First slide

Trigonometric equations

Question

The equation $a \sin x + b \cos x = c$ , where $|c| > \sqrt{a^{2} + b^{2}}$ has

Easy

A

a unique solution
B

infinite number of solutions
C

no solution
D

none of these

Solution

Let $a = r \cos α, b = r \sin α$ so that $r = \sqrt{a^{2} + b^{2}}$
The given equation can be written as $r \sin (x + α) = c \Rightarrow \sin (x + α) = \frac{c}{\sqrt{a^{2} + b^{2}}}$
$\Rightarrow |\sin (x + α)| = \frac{|c|}{\sqrt{a^{2} + b^{2}}} > 1$ is not possible as sinx always $\leq$ 1
hence $|c| > \sqrt{a^{2} + b^{2}}$ is not possible for any value of x

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