Question

The equation a sin x + b cos x = c , where | c | > root a² + b² has answer with full explanation ​

Answer 1

Answer:

$a \sin(x) + b \cos(x) \\ = \sqrt{ {a}^{2} + {b}^{2} } ( \frac{a}{\sqrt{ {a}^{2} + {b}^{2} }} \sin(x) + \frac{b}{\sqrt{ {a}^{2} + {b}^{2} }}) \\ let \: \: tan \: y \: = \frac{b}{a} \\ = \sqrt{ {a}^{2} + {b}^{2} }( \sin(x) \cos(y) + \cos(x) \sin(y) ) \\ = \sqrt{ {a}^{2} + {b}^{2} } \sin(x + y) \\ = c$

So

$\sqrt{ {a}^{2} + {b}^{2} } \sin(x + y) = c \\since \: \: |\sin( \theta) |\leqslant 1 \\ \sqrt{ {a}^{2} + {b}^{2} } | \sin(x + y) | = |c| \leqslant \sqrt{ {a}^{2} + {b}^{2} }$