Identity

$sin 2 A = 2 sin A cos A$

Trigonometry • Trigonometric Identity

Formula

sin 2 A = 2 sin A cos A

For every angle

A

sin 2 A = 2 sin A cos A

. Read from right to left it collapses a product into one ratio,

sin A cos A = \frac{1}{2} sin 2 A

; and replacing

A

\frac{θ}{2}

gives the half-angle form

sin θ = 2 sin \frac{θ}{2} cos \frac{θ}{2}

How to use it

1Use it in the direction that reduces the number of distinct angles in the expression.
2When an answer carries $sin \frac{θ}{2} cos \frac{θ}{2}$ and your derivation carries $sin θ$ , this identity is what matches the two — and the factor $2$ is where the match is usually lost.
3With $sin A$ known and the quadrant given, get $cos A$ from $sin^{2} A + cos^{2} A = 1$ first, then substitute.

sin A = \frac{3}{5}

and

A

is acute, find

sin 2 A

Step-by-step

1
$cos A = 1 - (\frac{3}{5})^{2} = \frac{4}{5}$
Why: $A$ is acute, so the cosine is taken positive.
2
$sin 2 A = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25}$

Answer

\frac{24}{25}

Watch out

sin 2 A

is not

2 sin A

: the factor

cos A

is not optional, and the two agree only where

cos A = 1

. Nor does doubling the angle double the ratio —

sin 2 A

still lies between

- 1

and

1

, and it falls as

A

passes

4 5^{\circ}

. If the angle is in degrees,

2 A

is in degrees too; the identity says nothing about units.

Builds on

Also known as

double angle formula for sine, sin 2A, 2 sin A cos A, sine of a double angle, half-angle product form of sine