Lipossig

2021-01-24

Let . Then find

Gennenzip

$\sqrt{{a}^{2}-{s}^{2}{\mathrm{sin}}^{2}\left(\theta \right)}=$
Factor out common term ${a}^{2}$
$=\sqrt{{a}^{2}\left(1-{\mathrm{sin}}^{2}\left(\theta \right)\right)}$
Apply radical rule $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$, assuming $\frac{a\ge 0}{b\ge 0}$
$=\sqrt{{a}^{2}}\sqrt{-{\mathrm{sin}}^{2}\left(\theta \right)+1}$
Apply radical rule $\sqrt[n]{{a}^{n}}=a$, assuming $a\ge 0$ Use the following identity: ${\mathrm{cos}}^{2}\left(x\right)+{\mathrm{sin}}^{2}\left(x\right)=1$
Therefore $1-{\mathrm{sin}}^{2}\left(x\right)={\mathrm{cos}}^{2}\left(x\right)$
$=a\sqrt{{\mathrm{cos}}^{2}\left(\theta \right)}=a\mathrm{cos}\left(\theta \right)$
Therefore $\sqrt{{a}^{2}-{a}^{2}{\mathrm{sin}}^{2}\left(\theta \right)}=a\mathrm{cos}\left(\theta \right)$
$\phantom{\rule{0.278em}{0ex}}⟹\phantom{\rule{0.278em}{0ex}}\mathrm{cos}\theta =\sqrt{{a}^{2}-{a}^{2}}\frac{{\mathrm{sin}}^{2}\left(\theta \right)}{a}$
$\phantom{\rule{0.278em}{0ex}}⟹\phantom{\rule{0.278em}{0ex}}\frac{a\mathrm{sin}\left(\theta \right)}{\sqrt{{a}^{2}-{a}^{2}{\mathrm{sin}}^{2}\left(\theta \right)}}=\mathrm{tan}\left(\theta \right)$
$\sqrt{{a}^{2}-{a}^{2}{\mathrm{sin}}^{2}\left(\theta \right)}=a\mathrm{cos}\left(\theta \right)$

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