Converting Between Polar and Rectangular Coordinates

Converting Between Polar and Rectangular Coordinates

Introduction to Coordinate Systems

In trigonometry, points in the plane can be represented using either rectangular (Cartesian) coordinates or polar coordinates. Understanding how to convert between these systems is essential for solving many trigonometric problems and for applications in physics and engineering.

• Rectangular Coordinates (x, y): Specify a point's horizontal and vertical distances from the origin.

• Polar Coordinates (r, θ): Specify a point's distance from the origin (r) and the angle (θ) measured from the positive x-axis.

Formulas for Conversion

• From Polar to Rectangular:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

• From Rectangular to Polar:

  • $r = \sqrt{x^2 + y^2}$

  • $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Example: Converting from Polar to Rectangular

Given the polar coordinate $(4, 45^\circ)$:

• $x = 4 \cos 45^\circ = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2}$

• $y = 4 \sin 45^\circ = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2}$

• Rectangular coordinate: $(2\sqrt{2}, 2\sqrt{2})$

Example: Converting from Rectangular to Polar

Given the rectangular coordinate $(-3, 3)$:

• $r = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$

• $\theta = \tan^{-1}\left(\frac{3}{-3}\right) = \tan^{-1}(-1) = 135^\circ$ (since the point is in the second quadrant)

• Polar coordinate: $(3\sqrt{2}, 135^\circ)$

Table: Summary of Conversion Formulas

From	To	Formulas
Polar $(r, \theta)$	Rectangular $(x, y)$	$x = r \cos \theta$ $y = r \sin \theta$
Rectangular $(x, y)$	Polar $(r, \theta)$	$r = \sqrt{x^2 + y^2}$ $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Applications

• Converting between coordinate systems is essential in calculus, physics (especially in dealing with circular motion), and engineering.

• Polar coordinates are particularly useful for describing curves and regions with circular symmetry.