2022-01-27

Given that
$5\mathrm{cos}\left(x\right)-12\mathrm{sin}\left(x\right)=13$
Im

becky4208fj

Expert

Hint By Cauchy Schwarz you have
$169={13}^{2}={\left(5\mathrm{cos}\left(x\right)-12\mathrm{sin}\left(x\right)\right)}^{2}\le \left({5}^{2}+{12}^{2}\right)\left({\mathrm{cos}}^{2}\left(x\right)+{\mathrm{sin}}^{2}\left(x\right)\right)=169$
Therefore, you must have equality in CS, and hence
$\frac{\mathrm{cos}\left(x\right)}{5}=\frac{\mathrm{sin}\left(x\right)}{-12}$
Now combine this equality with
${\mathrm{sin}}^{2}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)=1$

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