Half Angle Calculator

Understanding Half Angle Formulas in Trigonometry

The half angle formulas in trigonometry allow you to evaluate the trigonometric functions of an angle divided by two, written as $\theta/2$, based on the trigonometric values of the original angle $\theta$. Derived directly from the double-angle identities for cosine, these formulas are indispensable for simplifying algebraic expressions, solving calculus integrals, evaluating exact trigonometric ratios for non-standard angles like $15^\circ$ or $22.5^\circ$, and analyzing waveforms.

The Core Half Angle Identities

The primary half-angle formulas relate $\sin(\theta/2)$, $\cos(\theta/2)$, and $\tan(\theta/2)$ to $\cos(\theta)$:

Function	Primary Half Angle Formula	Alternative Forms
$\sin(\theta/2)$	$\pm \sqrt{\frac{1 - \cos\theta}{2}}$	Sign depends on the quadrant of $\theta/2$
$\cos(\theta/2)$	$\pm \sqrt{\frac{1 + \cos\theta}{2}}$	Sign depends on the quadrant of $\theta/2$
$\tan(\theta/2)$	$\pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$	$\frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$

How to Choose the Correct Sign ($\pm$)

The plus or minus sign in front of the radical does not mean there are two answers for a specific angle. Instead, it indicates that you must determine the sign based on the quadrant in which the half-angle $\theta/2$ terminates:

Quadrant I ($0^\circ < \theta/2 < 90^\circ$): $\sin(\theta/2) > 0$, $\cos(\theta/2) > 0$, $\tan(\theta/2) > 0$.
Quadrant II ($90^\circ < \theta/2 < 180^\circ$): $\sin(\theta/2) > 0$, $\cos(\theta/2) < 0$, $\tan(\theta/2) < 0$.
Quadrant III ($180^\circ < \theta/2 < 270^\circ$): $\sin(\theta/2) < 0$, $\cos(\theta/2) < 0$, $\tan(\theta/2) > 0$.
Quadrant IV ($270^\circ < \theta/2 < 360^\circ$): $\sin(\theta/2) < 0$, $\cos(\theta/2) > 0$, $\tan(\theta/2) < 0$.

Step-by-Step Example: Finding Exact Value of $\sin(22.5^\circ)$

Suppose you want to compute the exact value of $\sin(22.5^\circ)$. Notice that $22.5^\circ = 45^\circ / 2$, so $\theta = 45^\circ$.

Evaluate $\cos(45^\circ) = \frac{\sqrt{2}}{2}$.
Identify the quadrant of $22.5^\circ$: Since $22.5^\circ$ lies in Quadrant I, $\sin(22.5^\circ)$ is positive.
Apply the sine half angle formula: $$\sin(22.5^\circ) = \sqrt{\frac{1 - \cos(45^\circ)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \approx 0.382683$$

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Frequently Asked Questions

What is the difference between half-angle and double-angle formulas?

Double-angle formulas express trigonometric functions of $2\theta$ in terms of functions of $\theta$, such as $\sin(2\theta) = 2\sin\theta\cos\theta$. In contrast, half-angle formulas express trigonometric functions of $\theta/2$ using functions of the original angle $\theta$, such as $\cos(\theta/2) = \pm\sqrt{(1+\cos\theta)/2}$.

Why does the tangent half-angle formula have versions without radicals?

The alternative tangent half-angle identities $\tan(\theta/2) = \frac{1-\cos\theta}{\sin\theta} = \frac{\sin\theta}{1+\cos\theta}$ eliminate the square root and ambiguity about the sign. The sign is automatically preserved by the signs of $\sin\theta$ and $1-\cos\theta$.

How do I find $\sec(\theta/2)$ or $\csc(\theta/2)$?

Once you calculate $\sin(\theta/2)$ and $\cos(\theta/2)$, you can easily find the reciprocal trigonometric functions by taking their inverses: $\csc(\theta/2) = 1/\sin(\theta/2)$, $\sec(\theta/2) = 1/\cos(\theta/2)$, and $\cot(\theta/2) = 1/\tan(\theta/2)$.

Can the half-angle formula be used for radians?

Yes. The formulas apply identically whether the input angle is in degrees or radians. For example, $\pi/8 = (\pi/4)/2$, so you can use $\theta = \pi/4$ in the formulas to evaluate exact trig values for $\pi/8$.

Function	Primary Half Angle Formula	Alternative Forms
\(\sin(\theta/2)\)	\(\pm \sqrt{\frac{1 - \cos\theta}{2}}\)	Sign depends on the quadrant of \(\theta/2\)
\(\cos(\theta/2)\)	\(\pm \sqrt{\frac{1 + \cos\theta}{2}}\)	Sign depends on the quadrant of \(\theta/2\)
\(\tan(\theta/2)\)	\(\pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}\)	\(\frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}\)

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Understanding Half Angle Formulas in Trigonometry

The Core Half Angle Identities

How to Choose the Correct Sign (\(\pm\))

Step-by-Step Example: Finding Exact Value of \(\sin(22.5^\circ)\)

Related Trigonometric Tools

Frequently Asked Questions

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