Question

Is sin theta=5/6, what are the values of cos theta and tan theta?

Answer 1

let's bear in mind that sin(θ) in this case is positive, that happens only in the I and II Quadrants, where the cosine/adjacent are positive and negative respectively.

`\bf sin(\theta )=\cfrac{\stackrel{opposite}{5}}{\stackrel{hypotenuse}{6}}\qquad \impliedby \textit{let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \pm√(c^2-b^2)=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \pm√(6^2-5^2)=a\implies \pm√(36-25)\implies \pm √(11)=a \\\\[-0.35em] ~\dotfill`

`\bf cos(\theta )=\cfrac{\stackrel{adjacent}{\pm√(11)}}{\stackrel{hypotenuse}{6}} \\\\\\ tan(\theta )=\cfrac{\stackrel{opposite}{5}}{\stackrel{adjacent}{\pm√(11)}}\implies \stackrel{\textit{and rationalizing the denominator}~\hfill }{tan(\theta )=\pm\cfrac{5}{√(11)}\cdot \cfrac{√(11)}{√(11)}\implies tan(\theta )=\pm\cfrac{5√(11)}{11}}`