Question

Let (3, -2) be a point on the Terminal Side of θ. Find the exact values of Cosθ, Secθ, and Cotθ

Answer 1

`\bf (\stackrel{a}{3}~,~\stackrel{b}{-2})\qquad \impliedby \textit{let's find the hypotenuse} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=√(a^2+b^2) \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c = √(3^2+(-2)^2)\implies c=√(9+4)\implies c=√(13) \\\\[-0.35em] ~\dotfill`

`\bf cos(\theta )\implies \cfrac{\stackrel{adjacent}{3}}{\stackrel{hypotenuse}{√(13)}}\implies \cfrac{3}{√(13)}\cdot \cfrac{√(13)}{√(13)}\implies \cfrac{3√(13)}{13} \\\\\\ sin(\theta )\implies \cfrac{\stackrel{opposite}{-2}}{\stackrel{hypotenuse}{√(13)}}\implies \cfrac{-2}{√(13)}\cdot \cfrac{√(13)}{√(13)}\implies \cfrac{-2√(13)}{13} \\\\\\ cot(\theta )\implies \cfrac{\stackrel{adjacent}{3}}{\stackrel{opposite}{-2}}`