Question

(√3-1) cosx+(√3+1) sinx=2
Solve by general solution

Answer 1

Step-by-step explanation:

We have,

$( \sqrt{3} - 1) \cos(x) + ( \sqrt{3} + 1) \sin(x) = 2$

Dividing both side by 2√2

$( \frac{ \sqrt{3} - 1}{2 \sqrt{2} }) \cos(x) + ( \frac{ \sqrt{3} + 1 }{2 \sqrt{2} } ) \sin(x) = \frac{2}{2 \sqrt{2} }$

We know, sin(5π/12)= (√3 + 1)/(2√2) and cos(5π/12) = (√3 - 1)/(2√2)

$\cos( \frac{5\pi}{12} ) \cos(x) + \sin( \frac{5\pi}{12} ) \sin(x) = \cos( \frac{\pi}{4} )$

$\cos(x - \frac{5\pi}{12} ) = \cos( \frac{\pi}{4} )$

$= > x - \frac{5\pi}{12} = 2n\pi + \frac{\pi}{4} \: \: or \: \: x - \frac{5\pi}{12} = 2n\pi - \frac{\pi}{4}$

$= > x = 2n\pi + \frac{3\pi }{4} \: \: or \: \: x = 2n\pi + \frac{\pi }{6}$