Polar coordinates Knowledge-Based Study

Study notes

1. The Polar Coordinate System

   In a standard rectangular (Cartesian) coordinate system, points are located using horizontal and vertical directed distances, denoted by the pair $(x, y)$. The polar coordinate system provides an alternative framework where a point in the plane is located using a distance from a fixed reference point and an angle relative to a fixed reference direction.

   Foundational Elements

   • The Pole (Origin): A fixed central point, analogous to the origin $(0, 0)$ in Cartesian coordinates.
   • The Polar Axis: A horizontal ray directed to the right from the pole, corresponding to the positive $x$-axis.
   • Polar Coordinates $(r, \theta)$: The position of a point $P$ is designated by an ordered pair $(r, \theta)$, where:
     • $r$ represents the directed distance from the pole to point $P$.
     • $\theta$ represents the directed angle from the polar axis to the ray containing the line segment from the pole through $P$.

   Sign Conventions

   Angles are measured counterclockwise from the polar axis for positive values of $\theta$ and clockwise for negative values. If $r > 0$, the point lies on the terminal side of the angle $\theta$. If $r < 0$, the point is plotted by measuring the angle $\theta$ and then moving a distance $|r|$ in the direction opposite to the terminal side (along the ray pointing in direction $\theta + \pi$). If $r = 0$, the coordinates $(0, \theta)$ represent the pole regardless of the angle $\theta$.

   Non-Uniqueness in Polar Coordinates

   Unlike Cartesian coordinates, where each point has a single unique coordinate pair $(x, y)$, every point in the polar plane has infinitely many coordinate representations $(r, \theta)$. This occurs because angles separated by complete revolutions ($2\pi$ radians or $360^\circ$) share the same terminal side, and reflecting through the pole by changing the sign of $r$ corresponds to adding an odd multiple of $\pi$.

   General Formulas

   If a point is described by $(r, \theta)$, all possible representations for that same point can be expressed as:

   $$(r, \theta + 2k\pi) \quad \text{for any integer } k$$$$(-r, \theta + (2k+1)\pi) \quad \text{for any integer } k$$

   Example

   Consider the point $P\left(3, \frac{\pi}{4}\right)$. Alternative valid representations for point $P$ include:

   • Adding $2\pi$ to the angle: $\left(3, \frac{\pi}{4} + 2\pi\right) = \left(3, \frac{9\pi}{4}\right)$
   • Subtracting $2\pi$ from the angle: $\left(3, \frac{\pi}{4} - 2\pi\right) = \left(3, -\frac{7\pi}{4}\right)$
   • Negating $r$ and adding $\pi$: $\left(-3, \frac{\pi}{4} + \pi\right) = \left(-3, \frac{5\pi}{4}\right)$
   • Negating $r$ and subtracting $\pi$: $\left(-3, \frac{\pi}{4} - \pi\right) = \left(-3, -\frac{3\pi}{4}\right)$

   Converting Between Polar and Rectangular Forms

   Superimposing the polar coordinate system onto the Cartesian plane—aligning the pole with the origin and the polar axis with the positive $x$-axis—allows us to derive straightforward trigonometric relationships between $(r, \theta)$ and $(x, y)$.

   Polar to Rectangular Conversion

   Given the polar coordinates $(r, \theta)$, the corresponding rectangular coordinates $(x, y)$ are found using basic right-triangle trigonometry:

   $$x = r \cos \theta$$$$y = r \sin \theta$$

   Example: Convert $\left(4, \frac{2\pi}{3}\right)$ to rectangular coordinates.

   $$x = 4 \cos\left(\frac{2\pi}{3}\right) = 4\left(-\frac{1}{2}\right) = -2$$$$y = 4 \sin\left(\frac{2\pi}{3}\right) = 4\left(\frac{\sqrt{3}}{2}\right) = 2\sqrt{3}$$

   Rectangular to Polar Conversion

   Given the rectangular coordinates $(x, y)$, the polar coordinates $(r, \theta)$ are determined using:

   $$r^2 = x^2 + y^2 \implies r = \sqrt{x^2 + y^2}$$$$\tan \theta = \frac{y}{x} \quad (x \neq 0)$$

   Note: When choosing $\theta$, ensure that the angle terminates in the quadrant containing $(x, y)$. If $x < 0$, calculate $\theta = \arctan\left(\frac{y}{x}\right) + \pi$.

   Converting Equations Between Coordinate Systems

   Translating equations between rectangular and polar forms is a critical skill for analyzing curves such as circles, lines, and conics.

   Rectangular to Polar Equations

   To convert an equation from Cartesian to polar form, replace every occurrence of $x$ with $r \cos \theta$ and $y$ with $r \sin \theta$, then simplify using trigonometric identities where applicable.

   • Vertical line: $x = c \implies r \cos \theta = c \implies r = c \sec \theta$
   • Horizontal line: $y = d \implies r \sin \theta = d \implies r = d \csc \theta$
   • Circle centered at the origin: $x^2 + y^2 = a^2 \implies r^2 = a^2 \implies r = a$
   • Circle shifted along the $x$-axis: $x^2 + y^2 = 2ax \implies r^2 = 2ar \cos \theta \implies r = 2a \cos \theta$

   Polar to Rectangular Equations

   To convert from polar to rectangular form, apply algebraic manipulations—such as multiplying both sides by $r$ or taking trigonometric functions—to introduce terms like $r^2$, $r \cos \theta$, and $r \sin \theta$.

   Example: Convert $r = 6 \sin \theta$ into rectangular form.

   1. Multiply both sides by $r$: $$r^2 = 6r \sin \theta$$
   2. Substitute $r^2 = x^2 + y^2$ and $r \sin \theta = y$: $$x^2 + y^2 = 6y$$
   3. Rearrange and complete the square: $$x^2 + (y - 3)^2 = 9$$

   This reveals that $r = 6 \sin \theta$ represents a circle centered at $(0, 3)$ with radius $3$.