Question

Evaluate the cosine if the angle of rotation which contains the point (9, -3) on its terminal side

Answer 1

so we know the terminal point is at (9, -3), now, let's notice that's the IV Quadrant

`\bf (\stackrel{x}{9}~~,~~\stackrel{y}{-3})\impliedby \textit{let's find the \underline{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=√(9^2+(-3)^2)\implies c=√(81+9)\implies c=√(90) \\\\[-0.35em] ~\dotfill`

`\bf cos(\theta )=\cfrac{\stackrel{adjacent}{9}}{\stackrel{hypotenuse}{√(90)}}\implies \stackrel{\textit{rationalizing the denominator}}{\cfrac{9}{√(90)}\cdot \cfrac{√(90)}{√(90)}\implies \cfrac{9√(90)}{90}}\implies \cfrac{√(90)}{10}\implies \cfrac{3√(10)}{10}`