sin 2A (Double-Angle Formula)

sin 2A (Double-Angle Formula)

How to solve sin 2A (Double-angle formula) problems: formula, proof, example, and its solution.

Formula

sin 2A = 2 sin A cos A

sin 2A = 2 sin A cos A

Proof

sin 2A (Double-Angle Formula): Proof of the Formula

sin 2A = sin (A + A)
= sin A cos A + cos A sin A

sin (A + B)

Example

If sin theta = 3/4 and pi/2 <= theta <= pi, find the value of sin 2*theta.

π/2 ≦ θπ
So θ is in quadrant II.

sin θ = 3/4
And sine: SOH.

So draw a right triangle in quadrant II
whose opposite side is 3
and whose hypotenuse is 4.

The triangle above is a right triangle.

So 32 + (purple side)2 = 42.

Pythagorean theorem

The brown angle is the reference angle of ∠θ.

And cosine: CAH.

So cos θ = cos (brown)
= -√7/4.

sin θ = 3/4, cos θ = -√7/4

So sin 2θ = 2⋅(3/4)⋅(-√7/4).