Introduction to Polar Coordinates

Key Differences

• Rectangular Coordinates: Contains x and y variables.
• Polar Coordinates: Contains r and ( \theta ) (theta).
  • r: Radius of a circle.
  • ( \theta ): Angle measured from the positive x-axis.

Graphing Polar Coordinates

Example: Plotting ( (3, 45^{\circ}) )

1. Plot circles with radii 1, 2, and 3.
2. Draw a ray at a 45-degree angle to the third circle (radius 3).
   • Positive angles are measured counter-clockwise.

Example: Plotting ( (2, \frac{3\pi}{4}) )

1. Convert ( \frac{3\pi}{4} ) to degrees: 135 degrees.
2. Identify the quadrant: Quadrant II (between 90 and 180 degrees).
3. Stop at the second circle (radius 2).

Negative Radius Example: Plotting ( (-2, 60^{\circ}) )

1. Plot ( (2, 60^{\circ}) ) first.
2. For negative radius, plot in the opposite direction (add 180 degrees): 240 degrees.

Another Example: Plotting ( (-3, 120^{\circ}) )

1. Plot ( (3, 120^{\circ}) ) in quadrant II.
2. For negative radius, plot in the opposite direction: 300 degrees.

Finding Equivalent Polar Coordinates

Example: ( (2, 30^{\circ}) )

1. Original: ( (2, 30^{\circ}) )
2. Second: ( (2, -330^{\circ}) ) (subtract 360 degrees).
3. Third: ( (-2, 210^{\circ}) ) (add 180 degrees).
4. Fourth: ( (-2, -150^{\circ}) ) (add or subtract as needed).

Conversion Between Polar and Rectangular Coordinates

From Polar to Rectangular

• Equations:
  • ( x = r \cos(\theta) )
  • ( y = r \sin(\theta) )

Example: Convert ( (4, 60^{\circ}) )

• ( x = 4 \cos(60^{\circ}) = 2 )
• ( y = 4 \sin(60^{\circ}) = 2\sqrt{3} )
• Result: ( (2, 2\sqrt{3}) )

From Rectangular to Polar

• Equation for r: ( r = \sqrt{x^2 + y^2} )
• Equation for ( \theta ): ( \theta = \arctan(\frac{y}{x}) )

Example: Convert ( (2, -4) ) to Polar

1. Calculate ( r = \sqrt{2^2 + (-4)^2} = 2\sqrt{5} )
2. Calculate angle ( \theta ) using ( \arctan(\frac{-4}{2}) ) and adjust for quadrant.
3. Result: ( (2\sqrt{5}, 296.56^{\circ}) )

Example: Convert ( (-5, 5\sqrt{3}) ) to Polar

1. Calculate ( r = 10 )
2. Calculate angle ( \theta = 120^{\circ} ) (in quadrant II).
3. Result: ( (10, 120^{\circ}) ) or ( (10, \frac{2\pi}{3}) ) in radians.