Question

In standard position point (2, 3) is on the terminal side of angle θ. Find the values of sine, cosine, and tangent of θ? Please show all work.

Answer 1

so the point is at (2,3) namely x = 2, and y = 3, or one could say a = 2 and b = 3,


`\bf (\stackrel{a}{2}~~,~~\stackrel{b}{3})\impliedby \textit{first off, 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=√(2^2+3^2)\implies c=√(4+9)\implies c=√(13)`

bearing in mind that, the hypotenuse is just the radius unit in the angle, and therefore is never negative.


`\bf sin(\theta)=\cfrac{opposite}{hypotenuse} \qquad cos(\theta)=\cfrac{adjacent}{hypotenuse} \quad % tangent tan(\theta)=\cfrac{opposite}{adjacent}\\\\ -------------------------------`


`\bf sin(\theta)=\cfrac{3}{√(13)}\impliedby \textit{let's rationalize the denominator} \\\\\\ \cfrac{3}{√(13)}\cdot \cfrac{√(13)}{√(13)}\implies \cfrac{3√(13)}{√(13^2)} \implies \cfrac{3√(13)}{13} \\\\\\ cos(\theta)=\cfrac{2}{√(13)}\impliedby \textit{let's rationalize the denominator again} \\\\\\ \cfrac{2}{√(13)}\cdot \cfrac{√(13)}{√(13)}\implies \cfrac{2√(13)}{√(13^2)}\implies \cfrac{2√(13)}{13} \\\\\\ tan(\theta)=\cfrac{3}{2}`