Double Angle Formulas

Nothing New to Memorize

The double angle formulas are not a fresh set of rules. They are the addition formulas with both angles the same.

Set $B = A$ in $\sin(A + B)$ and you get $\sin(A + A) = \sin 2A$. That is all a “double angle” is.

The yellow ray turns at angle $\theta$. The green ray turns at $2\theta$, sweeping twice as fast. The two bars show $\sin 2\theta$ and $2\sin\theta\cos\theta$ locked together, equal at every instant. That equality is the formula.

Deriving Sine

Start from the addition formula and let the two angles be equal:

$\sin 2\theta = 2\sin\theta\cos\theta$

Deriving Cosine

Same move with the cosine addition formula:

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

Cosine Has Three Faces

This is where cosine gets useful. Using the Pythagorean identity $\cos^2\theta + \sin^2\theta = 1$, we can rewrite $\cos 2\theta$ in two more ways.

Replace $\cos^2\theta$ with $1 - \sin^2\theta$:

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

Replace $\sin^2\theta$ with $1 - \cos^2\theta$:

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

So all three of these are the same thing:

Pick the version that matches what you already have.

Tangent

Setting $B = A$ in the tangent formula gives:

$\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$

Don’t Fall for the Shortcut

Doubling the angle is not doubling the sine:

$\sin 2\theta \neq 2\sin\theta$

The green bar in the animation is $\sin 2\theta$, but the true value is $2\sin\theta\cos\theta$. That extra $\cos\theta$ factor is the whole point.

Example

Suppose $\sin\theta = \frac{3}{5}$ and $\theta$ is acute. Find $\sin 2\theta$.

First get $\cos\theta$ from the Pythagorean identity: $\cos\theta = \frac{4}{5}$.

Then:

$\sin 2\theta = 2\sin\theta\cos\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$

The Set to Know