Find a rectangular-coordinate equation for the curve by eliminating the parameter.

Question

$$\begin{align} x &= (5)\sin(t)\\ y &= (5)\cos(t) \end{align}$$

within $0 < t < \pi$

I know how to graph this but I need help with turning it into a cartesian-coordinate equation.

Any help is welcome.

Answer 1

$x=5\sin t\implies x^2=25\sin^2 t$

$y=5\cos t \implies y^2=25\cos^2 t$

$x^2+y^2=25\sin^2 t+25\cos^2 t\implies x^2+y^2=25$ because $\sin^2 t+\cos^2 t = 1$

Answer 2

Use the identity $\sin^2x+\cos^2x=1$

Given $x=5\sin t\implies x^2=25\sin^2t$

$y=5\cos t\implies y^2=25\cos^2t$

Now,

$\dfrac{x^2}{25}=\sin^2t$ and $\dfrac{y^2}{25}=\cos^2t$

$\dfrac{x^2}{25}+\dfrac{y^2}{25}=\sin^2t+\cos^2t$

$\dfrac{x^2}{25}+\dfrac{y^2}{25}=1$