Find the general solution for $cot( heta)$ on the unit circle

To find the general solution for $\cot(\theta)$ on the unit circle, first recall the definition of cotangent, which is $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.

We are looking for the values of $\theta$ where $\cot(\theta) = k$ for some constant $k$.

Since $\cot(\theta)$ is periodic with period $\pi$, the general solution for $\theta$ can be written as:

$\theta = \arccot(k) + n\pi$

where $n$ is any integer. This is because the cotangent function repeats every $\pi$ radians. Thus, the full general solution for $\cot(\theta) = k$ can be written as:

$\theta = \arccot(k) + n\pi \quad \text{for} \; n \in \mathbb{Z}$

Given $cot( heta) = k$, where $k$ is a constant, we need to find the general solution for $ heta$ on the unit circle.

Recall that:

$cot( heta) = frac{cos( heta)}{sin( heta)}$

This can be rewritten as:

$cos( heta) = k sin( heta)$

To find $ heta$, we solve for $ heta$ in terms of $k$. The cotangent function is periodic with period $pi$, so the general form of the solution is:

$ heta = arccot(k) + npi$

Since $cot( heta) = cot( heta + npi)$ for any integer $n$, the general solution is:

$ heta = arccot(k) + npi quad ext{for} ; n in mathbb{Z}$

The problem is to find the general solution for $cot( heta)$ on the unit circle. Given $cot( heta) = k$:

$ heta = arccot(k) + npi quad ext{for} ; n in mathbb{Z}$

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