Replace the polar equation with equivalent Cartesian equations. \frac{x^{2}}{9}+\frac{y^{2}}{4}=1

Equations
Replace the polar equation with equivalent Cartesian equations. $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$

2021-06-08
Step 1
We have given a cartesian equation $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$. The equation $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$ is a standard form of ellipse
$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ and generally, we take polar coordinates x as a $$\cos 0$$.
Here, a=3. So set $$x=3 \cos 0$$. Now, substitute $$x=3 \cos 0$$ in $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$
$$\frac{(3\cos 0)^{2}}{9}+\frac{y^{2}}{4}=1$$
$$\frac{9\cos^{2}0}{9}+\frac{y^{2}}{4}=1$$
Step 2
Use the identity $$1-\cos^{2}0=\sin^{2}0$$ to simplify it further.
$$\cos^{2}0+\frac{y^{2}}{4}=1$$
$$\frac{y^{2}}{4}=1-\cos^{2}0$$
$$=\sin^{2}0$$
$$y^{2}=4\sin^{2}0$$
Take square root on both sides to find y. Consider only positive sign.
$$\sqrt{y^{2}}=\pm \sqrt{4\sin^{2}0}$$
$$y=\pm 2\sin 0$$
$$y=2\sin 0$$ [Take only (+) sign]
Hence, the polar equation is $$x=3 \cos 0, y=2 \sin 0$$.