When should I use the half angle formula instead of just calculating the angle on a calculator?

Half angle formulas give you exact algebraic answers (with radicals) when calculators only give decimals. For homework, tests, or any work requiring exact values, these formulas are essential. Also, many equations and identities only simplify nicely if you work with the exact radical forms.

Why does the half angle formula for cosine use a plus sign but sine uses a minus sign?

Both formulas come from rearranging the double angle formulas \cos(2\alpha) = 2\cos^2(\alpha) - 1 = 1 - 2\sin^2(\alpha). When you solve for \sin^2(\alpha) you get the minus, and when you solve for \cos^2(\alpha) you get the plus. Deriving them yourself once will make the signs stick.

How do I know which root to pick if the formula has \pm?

Find the range of the half angle \theta/2 based on the given range of \theta. If \theta/2 falls in a quadrant where sine (or cosine) is positive, pick the + root; if negative, pick the - root. Writing out which quadrant the half angle is in takes 10 seconds and prevents most sign errors.

Can I derive the half angle formulas from other formulas I already know?

Yes! Start with \cos(2\alpha) = 1 - 2\sin^2(\alpha), rearrange to \sin^2(\alpha) = \frac{1 - \cos(2\alpha)}{2}, take the square root, then substitute \theta = 2\alpha to get the half angle form. This derivation takes a minute and is far more reliable than trying to memorize formulas.

Half Angle Formulas

Half angle formulas express trigonometric functions of half an angle in terms of the full angle, helping solve equations and simplify expressions.

\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}

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What each symbol means

What Half Angle Formulas takes

Symbol	Meaning
$\theta$	The full angle, measured in radians or degrees, whose cosine or other trigonometric value you know; confusing it with the half angle $\theta/2$ will make your formula give the wrong result.

When to use it

When you know a trigonometric function value of an angle and need to find the exact trig value of exactly half that angle.

Level

Usually taught in: Pre-Calculus

Worked examples

1. Find sine of a half angle with a decimal input

Problem

Given that

\cos(\theta) = 0.8

and

0 < \theta < \pi

, find

\sin(\theta/2)

$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$
Apply the half angle formula for sine.
$\text{Since } 0 < \theta < \pi \text{, we have } 0 < \frac{\theta}{2} < \frac{\pi}{2}$
This means $\theta/2$ is in the first quadrant, where sine is always positive.
$\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - 0.8}{2}}$
Substitute $\cos(\theta) = 0.8$ and choose the positive root.
$= \sqrt{\frac{0.2}{2}}$
Simplify the numerator: $1 - 0.8 = 0.2$ .
$= \sqrt{0.1}$
Divide: $0.2 / 2 = 0.1$ .
$= \sqrt{\frac{1}{10}} = \frac{\sqrt{10}}{10}$
Express as a fraction and rationalize the denominator.

Answer: $\sin\left(\frac{\theta}{2}\right) = \frac{\sqrt{10}}{10} \approx 0.316$

The half angle formula lets us find the exact sine value when we know only the cosine of the full angle. The key step is checking the domain of $\theta/2$ to confirm we need the positive root, since first-quadrant angles have positive sine values.

2. Find sine of a half angle with negative cosine and determine the correct sign

Problem

Given

\cos(\alpha) = -\tfrac{1}{2}

and

\tfrac{\pi}{2} < \alpha < \pi

, find

\sin(\alpha/2)

$\text{Since } \frac{\pi}{2} < \alpha < \pi, \text{ we have } \frac{\pi}{4} < \frac{\alpha}{2} < \frac{\pi}{2}$
Divide all parts of the inequality by 2 to find the range of the half angle.
$\text{This range is in the first quadrant, so } \sin\left(\frac{\alpha}{2}\right) > 0$
First quadrant means both sine and cosine are positive, so we use the positive root.
$\sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \cos(\alpha)}{2}}$
Apply the half angle formula with the positive root.
$= \sqrt{\frac{1 - \left(-\frac{1}{2}\right)}{2}}$
Substitute $\cos(\alpha) = -1/2$ into the formula.
$= \sqrt{\frac{1 + \frac{1}{2}}{2}}$
Simplify $1 - (-1/2) = 1 + 1/2$ .
$= \sqrt{\frac{\frac{3}{2}}{2}} = \sqrt{\frac{3}{4}}$
Simplify: $1 + 1/2 = 3/2$ , then $(3/2) / 2 = 3/4$ .
$= \frac{\sqrt{3}}{2}$
Take the square root: $\sqrt{3/4} = \sqrt{3}/\sqrt{4} = \sqrt{3}/2$ .

Answer: $\sin\left(\frac{\alpha}{2}\right) = \frac{\sqrt{3}}{2}$

This example shows why checking the domain is critical: even though the formula has a $\pm$ sign, the range of $\alpha/2$ forces us to pick the positive root. The result is one of the standard values you see on the unit circle.

