Question

If sin=2/3 and tan is less than 0, what is the value of cos

Answer 1

tangent is less than 0 or tan(θ) < 0, is another way to say tan(θ) is negative, well, that only happens on the II Quadrant and IV Quadrant, where sine and cosine are different signs, so we know θ is on the II or IV Quadrant.

$\bf sin(\theta )=\cfrac{\stackrel{opposite}{2}}{\stackrel{hypotenuse}{3}}\qquad \impliedby \textit{let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \pm\sqrt{c^2-b^2}=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases}$

$\bf \pm\sqrt{3^2-2^2}=a\implies \pm\sqrt{5}=a\implies \stackrel{\textit{II Quadrant}}{-\sqrt{5}=a} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill cos(\theta )=\cfrac{\stackrel{adjacent}{-\sqrt{5}}}{\stackrel{hypotenuse}{3}}~\hfill$