# Find the lengths of the missing sides if side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse. tan (A)=5/12, b=2

Darryl English 2022-07-26 Answered
Find the lengths of the missing sides if side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse.
$\mathrm{tan}\left(A\right)=\frac{5}{12},b=2$
$a=?$
$c=?$
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Step 1

Given $\mathrm{tan}\left(A\right)=\frac{5}{12},b=2$
But from triangle ABC
$\mathrm{tan}A=\frac{a}{b}$
$⇒\frac{5}{12}=\frac{a}{b}\phantom{\rule{0ex}{0ex}}⇒\frac{5}{12}=\frac{a}{2}\phantom{\rule{0ex}{0ex}}⇒a=\frac{5}{12}×2=\frac{5}{6}\phantom{\rule{0ex}{0ex}}a=\frac{5}{6}$
Step 2
We know that pythagorus theorem
${c}^{2}={a}^{2}+{b}^{2}\phantom{\rule{0ex}{0ex}}{c}^{2}=\left(\frac{5}{6}{\right)}^{2}+\left(2{\right)}^{2}\phantom{\rule{0ex}{0ex}}c=\sqrt{\frac{25}{36}}+4\phantom{\rule{0ex}{0ex}}c=\sqrt{\frac{169}{36}}\phantom{\rule{0ex}{0ex}}c=\frac{13}{6}$

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