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Bleck3
Find cot θ if csc θ = -sqrt37/6 and tan θ > 0.
options are 1/6 sqrt37/37 sqrt37 6
Think about the triangle required to get the numbers given. If the csc theta is -sqrt(37)/6, the unscaled triangle is a 1/6/sqrt(37) triangle. Then cot theta will be 1/6 (same sign as tan).
Another way to think about it.\[\csc \theta = -\frac{\sqrt{37}}{6} \implies \sin \theta =-\frac{6}{\sqrt{37}} \implies \cos \theta= \pm \frac{1}{\sqrt{37}}\]Since tan theta and cot theta are reciprocals, they have to have the same sign. That means\[\cot \theta = \frac{1}{6}\]