Calculate tg a if a belongs to interval (pi, 3pi/2) and sin a = -4/5?

If a belongs to the third qudrant, that means that the values of the tangent function are positive.

The tangent function is a ratio:

tg a = sin a / cos a

We have the value of sin a but we don't know the value of cos a.

We can calculate the value of cos a, using the fundamental formula of trigonometry:

(sina)^2 + (cosa)^2 = 1

Because a is in the third quadrant, cos a<0.

cos a = -sqrt[1 - (sina)^2]

cos a = -sqrt(1 - 16/25)

cos a = -sqrt [(25-16)/25]

cos a = -sqrt (9/25)

cos a = - (3/5)

tg a = sin a / cos a

tg a = (-4/5) / (-3/5)

tg a = 4/3