First slide

Trigonometric equations

Question

If $a \cos A - b \sin A = c$ , then $a \sin A + b \cos A$ is equal to

Moderate

A

$\sqrt{a^{2} + b^{2} - c^{2}}$
B

$\sqrt{b^{2} + c^{2} - a^{2}}$
C

$\sqrt{c^{2} + a^{2} - b^{2}}$
D

$\sqrt{a^{2} + b^{2} + c^{2}}$

Solution

Put $a \sin A + b \cos A = x$ , then
$c^{2} + x^{2} = (a \cos A - b \sin A)^{2} + (a \sin A + b \cos A)^{2}$
$\begin{array}{l} = a^{2} + b^{2} \\ \Rightarrow x = \sqrt{a^{2} + b^{2} - c^{2}} \end{array}$

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