Question

How do you convert $r=2\sin \theta$ to rectangular form?

Answer 1

We know that Cartesian or rectangular and polar are two different coordinate systems. One implements $\left( x,y \right)$ and the other implements $\left( r,\theta \right)$ . Conversion or transformation of one coordinate system to another can be done using two basic formula
$\begin{aligned} & x=r\cos \theta ....\text{formula }1 \\\ & y=r\sin \theta ....\text{formula }2 \\\ \end{aligned}$
At first, we replace the $\sin \theta$ term with $\dfrac{y}{r}$ using the above formula and then, we get the equation in terms of ${{r}^{2}}$ and $y$ . Replacing the ${{r}^{2}}$ term with ${{x}^{2}}+{{y}^{2}}$ using the above formula, we finally get an equation in the rectangular coordinate system.

Complete step-by-step solution:
Coordinate systems are basically of two major types, Cartesian (or rectangular) and polar. Cartesian coordinates are the ones represented by $\left( x,y \right)$ and polar coordinates are those which are represented by $\left( r,\theta \right)$ . Conversion from rectangular coordinate system to polar coordinate system can be done easily using the two transformation formula
$\begin{aligned} & x=r\cos \theta ....\text{formula }1 \\\ & y=r\sin \theta ....\text{formula }2 \\\ \end{aligned}$
The given equation is
$r=2\sin \theta$
$formula1$ can be rearranged as $\sin \theta =\dfrac{y}{r}$ . At first, we replace $\sin \theta$ in the given equation with $\dfrac{y}{r}$ using $\text{formula }1$ . The equation thus becomes,
$\Rightarrow r=2\dfrac{y}{r}$
Multiplying both sides of the above equation with $r$ , we get
$\Rightarrow {{r}^{2}}=2y....\text{equation }1$
$formula1$ and $\text{formula }2$ can be combined to get another formula. This is done by squaring and adding the two formulas. Thus, the third formula becomes,
${{x}^{2}}+{{y}^{2}}={{r}^{2}}\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)$
We know that, $\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)=1$ . So, the formula becomes,
$\Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}....\text{formula }3$
Using $\text{formula }3$ in $\text{equation }1$ , we get,
$\Rightarrow {{x}^{2}}+{{y}^{2}}=2y$
Therefore, we can conclude that the equation in polar form $r=2\sin \theta$ can be converted to the equation ${{x}^{2}}+{{y}^{2}}=2y$ which is in rectangular form.

Note: We must be careful while applying the basic formulas and students often mistake the original formulas with $x\cos \theta =r,y\sin \theta =r$ . This leads to wrong results. If we were given any terms including $\tan \theta$ , we could replace it by $\dfrac{y}{x}$ . We should always replace the $r$ and $\theta$ terms separately, as this method avoids errors.