Question

Consider that (x, y) = (3, –4). determine all six trigonometric functions for the angle.

Answer 1

`(\stackrel{a}{3}~~,~~\stackrel{b}{-4})\qquad \textit{let's find the \underline{hypotenuse} for such point} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=√(a^2 + b^2) \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{3}\\ b=\stackrel{opposite}{-4}\\ \end{cases} \\\\\\ c=√((3)^2 + (-4)^2)\implies c=√(9+16)\implies c=√(25)\implies c=5 \\\\[-0.35em] ~\dotfill`

`sin(\theta )=\cfrac{\stackrel{opposite}{-4}}{\underset{hypotenuse}{5}}~\hfill cos(\theta )=\cfrac{\stackrel{adjacent}{3}}{\underset{hypotenuse}{5}}~\hfill tan(\theta )=\cfrac{\stackrel{opposite}{-4}}{\underset{adjacent}{3}} \\\\\\ cot(\theta )=\cfrac{\stackrel{adjacent}{3}}{\underset{opposite}{-4}}~\hfill sec(\theta )=\cfrac{\stackrel{hypotenuse}{5}}{\underset{adjacent}{3}}~\hfill csc(\theta )=\cfrac{\stackrel{hypotenuse}{5}}{\underset{opposite}{-4}}`