Polar Coordinates

Course Resources

  • PDF notes available at the end of the class on Blackboard.

  • Homework assignments will also be posted.

Graphing in Polar Coordinates

  • Understanding Polar Coordinates:

    • Polar coordinates are represented as (r, θ).

    • r indicates the distance from the pole (origin), and θ indicates the angle.

    • For example, the point (2, π/4) means:

      • Move along the angle π/4.

      • Move a distance of 2.

  • Graphing Points:

    • Locate the angle first.

    • For (2, π/4):

      • Find π/4 on the unit circle.

      • Move 2 units towards this angle.

    • For negative radius values like (-2, π/4):

      • Start at angle π/4, then move 2 units away from the angle.

  • Coterminal Angles:

    • Coterminal angles can be found by adding or subtracting 2π.

    • For example, π/4 and 9π/4 represent the same point on the graph.

  • Negative Angles:

    • For negative angles like (-2, -π/4):

      • Find the angle -π/4, then move 2 units towards the opposite direction of the angle.

  • Estimation on the Graph:

    • Points do not have to be on the unit circle (e.g., 1.5π/3).

    • Use interpolation (splitting the difference) to find where to place the point.

Converting Between Polar and Rectangular Coordinates

  • Conversion Basics:

    • Polar coordinates (r, θ) can be converted to rectangular coordinates (x, y) using:

      1. x = r * cos(θ)

      2. y = r * sin(θ)

    • To convert back, use:

      1. r = √(x² + y²)

      2. θ = tan⁻¹(y/x)

  • Converting Polar to Rectangular:

    • For (4, π/3):

      • Set up a right triangle where:

        • r = 4 (hypotenuse)

        • Use trig ratios to find x and y:

          • x = 4 * cos(π/3) = 2

          • y = 4 * sin(π/3) = 2√3

  • Example Problems:

    • For (2, -π/4): Find the angle first, determine coterminal angles if needed, and factor in the negative radius.

Essential Conversion Formulas

  1. x = r * cos(θ)

  2. y = r * sin(θ)

  3. r = √(x² + y²)

  4. θ = tan⁻¹(y/x)

Rectangular to Polar Conversion Examples

  • Given a rectangular point (-2, 3):

    1. Calculate r:

      • r² = (-2)² + (3)² = 13 → r = ±√13.

    2. Calculate θ:

      • θ = tan⁻¹(3/-2). Check the quadrant to ensure it points in the correct direction.

Graphing Polar Equations

  • Identify the nature of the equation.

  • Rearrange or translate into rectangular form when necessary.

  • Examples:

    • r = 2: Represents a circle with radius 2 around the pole.

    • r = 4csc(θ): Convert to rectangular: y = 4 (after identifying r = 4sin(θ)).

      • Use the relationship between sine and y values to express the polar equation in rectangular form.

Common Graphing Problems & Solutions

  • For equations in polar form, ensure they are solved for r to graph correctly.

  • Apply properties like the Pythagorean theorem to relate x and y while solving equations.

  • For lines, ensure to express in a format only containing x's and y's.