Mathematics Coordinate Geometry Polar Coordinate System | Conversion between Rectangular and Polar...

Polar Coordinate System | Conversion between Rectangular and Polar Equations

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Cartesian and polar coordinate system we have discussed earlier, see here ». In this article we will discuss about the conversion between them.[mathjax]

Polar coordinate System


In polar coordinates, we determine position of a Point or any Geometric Element on a plane by radical distance $r$ from the pole $0$, and rotational angle $θ$ from the polar axis. In cartesian coordinates, we specify any position by left-right, up-down distance $(x, y)$ from the origin $(0, 0)$.

Polar Coordinate System - csschool

let $P$ is a point in polar coordinates, $P(r, θ)$. if we superposition the polar axis on a Cartesian plane, then the position of $P$ in Cartesian coordinates, $P(x, y)$. Therefore, according to trigonometric ratio, $sin θ=\frac{y}{r}$, $cos θ=\frac{x}{r}$, $tan θ=\frac{y}{x}$.

Relation Between Cartesian and Polar Coordinates


According to Pythagorean-theorem, for any right-triangle, $r^{2}=x^{2}+y^{2}$. $r$ is the hypotenuse of right-triangle. $x$ is called Adjacent, and $y$ is the Opposite. polar coordinate system - right triangle - cs school

$tan θ=\frac{opposite}{adjacent}$, Or, $tan θ=\frac{y}{x}$.

Therefore, in our case, $r$ is the radical distance is polar-coordinates, $θ$, is the angle, and $x, y$ is the cartesian coordinate of the point, $P (x, y)$. So the basic relationship between polar and cartesian coordinates,

$x=r cos θ, \quad\text{} y=r sinθ$,
$r^{2}=x^{2}+y^{2}$, and
$tan θ=\frac{y}{x}$.

Converting between rectangular and Polar – Equations


As-we-know the Relationship between Cartesian, and Polar Coordinate-system, therefore conversion between them is now easy. Let see some examples.

Converting rectangular to polar – Equations


Example 1: Given an equation, $x^{2}+y^{2}=1$, transform it to Polar form.

Solution:  We know, $r^{2}=x^{2}+y^{2}$, so, $r^{2}=1, \text{Or, }r=±1$. Hence, this is the polar form of given rectangular equation.

Example 2: Given rectangular Equation $y=2x+1$, convert it to polar equation.

Solution: Similarly, from above relationship, $x= r cos θ, \quad\text{}y=r sin θ$. Substitute these value  to given equation, $r sin θ=2(r cos θ)+1,$ ⇒ $r sin θ-2r cos θ=1.$ Taking $r$ out, $r(sin θ-2cos θ)=1,$ finally,

$$r=\frac{1}{sin θ-2cos θ}.$$

This is the polar form of given rectangular equation.

Example 3: Given rectangular equation, $y=\frac{3}{x},$ convert it to polar equation.

Solution: Substitute, $x= r cos θ, \quad\text{}y=r sin θ,$ to above equation gives,

$$r sin θ=\frac{3}{r cos θ}, ⇒ r^{2}=\frac{3}{cos θ. sin θ}.$$

Similarly, This is the solution.

Converting polar to rectangular – Equations


Example 1: Given Polar Equation, $r=3cos θ,$ convert it to rectangular form.

Solution: multiplying $r$ both side of the equation, $r.r = 3r.cos θ$.
As we know, $x= r cos θ\text{, }r=x^{2}+y^{2},$ therefore, $x^{2}+y^{2}=3x$. Finally, $x^{2}+y^{2}-3x=0$, is the rectangular form of the polar equation.

Example 2: Polar equation, $r^{2}=-3sec θ$, transform it to rectangular equation.

Solution: $r^{2}=-\frac{3}{cos θ},$ or, $\frac{1}{r}.r^{2}=-\frac{3}{r. cos θ}$. Or, $r=\frac{3}{x},$ because, $x=r.cos θ.$ Again, $r=\sqrt{x^{2}+y^{2}}$. Therefore, the rectangular equation, $\sqrt{x^{2}+y^{2}}=\frac{3}{x},$ or, $x^{2}+y^{2}=\frac{9}{x^{2}}.$

This is the end of discussion on cartesian or rectangular and polar coordinate system. And equation-conversion between them.

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