# How do you find the quadrants in which the terminal side of theta must lie given costheta and cottheta have the same sign?

Dec 22, 2017

If $\cos \left(\theta\right)$ and $\cot \left(\theta\right)$ have the same sign, then the terminal side of $\theta$ must be in QI or QII.

#### Explanation:

$\cos \left(\theta\right)$ is positive anywhere $x > 0$, so QI and QIV.
$\cos \left(\theta\right)$ is negative when $x < 0$, so in QII and QIII.

$\cot \left(\theta\right)$ is positive anywhere $x$ and $y$ have the same sign, so QI and QIII.
$\cot \left(\theta\right)$ is negative when $x$ and $y$ have opposite signs, so QII and QIV.

From this we know $\cos \left(\theta\right) > 0$ and $\cot \left(\theta\right) > 0$ in QI.
Both $\cos \left(\theta\right)$ and $\cot \left(\theta\right)$ are negative in QII.