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.