Question

How do you find the value of `sin2theta` given `cottheta=4/3` and `pi<theta<(3pi)/2`?

Answer 1

`24/25`

Explanation:

`cot theta = 4/3`
t is an angle of a right triangle that has 3 sides:
- opposite leg = 3
- Adjacent leg = 4
- hypotenuse = 5
In this triangle,
sin theta = (opposite leg)/hypotenuse `= 3/5`
cos theta = (adjacent leg)/hypotenuse = `4/5`
This is for the principal value of `theta`.
As `theta in (pi, 3pi/2)` wherein cot > 0 but both sin and cos are < 0,
our `theta= pi +` this principal value of `theta`. Accordingly, here,

`sin theta = -3/5 and cos theta =-4/5`

`sin 2theta = 2sin theta cos theta = 2(-3/5)(-4/5) = 24/25`