# How do you convert r=6sectheta into cartesian form?

Jul 13, 2016

$r = 6 \sec \left(\theta\right) \to x = 6$

#### Explanation:

To convert from polar to cartesian, we can make use the following:

$x = r \cos \left(\theta\right)$
$y = r \sin \left(\theta\right)$
${r}^{2} = {x}^{2} + {y}^{2}$

Since $r = 6 \sec \left(\theta\right)$, we now have $x = {\underbrace{6 \sec \left(\theta\right)}}_{r} \cos \left(\theta\right)$

We can simplify this expression even further since

$\sec \left(\theta\right) = \frac{1}{\cos \left(\theta\right)}$, so our expression becomes

$x = \frac{6}{\cancel{\cos \left(\theta\right)}} \cdot \cancel{\cos \left(\theta\right)} = 6$

So our final answer is $x = 6$.

We can check this by graphing, for example.

In this graph, $x = 6$ and $r = 6 \sec \left(\theta\right)$ both overlap because they are essentially the same.