Equation Conversions: Polar and Rectangular Coordinates

Polar and Rectangular Coordinates

Overview of Coordinate Systems

  • Rectangular Coordinates (x, y)

    • Also known as Cartesian coordinates.

    • Points are identified by a pair of numerical coordinates, (x, y).

  • Polar Coordinates (r, θ)

    • Points are identified by a distance from the origin (r) and an angle (θ).

    • Relationship to rectangular coordinates:

    • x=rcos⁡(θ)x = r \cos(\theta)

    • y=rsin⁡(θ)y = r \sin(\theta)

    • The relationship between r, x, and y can also be expressed as:

      • r=x2+y2r = \sqrt{x^2 + y^2}

      • tan⁡(θ)=yx\tan(\theta) = \frac{y}{x}

Conversion from Polar to Rectangular Coordinates

  • General formula to convert polar to rectangular:

    • x=rcos⁡(θ)x = r\cos(\theta)

    • y=rsin⁡(θ)y = r\sin(\theta)

Example Conversions

  1. Convert to Rectangular Coordinates:

    • Given polar coordinates (r, θ) = (4, 3π/4)

      • Calculate x:
        x=4⋅(−22)=−22x = 4 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -2\sqrt{2}

      • Calculate y:
        y=4⋅sin⁡(3π/4)=4⋅(22)=22y = 4 \cdot \sin(3\pi/4) = 4 \cdot \left(\frac{\sqrt{2}}{2}\right) = 2\sqrt{2}

      • Final result: (x, y) = (-2√2, 2√2)

  2. Convert to Rectangular Coordinates:

    • Given r=5r = 5 and angle θ=π3\theta = \frac{\pi}{3}

      • Calculate x:
        x=5cos⁡(π3)=5⋅12=2.5x = 5\cos(\frac{\pi}{3}) = 5\cdot\frac{1}{2} = 2.5

      • Calculate y:
        y=5sin⁡(π3)=5⋅32=532y = 5\sin(\frac{\pi}{3}) = 5\cdot\frac{\sqrt{3}}{2} = 5\frac{\sqrt{3}}{2}

      • Final result: (x, y) = (2.5, 5√3/2)

Conversion from Rectangular to Polar Coordinates

  • General formula to convert rectangular to polar:

    • Calculate r:

    • r=x2+y2r = \sqrt{x^2 + y^2}

    • Calculate θ:

    • θ=tan⁡−1(yx)\theta = \tan^{-1}\left(\frac{y}{x}\right)

Example Conversions

  1. Convert to Polar Coordinates:

    • Given rectangular coordinates (x, y) = (√3, 1)

      • Calculate r:
        r=(3)2+12=3+1=4=2r = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2

      • Calculate θ:
        θ=tan⁡−1(13)=π6\theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}

      • Final result: (r, θ) = (2, π/6)

  2. Convert to Polar Coordinates:

    • Given (x, y) = (4, -1)

      • Calculate r:
        r=42+(−1)2=16+1=17r = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}

      • Calculate θ:
        θ=tan⁡−1(−14)\theta = \tan^{-1}\left(\frac{-1}{4}\right)

      • Final result: (r, θ) = (√17, θ)

Additional Examples of Conversion Techniques

  • Converting more complex equations involving polar coordinates:

    • For an expression like r=−3sin⁡(θ)r = -3\sin(\theta),

    • To convert to rectangular form, derive:

      • Substitute y=rsin⁡(θ)y = r\sin(\theta) into the equation.

      • This leads to r2=−3yr^2 = -3y with modifications leading to the standard form.

Summary of Problem Assignments

  • Problems from worksheet labeled WS #11-21, including odd numbers: 33, 35, 39, 41, 49-55, odd; 61, 63.

Key Mathematical Relationships

  • Key relationships include:

    • x=rcos⁡(θ)x = r\cos(\theta)

    • y=rsin⁡(θ)y = r\sin(\theta)

    • r=x2+y2r = \sqrt{x^2 + y^2}

    • tan⁡(θ)=yx\tan(\theta) = \frac{y}{x}