Question

Given:
sin (C) 4/9

Find tan (C)

Answer 1

`sin(C)=\cfrac{\stackrel{opposite}{4}}{\underset{hypotenuse}{9}}\qquad \textit{let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies √(c^2 - b^2)=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ √(9^2 - 4^2)=a\implies √(65)=a \\\\[-0.35em] ~\dotfill`

`tan(C)=\cfrac{\stackrel{opposite}{4}}{\underset{adjacent}{√(65)}}\implies tan(C)=\cfrac{4}{√(65)}\cdot \cfrac{√(65)}{√(65)}\implies tan(C)=\cfrac{4√(65)}{65}`