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natali 33 [55]
3 years ago
9

How do you find the cot θ if csc θ = square root of five divided by two and tan θ > 0.

Mathematics
1 answer:
Sladkaya [172]3 years ago
6 0
\csc\theta=\dfrac{\sqrt5}2\implies \sin\theta=\dfrac1{\csc\theta}=\dfrac2{\sqrt5}

Since \tan\theta=\dfrac{\sin\theta}{\cos\theta}>0, and we know that \sin\theta=\dfrac2{\sqrt5}>0, we then also know that \cos\theta>0.

Recall that

\sin^2\theta+\cos^2\theta=1\implies\cos\theta=\pm\sqrt{1-\sin^2\theta}

We take the positive root because we know that \cos\theta>0. Then

\cos\theta=\sqrt{1-\left(\dfrac2{\sqrt5}\right)^2}=\sqrt{1-\dfrac45}=\dfrac1{\sqrt5}

Now,

\cot\theta=\dfrac{\cos\theta}{\sin\theta}=\dfrac{\frac1{\sqrt5}}{\frac2{\sqrt5}}=\dfrac12
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