Answer :
first off, let's notice something, on the III Quadrant, both x,y or cosine, sine are negative values.
now, in a right-triangle the hypotenuse is never negative, since it's just a radius unit, and therefore is always just a positive value.
[tex]\bf sin(\theta )=\cfrac{\stackrel{opposite}{-1}}{\stackrel{\stackrel{never~negative}{hypotenuse}}{6}}\impliedby \textit{let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2-b^2}=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \pm\sqrt{6^2-(-1)^2}=a\implies \pm\sqrt{35}=a\implies \stackrel{III~Quadrant}{-\sqrt{35}=a} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf \pm\sqrt{6^2-(-1)^2}=a\implies \pm\sqrt{35}=a\implies \stackrel{III~Quadrant}{-\sqrt{35}=a} \\\\[-0.35em] ~\dotfill\\\\ cot(\theta )=\cfrac{\stackrel{adjacent}{-\sqrt{35}}}{\stackrel{opposite}{-1}}\implies cot(\theta )=\sqrt{35}[/tex]
The value of the cot(0) is √35 in the third quadrant if the value of sin(0) is -1/6
What is trigonometry?
Trigonometry is a branch of mathematics that deals with the relationship between sides and angles of a right-angle triangle.
We have:
Sin(0) = -1/6
A cot is the ratio of the adjacent side to the opposite side.
Adjacent side = √(6²-1²) = √35
Cot(0) = √35/1 =√35 (in the Quadrant III)
Thus, the value of the cot(0) is √35 in the third quadrant if the value of sin(0) is -1/6
Learn more about trigonometry here:
brainly.com/question/26719838
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