Question

Given a function value of an acute​ angle, find the other five trigonometric function values

Answer 1

well, we know that θ is an acute angle, that means is in the I Quadrant, that means that cosine and sine are both positive and likewise adjacent and opposite sides of θ, so let's use the pythagorean theorem to find the missing side

`sin(\theta )=\cfrac{\stackrel{opposite}{5}}{\underset{hypotenuse}{13}}\hspace{5em}\textit{let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies √(c^2 - b^2)=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \pm√(13^2 - 5^2)=a\implies \pm 12=a\implies \stackrel{I~Quadrant}{+12=a} \\\\[-0.35em] ~\dotfill`

`cos(\theta )=\cfrac{\stackrel{adjacent}{12}}{\underset{hypotenuse}{13}}~\hfill tan(\theta )=\cfrac{\stackrel{opposite}{5}}{\underset{adjacent}{12}} ~\hfill cot(\theta )=\cfrac{\stackrel{adjacent}{12}}{\underset{opposite}{5}}~\hfill \\\\\\ sec(\theta )=\cfrac{\stackrel{hypotenuse}{13}}{\underset{adjacent}{12}}\hspace{3em} csc(\theta )=\cfrac{\stackrel{hypotenuse}{13}}{\underset{opposite}{5}}`