tan A/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.