Question

Find the tangent of

Answer 1

Check the picture below.

`\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2 \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{√(21)}\\ a=\stackrel{adjacent}{YZ}\\ o=\stackrel{opposite}{3} \end{cases} \\\\\\ (√(21))^2= (YZ)^2 + (3)^2\implies 21=YZ^2+9 \\\\\\ 12=YZ^2 \implies √(12)=YZ \\\\[-0.35em] ~\dotfill`

`\tan(Y )=\cfrac{\stackrel{opposite}{3}}{\underset{adjacent}{√(12)}}\implies \tan(Y )=\cfrac{3}{√(12)}\cdot \cfrac{√(12)}{√(12)}\implies \tan(Y )=\cfrac{3√(12)}{12} \\\\\\ \tan(Y )=\cfrac{3√(2^2\cdot 3)}{12}\implies \tan(Y )=\cfrac{6√(3)}{12}\implies \tan(Y )=\cfrac{√(3)}{2}`