# How do you find the solution to cottheta+8=3cottheta+2 if 0<=theta<2pi?

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Alan P. Share
Jan 10, 2017

#### Answer:

$\theta \in \left\{0.322 , 3.463\right\}$ (approx.)

#### Explanation:

$\textcolor{b l u e}{\cot \left(\theta\right)} + 8 = 3 \textcolor{b l u e}{\cot \left(\theta\right)} + 2$

$\rightarrow 2 \textcolor{b l u e}{\cot \left(\theta\right)} = 6$

$\rightarrow \textcolor{b l u e}{\cot \left(\theta\right)} = 3$

If you have a calculator that evaluates $\arccos$ directly then
the primary value for $\textcolor{red}{\theta} = \textcolor{red}{\arccos \left(3\right)}$
My calculator won't do that so I needed to use
$\textcolor{w h i t e}{\text{XX}} \textcolor{g r e e n}{\tan \left(\theta\right) = \frac{1}{\cot \left(\theta\right)} = \frac{1}{3}}$
and then
$\textcolor{w h i t e}{\text{XX}} \textcolor{red}{\theta} = \textcolor{red}{\arctan \left(\frac{1}{3}\right)}$
for the primary value of $\textcolor{red}{\theta}$

This gave: $\textcolor{red}{\theta \approx 0.321750554}$
as the primary value (in Quadrant 1).
Based on CAST notation for the 4 quadrants we know that a secondary value for $\textcolor{red}{\theta}$ will also exist in Quadrant 3 at
$\textcolor{w h i t e}{\text{XX}} \textcolor{red}{\theta} \approx \textcolor{red}{0.32175055 + \pi} = \textcolor{red}{3.463343208}$

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