How do you convert the equation involving x and y into polar form using the variables r and θ?

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Converting Cartesian Coordinates to Polar Form

To convert an equation from Cartesian coordinates (in terms of x and y) to polar coordinates (in terms of r and θ), we use the following relationships:

  • x = r * cos(θ)
  • y = r * sin(θ)
  • r = sqrt(x² + y²)
  • θ = atan2(y, x)

Let’s go through a step-by-step example to illustrate this.

Example: Convert the equation x² + y² = 1 to Polar Form

1. Start with the given equation:

x² + y² = 1

2. Substitute the polar relationships into the equation:

(r * cos(θ))² + (r * sin(θ))² = 1

3. Simplify the equation:

r² * (cos²(θ) + sin²(θ)) = 1

Since by the Pythagorean identity, we know that cos²(θ) + sin²(θ) = 1, the equation simplifies to:

r² = 1

4. Taking the square root of both sides yields:

r = 1

Now, this tells us that the equation describes a circle of radius 1 centered at the origin in polar coordinates.

Conclusion

In conclusion, converting equations from Cartesian to polar form involves substituting x and y with their polar equivalents, followed by simplification. This example illustrates how to change the form of an equation to understand its geometry more clearly in the polar coordinate system.

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