# tan A/2

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

## Formula

### Formula

tan A/2 = ±√(1 - cos A)/(1 + cos A)

This is the half-angle formula of cosine.

(1 - cos A) part is from sin A/2:
sin A/2 = ±√(1 - cos A)/2.

(1 + cos A) part is from cos A/2:
cos A/2 = ±√(1 + cos A)/2.

## Example

### Solution

First find the sign of tan θ/2.

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

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 IV,
then cos is plus.
(Sine and tangent are minus.)

π/2 ≤ θ/2 ≤ 3π/4

So draw a terminal side

Reference Angle

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

So tan θ/2 is minus.

Next, find cos θ.

π ≤ θ ≤ 3π/2

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

tan θ = 3/4 = (-3)/(-4)

Tangent is TOA:
Tangent,
Opposite side (-3),

So draw a right triangle
whose opposite side is -3
and whose adjacent side is -4.

See the right triangle.

The sides are -3, -4, (hypotenuse).

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

So the hypotenuse is 5.

Pythagorean Triple

Find cos θ.

Cosine is CAH:
Cosine,
Hypotenuse (5).

So cos θ = -4/5.

tan θ/2 is minus.
cos θ = -4/5

So tan θ/2 = -√(1 - [-4/5])/(1 + [-4/5]).

1 - [-4/5] = 1 + 4/5

1 + [-4/5] = 1 - 4/5

Multiply 5
to both of the numerator and the denominator.

(1 + 4/5)⋅5 = 5 + 4

(1 - 4/5)⋅5 = 5 - 4

5 + 4 = 9

5 - 4 = 1

-√9/1
= -√9
= -3

Square Root

So tan θ/2 = -3.