sin 2A

How to find sin 2A by using its formula (double-angle formula): formula, 1 example, and its solution.

Formula

Formula

sin 2A = 2 sin A cos A

This is the double-angle formula of sine.

To prove this formula,
put A and A into sin (A + B) formula:
sin (A + A)
= sin A cos A + cos A sin A
= 2 sin A cos A.

Example

Example

Solution

It says to find sin 2θ.
And sin θ is given.
So find cos θ.

It says π/2 ≤ θ ≤ π.
So draw a terminal side
in quadrant II.

Reference Angle

sin θ = 3/5

Sine is SOH:
Sine,
Opposite side (3),
Hypotenuse (5).

So draw a right triangle
whose opposite side is 3
and whose hypotenuse is 5.

See the right triangle.

The sides are 3, (adjacent side), and 5.

So this is a [3, 4, 5] right triangle.

So the adjacent side is -4.

Pythagorean Triple

Find cos θ.

Cosine is CAH:
Cosine,
Adjacent side (-4),
Hypotenuse (5).

So cos θ = (-4)/5 = -4/5.

sin θ = 3/5
cos θ = -4/5

So sin 2θ = 2⋅[3/5]⋅[-4/5].

2⋅3⋅4 = 6⋅4 = 24

5⋅5 = 25

So -24/25 is the answer.