Question

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

Answer 1

$\csc\theta=\dfrac{\sqrt5}2\implies \sin\theta=\dfrac1{\csc\theta}=\dfrac2{\sqrt5}$

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,

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