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=\frac{\sqrt5}2\implies \sin\theta=\frac1{\csc\theta}=\frac2{\sqrt5}`

Since
`\tan\theta=(\sin\theta)/(\cos\theta)>0`, and we know that
`\sin\theta=\frac2{\sqrt5}>0`, we then also know that
`\cos\theta>0`.

Recall that


`\sin^2\theta+\cos^2\theta=1\implies\cos\theta=\pm√(1-\sin^2\theta)`

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


`\cos\theta=\sqrt{1-\left(\frac2{\sqrt5}\right)^2}=√(1-\frac45)=\frac1{\sqrt5}`

Now,


`\cot\theta=(\cos\theta)/(\sin\theta)=\frac{\frac1{\sqrt5}}{\frac2{\sqrt5}}=\frac12`