Question

Find all the solutions for √3 sin x = cos ( x - pi/3 ) ?

Answer 1

Solve `sqrt3sin x = cos (x - pi/3)`(1)

Ans: `pi/6 + kpi`

Explanation:

First, develop `cos (x - pi/3) = cos x.cos ((pi)/3) + sin ((pi)/3).sin x =`
`= (1/2)cos x + (sqrt3/2) sin x.`
Bring equation (1) to standard form, then, simplify
`sqrt3sinx - (sqrt3sin x)/2 - (1/2)cos x = 0`
`(sqrt3)sin x - cos x = 0` (2).
`sin x - (1/sqrt3)cos x = 0`
Replace `(1/sqrt3) = tan ((pi)/6) = (sin (pi/6))/(cos (pi/6))`
Equation (2) -->`sin x.cos ((pi)/6) - sin ((pi)/6).cos x = sin (x - pi/6) = 0`
`sin (x - pi/6) = 0` --> `x = 0; x = pi; x = 2pi`
a. `x - pi/6 = 0` --> `x = pi/6`
b.`x - pi/6 = pi` --> `x = pi + pi/6 = (7pi)/6`
c. `x - pi/6 = 2pi` --> `x = 2pi + pi/6 = (pi)/6`
Check by calculator
`x = pi/6` -> `sqrt3sin x = sqrt3/2`; `cos (pi/6 - pi/3) = cos (-pi/6) = sqrt3/2`. OK