Half Angle Formulas

Running the Machine Backwards

Double angle took us from $\theta$ up to $2\theta$. Half angle goes the other way: from a full angle $x$ down to $\frac{x}{2}$.

And we do not need anything new. The half angle formulas come straight out of rearranging the cosine double angle forms.

The green ray turns at $x$. The yellow ray turns at $\frac{x}{2}$, half as fast. The bars show that the height $\sin\frac{x}{2}$ exactly matches $\sqrt{\frac{1 - \cos x}{2}}$ at every step. That match is the formula.

Deriving the Sine Half Angle

Recall one of the cosine double angle forms:

$\cos 2\theta = 1 - 2\sin^2\theta$

Solve it for $\sin^2\theta$:

$\sin^2\theta = \frac{1 - \cos 2\theta}{2}$

This is true for any angle. So replace $\theta$ with $\frac{x}{2}$, which turns $2\theta$ into $x$:

$\sin^2\frac{x}{2} = \frac{1 - \cos x}{2}$

Take the square root:

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

Deriving the Cosine Half Angle

Start from the other cosine form:

$\cos 2\theta = 2\cos^2\theta - 1$

Solve for $\cos^2\theta$, then swap $\theta$ for $\frac{x}{2}$:

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

$\cos\frac{x}{2} = \pm\sqrt{\frac{1 + \cos x}{2}}$

Same shape as the sine version, just a plus sign under the root instead of a minus.

What the ± Means

The square root forces a choice: is the answer positive or negative?

The sign is decided by the quadrant of $\frac{x}{2}$, not the quadrant of $x$.

Work out where $\frac{x}{2}$ lands, then pick the sign that sine (or cosine) has in that quadrant. The formula gives the size; the quadrant gives the sign.

The Tangent Half Angle

Dividing sine by cosine gives the tangent version:

$\tan\frac{x}{2} = \pm\sqrt{\frac{1 - \cos x}{1 + \cos x}}$

But tangent has two cleaner forms that carry the correct sign on their own, so you never have to guess:

$\tan\frac{x}{2} = \frac{1 - \cos x}{\sin x} = \frac{\sin x}{1 + \cos x}$

These are the ones to reach for in practice.

Example

Suppose $\cos x = \frac{3}{5}$ and $x$ is acute, so $\frac{x}{2}$ is also in the first quadrant (everything positive).

Sine half angle:

Cosine half angle:

We took the positive root for both because $\frac{x}{2}$ sits in the first quadrant.

The Set to Know

Minus under the root for sine, plus for cosine, and let the quadrant of $\frac{x}{2}$ settle the sign.