logiski9s

2022-07-09

If $\frac{\mathrm{sin}\left(x\right)}{a}=\frac{\mathrm{cos}\left(x\right)}{b}$ then $a\mathrm{sin}\left(2x\right)+b\mathrm{cos}\left(2x\right)=?$
$b\mathrm{sin}\left(x\right)=a\mathrm{cos}\left(x\right)$
$\mathrm{tan}\left(x\right)=\frac{a}{b}$
I couldn't simplify after that.

Dayana Zuniga

Expert

Though there are lots of answers here, I'd like to put an answer with almost no computations.
$\begin{array}{rl}a\mathrm{sin}\left(2x\right)+b\mathrm{cos}\left(2x\right)& =\frac{a}{\mathrm{sin}\left(x\right)}\left(\mathrm{sin}\left(x\right)\mathrm{sin}\left(2x\right)+\mathrm{cos}\left(x\right)\mathrm{cos}\left(2x\right)\right)\\ & =\frac{a}{\mathrm{sin}\left(x\right)}\mathrm{cos}\left(x\right)=b\end{array}$

Sylvia Byrd

Expert

$\mathrm{tan}x=\frac{a}{b}$ and from here
$a\mathrm{sin}2x+b\mathrm{cos}2x=\frac{\frac{2{a}^{2}}{b}}{1+\frac{{a}^{2}}{{b}^{2}}}+\frac{b-\frac{{a}^{2}}{b}}{1+\frac{{a}^{2}}{{b}^{2}}}=b$

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