Question

Solve the equation: sin x + √3cos x = √2

Answer 1

To solve the given equation

sin x + √3cos x = √2

Multiply both side by 1/2

$\frac{1}{2} \: sin \: x + \frac{ \sqrt{3} }{2} \: cos \: x = \frac{ \sqrt{2} }{2}$
as we know that
$sin \: 30° = \frac{1}{2} \\ \\ cos \: 30° = \frac{ \sqrt{3} }{2}$
so

$sin \: 30°\: sin \: x + cos \: 30° \: cos \: x = \frac{ 1}{ \sqrt{2} }$

As we know that

$cos \: (A - B) = sin \: A \: \: sin \: B + cos \: A \: cos \: B$
so
$cos \: (30° - x) = sin \: 30° \: \: sin \: x + cos \: 30° \: cos \: x \\ \\ cos \: (30° - x) = \frac{1}{ \sqrt{2} } \\ \\ 30° - x = {cos}^{ - 1} ( \frac{1}{ \sqrt{2} } ) \\ \\ 30° - x = {cos}^{ - 1}(cos \: 45°) \\ \\ 30° - x = 45 \\ \\ x = -15°$