Question

Can someone show me how to solve this?

Answer 1

first off, let's notice that the angle θ is between 0 ⩽ θ ⩽ π/2, that's another way of saying in the 1st Quadrant, where cosine and sine are both positive, and namely the adjacent and opposite sides are positive, so

`\tan(\theta )=\cfrac{\stackrel{opposite}{8}}{\underset{adjacent}{6}} \hspace{5em}\textit{let's find the \underline{hypotenuse}} \\\\\\ \begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=√(a^2 + o^2) \end{array} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{6}\\ o=\stackrel{opposite}{8} \end{cases} \\\\\\ c=√( 6^2 + 8^2)\implies c=√( 36 + 64 ) \implies c=√( 100 )\implies c=10 \\\\[-0.35em] ~\dotfill`

`\sin(\theta )=\cfrac{\stackrel{opposite}{8}}{\underset{hypotenuse}{10}}\implies \sin(\theta )=\cfrac{4}{5} \\\\\\ \cos(\theta )=\cfrac{\stackrel{adjacent}{6}}{\underset{hypotenuse}{10}}\implies \cos(\theta )=\cfrac{3}{5} \\\\\\ \cot(\theta )=\cfrac{\stackrel{adjacent}{6}}{\underset{opposite}{8}}\implies \cot(\theta )=\cfrac{3}{4}`