# Thread: How do i convert this into a rectangular equation

1. ## How do i convert this into a rectangular equation

y = cos(2t), y = sin(2t);

t is greater than or equal to -pie and less than or equal to pie.

2. Originally Posted by cityismine
y = cos(2t), y = sin(2t);

t is greater than or equal to -pie and less than or equal to pie.
$\displaystyle y^2 = \cos^2 (2t)$.
$\displaystyle x^2 = \sin^2 (2t)$.

Therefore $\displaystyle x^2 + y^2 = 1$.

Domain is [-1, 1] and range is [-1, 1] so it's the entire circle.

3. Just so you know, given

$\displaystyle x=a\cos(n\theta)$

and $\displaystyle y=a\sin(n\theta)$

You will always get a circle of radius a

$\displaystyle x^2+y^2=a^2\cos^2(n\theta)+a^2\sin^2(n\theta)=a^2$

and if you have

$\displaystyle x=a\cos(n\theta)$

and

$\displaystyle y=b\sin(n\theta)$

You will get an ellipse

$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{a^2\cos^2(n\ theta)}{a^2}+\frac{b^2\sin^2(n\theta)}{b^2}=1$

4. -pie and less than or equal to pie.
Come on...this is math, not a bakery.

5. The answer in the back of the book is: y=(1/2)x^2-1

Is the answer in the book wrong?

6. Originally Posted by cityismine
The answer in the back of the book is: y=(1/2)x^2-1

Is the answer in the book wrong?
Yes

For we have that our equation is

$\displaystyle x^2+y^2=1\Rightarrow{y=\pm\sqrt{{\color{red}1-x^2}}}$

7. Originally Posted by cityismine
The answer in the back of the book is: y=(1/2)x^2-1

Is the answer in the book wrong?
If the parametric equations were actually $\displaystyle x = \cos t$ and $\displaystyle y = \cos (2t)$ then the cartesian equation would be $\displaystyle y = 2x^2 - 1$ ........

If the parametric equations were actually $\displaystyle x = 2 \sin t$ and $\displaystyle y = \cos (2t)$ then the cartesian equation would be $\displaystyle y = 1 - \frac{x^2}{2}$ ........

Are the equations you posted correct?

8. Yes, I posted them correctly, I just double checked. I guess it's a typo in the book.