Understanding the Formula
A point in polar coordinates is represented as \( P(\rho, \phi) \), where:
- \( \rho \) is the radial distance from the origin
- \( \phi \) is the angle measured from the positive x-axis (in degrees)
To find the distance between two points \( A(\rho_1, \phi_1) \) and \( B(\rho_2, \phi_2) \):
Step 1: Convert polar coordinates to Cartesian coordinates:
$$ x_1 = \rho_1 \cos(\phi_1), \quad y_1 = \rho_1 \sin(\phi_1) $$
$$ x_2 = \rho_2 \cos(\phi_2), \quad y_2 = \rho_2 \sin(\phi_2) $$
Step 2: Apply the Euclidean distance formula:
$$ d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} $$
Step 3: Substitute the expressions:
$$ d = \sqrt{[\rho_1 \cos(\phi_1) - \rho_2 \cos(\phi_2)]^2 + [\rho_1 \sin(\phi_1) - \rho_2 \sin(\phi_2)]^2} $$
which simplifies to
$$ d = \sqrt{\rho_1^2 + \rho_2^2 - 2\rho_1\rho_2\cos(\phi_1 - \phi_2)} $$
Note: Angles are entered in degrees. The calculator automatically converts them to radians for computation.