Question

How do you convert $r=4\sin \theta$ into cartesian form?

Answer 1

Substitute the value of $\sin \theta$ in terms of y and r using the relation: - $y=r\sin \theta$. Here, y – denotes the y – coordinate in cartesian form. Now, simplify the relation by cross – multiplication and use the cartesian formula: - ${{r}^{2}}={{x}^{2}}+{{y}^{2}}$ to get the required equation in cartesian form.
Now, we can see that the given relation is in polar form because it relates the radius vector (r) and angle ($\theta$). We need to change it into cartesian form that means we have to obtain the relationship between x and y – coordinate.

Complete step by step answer:
Let us consider a point (P) with cartesian coordinates P (x, y) and polar coordinates $\left( r,\theta \right)$, then the relation between the two-coordinate system is given as: -
$\Rightarrow x=r\cos \theta$ and $y=r\sin \theta$
Squaring both sides of the two equations and adding, we get,

& \Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}{{\cos }^{2}}\theta +{{r}^{2}}{{\sin }^{2}}\theta \\\ & \Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right) \\\ \end{aligned}$$ Using the identity: - $${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$$, we get, $$\Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}$$ - (1) Now, let us come to the question. So, we have, $$\Rightarrow r=4\sin \theta $$ Using the relation $$y=r\sin \theta $$ to replace the value of $$\sin \theta $$, we get, $$\Rightarrow r=4\times \dfrac{y}{r}$$ By cross – multiplication, we get, $$\Rightarrow {{r}^{2}}=4y$$ Using equation (1), we get, $$\begin{aligned} & \Rightarrow {{x}^{2}}+{{y}^{2}}=4y \\\ & \Rightarrow {{x}^{2}}+{{y}^{2}}-4y=0 \\\ \end{aligned}$$ Hence, the above relation represents the given equation in cartesian form. **Note:** One may note that if we are asked which type of curve the obtained equation represents then we can say that it represents a circle with centre (0, 2). You may directly remember that $$r=a\cos \theta $$ or $$r=a\sin \theta $$ generally represents a circle. You must remember the relationship between the cartesian coordinates and the polar coordinates to solve the above question. Also, remember the important relation: - $${{x}^{2}}+{{y}^{2}}={{r}^{2}}$$ as it will be used further in the coordinate geometry of circles.