Question

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

Answer 1

Hint: 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,
\[\begin{align}
  & \Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}{{\cos }^{2}}\theta +{{r}^{2}}{{\sin }^{2}}\theta \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right) \\
\end{align}\]
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,
\[\begin{align}
  & \Rightarrow {{x}^{2}}+{{y}^{2}}=4y \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}-4y=0 \\
\end{align}\]
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.