3. Find cosine of a half angle when given sine of the full angle

Problem

Given

\sin(\theta) = \tfrac{3}{5}

with

0 < \theta < \tfrac{\pi}{2}

, find

\cos(\theta/2)

$\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{3}{5}\right)^2$
Use the Pythagorean identity to find $\cos(\theta)$ from $\sin(\theta)$ .
$= 1 - \frac{9}{25} = \frac{16}{25}$
Calculate: $1 = 25/25$ , so $25/25 - 9/25 = 16/25$ .
$\cos(\theta) = \frac{4}{5}$
Since $0 < \theta < \pi/2$ , angle $\theta$ is in the first quadrant where cosine is positive, so take the positive square root.
$\text{Since } 0 < \theta < \frac{\pi}{2}, \text{ we have } 0 < \frac{\theta}{2} < \frac{\pi}{4}$
The half angle is also in the first quadrant, so we use the positive root of the half angle formula.
$\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}}$
Apply the half angle formula for cosine (note: this formula has a $+$ inside, unlike the sine formula).
$= \sqrt{\frac{1 + \frac{4}{5}}{2}}$
Substitute $\cos(\theta) = 4/5$ into the formula.
$= \sqrt{\frac{\frac{9}{5}}{2}} = \sqrt{\frac{9}{10}}$
Simplify: $1 + 4/5 = 9/5$ , then $(9/5) / 2 = 9/10$ .
$= \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10}$
Take the square root and rationalize: $\sqrt{9/10} = 3/\sqrt{10}$ , multiply by $\sqrt{10}/\sqrt{10}$ .

Answer: $\cos\left(\frac{\theta}{2}\right) = \frac{3\sqrt{10}}{10}$

This problem chains two tools: the Pythagorean Identity to recover a missing trig value, then the half angle formula. Notice the half angle formula for cosine uses $1 + \cos(\theta)$ instead of $1 - \cos(\theta)$ , which is a common place to make errors.

Common mistakes

Where Half Angle Formulas usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\sin(\theta/2) = \sqrt{\frac{1 - \cos(\theta)}{2}}$ without the $\pm$ sign, or always choosing the positive root.	The half angle could lie in any quadrant depending on where $\theta$ itself is, and sine is negative in the third and fourth quadrants, so the sign of the answer depends on the domain of $\theta/2$ .	Always write the $\pm$ sign, then determine which quadrant $\theta/2$ falls into to decide whether to use $+$ or $-$ .
Using $\sin(\theta) = \sqrt{\frac{1 - \cos(\theta)}{2}}$ to find $\sin(\theta)$ itself instead of $\sin(\theta/2)$ .	The half angle formula finds the trig value of half the given angle, not the angle itself; this confuses which angle you are solving for.	Read the problem carefully: if you know $\cos(\theta)$ and need to find $\sin(\theta/2)$ , use the half angle formula. If you know $\cos(\theta)$ and need $\sin(\theta)$ , use the Pythagorean identity instead.
Writing $\cos(\theta/2) = \sqrt{\frac{1 - \cos(\theta)}{2}}$ when looking for the cosine of the half angle.	The half angle formula for cosine has a $+$ in the numerator, not a $-$ : $\cos(\theta/2) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$ .	Memorize both formulas side by side: sine uses $1 - \cos(\theta)$ while cosine uses $1 + \cos(\theta)$ , or derive them from the double angle formulas to remember which is which.

Tips and when to use something else

Always determine the quadrant of the half angle before choosing the sign: is $\theta/2$ in a quadrant where the trig function should be positive or negative?
The formula for tangent of a half angle, $\tan(\theta/2) = \frac{\sin(\theta)}{1 + \cos(\theta)}$ , has no $\pm$ sign and is sometimes easier to use; try it if the sine formula feels complicated.
These formulas are the inverse of Double Angle Formulas: if you know $\sin(2\alpha)$ , you can find $\sin(\alpha)$ by treating $\alpha$ as the half angle, so the two formula families work together.
When solving trigonometric equations, half angle formulas let you rewrite equations in terms of smaller angles, which often leads to angles with standard values that you can solve exactly rather than with a calculator.

Related formulas

Double Angle FormulasDouble angle formulas express trig functions of 2θ in terms of single angles, letting you simplify complex expressions and solve trigonometric equations.Pythagorean IdentityThe Pythagorean Identity states that sine squared plus cosine squared always equals one, helping you simplify trigonometric expressions and solve equations.Unit CircleThe unit circle is a fundamental tool that shows how angles map to sine and cosine coordinates: the point (cos θ, sin θ) always lies on x² + y² = 1.Solving Trigonometric EquationsFind all angles that produce a specific sine value by using inverse sine and the periodic nature of trigonometric functions.

Frequently asked questions

When should I use the half angle formula instead of just calculating the angle on a calculator?: Half angle formulas give you exact algebraic answers (with radicals) when calculators only give decimals. For homework, tests, or any work requiring exact values, these formulas are essential. Also, many equations and identities only simplify nicely if you work with the exact radical forms.
Why does the half angle formula for cosine use a plus sign but sine uses a minus sign?: Both formulas come from rearranging the double angle formulas $\cos(2\alpha) = 2\cos^2(\alpha) - 1 = 1 - 2\sin^2(\alpha)$ . When you solve for $\sin^2(\alpha)$ you get the minus, and when you solve for $\cos^2(\alpha)$ you get the plus. Deriving them yourself once will make the signs stick.
How do I know which root to pick if the formula has $\pm$ ?: Find the range of the half angle $\theta/2$ based on the given range of $\theta$ . If $\theta/2$ falls in a quadrant where sine (or cosine) is positive, pick the $+$ root; if negative, pick the $-$ root. Writing out which quadrant the half angle is in takes 10 seconds and prevents most sign errors.
Can I derive the half angle formulas from other formulas I already know?: Yes! Start with $\cos(2\alpha) = 1 - 2\sin^2(\alpha)$ , rearrange to $\sin^2(\alpha) = \frac{1 - \cos(2\alpha)}{2}$ , take the square root, then substitute $\theta = 2\alpha$ to get the half angle form. This derivation takes a minute and is far more reliable than trying to memorize formulas.

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Reviewed 2026-09-18