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 = r \cos(\theta)

  • y = r \sin(\theta)

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

    • r = \sqrt{x^2 + y^2}

    • \tan(\theta) = \frac{y}{x}

Conversion from Polar to Rectangular Coordinates

• General formula to convert polar to rectangular:

  • x = r\cos(\theta)

  • y = r\sin(\theta)

Example Conversions

1. Convert to Rectangular Coordinates:

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

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

     • Calculate y:
       y = 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 = 5 and angle \theta = \frac{\pi}{3}

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

     • Calculate y:
       y = 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 = \sqrt{x^2 + y^2}

  • Calculate θ:

  • \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 = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2

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

     • Calculate θ:
       \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 = -3\sin(\theta) ,

  • To convert to rectangular form, derive:

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

    • This leads to r^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 = r\cos(\theta)

  • y = r\sin(\theta)

  • r = \sqrt{x^2 + y^2}

  • \tan(\theta) = \frac{y}{x}