The polar coordinate system offers a unique way to represent points and curves in two dimensions. Instead of using x and y coordinates, it uses distance from the origin and angle. This system is particularly useful for describing circular or spiral shapes.

Converting between polar and rectangular coordinates is a key skill. It allows us to switch between systems, choosing the one that makes a particular problem easier to solve. Graphing polar equations often reveals beautiful, symmetrical patterns not easily seen in rectangular form.

Polar Coordinate System

Plotting in polar coordinates

• Polar coordinates defined by distance $r$ from origin (pole) to point and angle $\theta$ formed by line segment from origin to point and positive x-axis
• Polar point denoted as $(r, \theta)$
• Angle $\theta$ measured in radians or degrees
  • Positive angles measured counterclockwise from positive x-axis (45°, 90°)
  • Negative angles measured clockwise from positive x-axis (-30°, -120°)
• To plot point in polar coordinates, draw line segment from origin at given angle $\theta$ and measure distance $r$ units along line segment to locate point (2, 60°)
• Polar grid: A coordinate system consisting of concentric circles and radial lines used for plotting points in polar coordinates

Conversion of coordinate systems

• Convert polar coordinates $(r, \theta)$ to rectangular coordinates $(x, y)$ using $x = r \cos(\theta)$ and $y = r \sin(\theta)$
  • (3, $\frac{\pi}{4}$) converts to ($\frac{3\sqrt{2}}{2}$, $\frac{3\sqrt{2}}{2}$)
• Convert rectangular coordinates $(x, y)$ to polar coordinates $(r, \theta)$ using $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(\frac{y}{x})$, adjusting for quadrant based on signs of $x$ and $y$
  • (2, 2) converts to ($2\sqrt{2}$, $\frac{\pi}{4}$)
• Complex plane: A two-dimensional representation of complex numbers where the real part is plotted on the horizontal axis and the imaginary part on the vertical axis, often used in conjunction with polar coordinates

Polar Equations and Graphs

Transformation of polar equations

• Convert polar equation to rectangular form by substituting $x = r \cos(\theta)$ and $y = r \sin(\theta)$ into polar equation and simplifying to express in terms of $x$ and $y$
  • $r = 2\cos(\theta)$ becomes $x^2 + y^2 = 2x$
• Convert rectangular equation to polar form by substituting $x = r \cos(\theta)$ and $y = r \sin(\theta)$ into rectangular equation, simplifying to express in terms of $r$ and $\theta$, and using trigonometric identities like $\cos^2(\theta) + \sin^2(\theta) = 1$
  • $x^2 + y^2 = 4$ becomes $r = 2$
• Parametric equations: A method of representing curves using separate equations for x and y in terms of a parameter, often used in conjunction with polar coordinates

Graphing polar equations

• Graph polar equation by creating table of $\theta$ values (usually in interval $[0, 2\pi]$ or $[0, \pi]$), calculating corresponding $r$ values for each $\theta$, plotting points $(r, \theta)$ in polar coordinate system, and connecting points smoothly to form graph
• Common polar equation graphs include circle $r = a$ (constant radius), cardioid $r = a(1 \pm \cos(\theta))$ (heart-shaped), rose curves $r = a \cos(n\theta)$ or $r = a \sin(n\theta)$ (petals), limaçon $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$ (inner loop), and lemniscate $r^2 = a^2 \cos(2\theta)$ (figure-eight)
• Conic sections: Curves formed by the intersection of a plane and a cone, which can be represented in polar form (e.g., ellipses, parabolas, and hyperbolas)

Polar vs rectangular representations

• Some curves have simpler equations in polar form than rectangular form
  • Cardioid $r = a(1 + \cos(\theta))$ more complex in rectangular coordinates
• Symmetry in polar equations
  • Graph symmetric about polar axis if equation unchanged when $\theta$ replaced by $-\theta$
  • Graph symmetric about pole if equation unchanged when $\theta$ replaced by $\theta + \pi$
• Periodicity in polar equations
  • Graph has $n$ petals or lobes if equation repeats every $\frac{2\pi}{n}$ radians, where $n$ is an integer (3 petals for $r = \cos(3\theta)$)
• Polar symmetry: The property of a polar curve remaining unchanged under certain transformations, such as reflection or rotation