# Parametric Equation with a Parameter and an Angle

So, just now I was doing computing some parametric equations and graphing them when I came across a weird problem. I attempted to search for a solution, but I don't think I was able to describe what I meant well enough to get any useful results.

$x=t\cos(\theta)$ and $y = t \sin(\theta)$, $t>0$ over the interval $[0,\pi]$

I was able to eliminate the parameter to get $y = x \tan(\theta)$. The problem I am having is what do I do with the $\theta$?

All help is greatly appreciated.

• Looks like you have a curve in polar form. Yes you can convert it to Cartesian form, but it would really help to know how $t$ is defined. Substitute $t = \sqrt{x^2+y^2}$ in the definition of $t.$ With an arbitrary $t$ as it is, your curve could be just about anything. – Doug M Sep 2 '16 at 16:58
• Should I make a definition of $\theta$ relative to $t$ and graph that? – Stephen Fratamico Sep 2 '16 at 17:17
• There may not be a nice expression in Cartesian form. Easiest thing to do is take a ruler and a protractor and measure $t$ units out from your origin for each angle of $\theta.$ But, maybe some examples would help. What is $t$? – Doug M Sep 2 '16 at 17:32
• That's the polar equation of a circle of radius $t$, where the tracing point is at an angle$\theta$. So if you keep $t$ constant you have a circle, if you keep $\theta$ constant you have a line ($y=tan(\theta) x$), if both are variable you have all the plane. – G Cab Sep 2 '16 at 22:31

## 1 Answer

HINT

$\theta$ is a parameter. We have equation $$x=0$$ when $\cos\theta=0$ and $$y=x\tan\theta$$ when $\cos\theta\not=0.$