sine, cosine, or tangent is positive or negative in that quadrant. For instance, if θ = 210°, then θ/2 = 105° is in quadrant II, where cosine is negative.

Mistake: Forgetting the square root in the sine or cosine half angle formulas, writing sin(θ/2) = (1 − cos θ)/2 instead of the correct form.

Correction: The correct formula is sin(θ/2) = ±√((1 − cos θ)/2). The expression without the square root, (1 − cos θ)/2, actually equals sin²(θ/2), not sin(θ/2). Always include the radical.

Related Terms

Trig Identities — Broad category containing half angle identities
Double Angle Identities — Source identities from which half angle formulas are derived
Sine — One of the three functions with a half angle formula
Cosine — One of the three functions with a half angle formula
Tangent — Has alternate half angle forms without ± sign
Angle — The geometric quantity being halved