# Relating X and Y

• Nov 18th 2008, 08:42 AM
gearshifter
Relating X and Y
Eliminate t to give an equation that relates x and y:

y= ??
• Nov 18th 2008, 10:01 AM
Aryth
Remember the identity:

$\displaystyle \sin^2{t} + \cos^2{t} = 1$

So:

$\displaystyle sin^2{t} = 1 - \cos^2{t}$

$\displaystyle y = 1 - \cos^2{t} - 3$

$\displaystyle y = -\cos^2{t} - 2$

Now we look at x:

$\displaystyle x = \cos{t}$

Square both sides:

$\displaystyle x^2 = \cos^2{t}$

Multiply both sides by -1:

$\displaystyle -x^2 = -\cos^2{t}$

Now, subtract both sides by 2:

$\displaystyle -x^2 - 2 = -\cos^2{t} - 2$

We know that:

$\displaystyle y = -\cos^2{t} - 2$

So, we get:

$\displaystyle y = -x^2 - 2$

And there you go.
• Nov 18th 2008, 10:16 AM
earboth
Quote:

Originally Posted by gearshifter
Eliminate t to give an equation that relates x and y:

y= ??

Here is a slightly different attempt:

$\displaystyle x = \cos(t)~\implies~x^2=\cos^2(t)$

$\displaystyle y = \sin^2(t)-3~\implies~y+3=\sin^2(t)$

$\displaystyle y+3+x^2=\sin^2(t)+\cos^2(t)=1~\implies~\boxed{y=-x^2-2}$