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