Question

This is my next Trigonometry question I am needed help on- I just want to be sure I've got it right, or if not, then the steps I need to take to get it right. Thank you.

Answer 1

$\bf \textit{simmetry identities}\\\\ sin(-\theta)=-sin(\theta)\\\\ -------------------------------\\\\ sin(-\theta)=\cfrac{1}{5}\implies -sin(\theta)=\cfrac{1}{5}\implies sin(\theta)=\cfrac{-1}{5}\cfrac{\leftarrow opposite=b}{\leftarrow hypotenuse=c} \\\\\\ c^2=a^2+b^2\implies \pm\sqrt{c^2-b^2}=a\implies \pm\sqrt{5^2-(-1)^2}=a \\\\\\ \pm\sqrt{24}=a\implies \pm2\sqrt{6}=a$

so. hmm which is it? the +/- ?   well, neverminding for a second, the value of tangent, just looking at the sign, the tangent is positive,

tangent = opposite/adjacent.... so that only happens, when both are the same sign, + or - both

now, we know the sine is -1/5.. if the sine is negative, the cosine also has t to be negative, so we'd use the -2√(6) = a

thus     $\bf cos(\theta)=\cfrac{adjacent}{hypotenuse}\implies cos(\theta)=\cfrac{-2\sqrt{6}}{5}$