# sin A/2

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

## Formula

### Formula

sin A/2 = ±√(1 - cos A)/2

This is the half-angle formula of sine.

To prove this formula,
put A/2 into cos 2A formula:
cos A = 1 - 2 sin2 A/2.

## Example

### Solution

First find the sign of sin θ/2.

It says
3π/2 ≤ θ ≤ 2π.
Then
3π/4 ≤ θ/2 ≤ π.

Draw a simple coordinate plane.
write all, sin, tan, cos

This shows
when the trigonometric function is plus.

For example,
if an angle is in quadrant III,
then tan is plus.
(Sine and cosine are minus.)

3π/4 ≤ θ/2 ≤ π

So draw a terminal side

Reference Angle

The terminal side is on [sin].
Then sine is plus.

So sin θ/2 is plus.

Next, find cos θ.

3π/2 ≤ θ ≤ 2π

So draw another coordinate plane.
And draw a terminal side in quadrant IV.

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,
Hypotenuse (5).

So cos θ = 4/5.

sin θ/2 is plus.
cos θ = 4/5

So sin θ/2 = √(1 - 4/5)/2.

Multiply 5
to both of the numerator and the denominator.

(1 - 4/5)⋅5 = 5 - 4

2⋅5 = 10

5 - 4 = 1

1/10 = 1/√10