Question

Solve the equation √3 cos x - sin x = 1

Answer 1

Solution:

on multiplying both side by 1/2 .

$\frac{\sqrt{3}}{2}\times cos \: x- \sin(x) \times \frac{1}{2} = \frac{1}{2} \\ \\ \cos(x) \cos(30°) - \sin(x) \sin(30°) = \frac{1}{ 2 }$

now we are applying the formula of
cos a cos b - sin a sin b

$\cos(A) \cos(B) - \sin(A) \sin(B) = \cos(A + B)$

$\cos(x) \cos(30°) - \sin(x) \sin(30°) = \frac{1}{ 2 } \\ \\ \cos(x + 30°) = \frac{1}{ 2 } \\ \\ x + 30° = {cos}^{ - 1}( \frac{1}{ 2 } ) \\ \\ x + 30° = {cos}^{ - 1}( \cos(60°) ) \\ \\ x + 30° = 60° \\ \\ x = 60° - 30° \\ \\ x = 30°\\\\x =\frac{π}{ 6 }$