Question

5sinx=5sqrt3cosx 0<x<2pi What are all possible solutions?

Answer 1

solve `5sin x = 5sqrt3cos x`

Ans: `pi/3 and (4pi)/3`

Explanation:

There are two ways.
1. First way. Divide both side by 5cos x (condition `cos x != 0, or x != pi/2 and x !=3pi/2`)
`tan x = sqrt3` --> `x = pi/3` and `x = pi/3 + pi = (4pi)/3`
2. Second way. Simplify both sides by 5
`sin x - sqrt3cos x = 0`
Replace in the equation (sqrt3) by `tan (pi)/3 = (sin (pi/3)/(cos pi/3))`
`sin x.cos ((pi)/3) - sin ((pi)/3).cos x = 0`
`sin (x - pi/3) = 0`
`(x - pi/3) = 0 -->` x = pi/3# and #(x - pi/3) = pi`-->`x = pi + pi/3 = (4pi)/3#