Solve the equation `sinx +sqrt(3) cosx=sqrt(2)`

Explanation

Before we set out to solve the equation, we must show that it needed has a solution.
Now, `|1.sinx + sqrt(3) cosx| le sqrt(1+sqrt(3))^(2) = sqrt(1+3)=2`
As `sqrt(2) lt 2`, we have shown that the solutions exists.
Now, `sinx + sqrt(3) cosx = sqrt(2)`
`=1/2 sinx+sqrt(3)/2cosx = sqrt(2)/2` (Dividing both sides by `sqrt(1+3)=2`)
`rArr sinpi/6 sinx + cospi/6 cosx=1/sqrt(2)`
`rArr cos(x-pi/6)= cospi/4`
`rArr x-pi/6 = 2npi +-pi/4`
Taking positive sign
`x=2npi + pi/4 +pi/6 = 2npi +(5pi)/12, n int Z`
Taking negative sign
`x=2npi-pi/4 + pi/6 = 2npi- pi/12, n int Z`