Question

What are the exact values of sin theta cos theta tan theta if (3,-4) is a point on the terminal side of theta?

Answer 1

$\bf (\stackrel{a}{3}~,~\stackrel{b}{-4})\impliedby \textit{now let's find the \underline{hypotenuse}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c=\sqrt{3^2+(-4)^2}\implies c=\sqrt{25}\implies \boxed{c=5}$

$\bf -------------------------------\\\\ sin(\theta )=\cfrac{\stackrel{opposite}{-4}}{\stackrel{hypotenuse}{5}}\qquad cos(\theta )=\cfrac{\stackrel{adjacent}{3}}{\stackrel{hypotenuse}{5}}\qquad tan(\theta )=\cfrac{\stackrel{opposite}{-4}}{\stackrel{adjacent}{3}}$