# tan A/2

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

## Formula

## Exampletan θ = 3/4, π ≤ θ ≤ 3π/2, tan θ/2 = ?

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.

Starting fron quadrant I,

write all, sin, tan, cos

in each quadrant.

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

in quadrant II.

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),

Adjacent side (-4).

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,

Adjacent side (-4),

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.