Convert a Rectangular Equation to a Polar Equation (Multiple Examples)

Описание: Learn how to convert rectangular equations (with x and y) into polar equations (with r and θ)! This video will guide you through the essential relationships between these coordinate systems and show you how to transform equations step-by-step. We'll work through four clear examples together.

In this tutorial, you will discover:

Rectangular vs. Polar Coordinates: Understanding (x, y) vs. (r, θ) and their geometric representation.
Key Conversion Relationships:
x^2 + y^2 = r^2 (from Pythagorean Theorem)
tan(θ) = y/x
x = r cos(θ)
y = r sin(θ)
Step-by-Step Conversion Examples:
Example 1: Circle (x^2 + y^2 = 121) - See how a simple substitution transforms it into polar form.
Example 2: Horizontal Line (y = 7) - Learn to substitute y = r sin(θ) and solve for r.
Example 3: Diagonal Line through Origin (y = √3x) - Substitute both x and y and simplify to find θ.
Example 4: Vertical Line (x = 6) - Substitute x = r cos(θ) and solve for r.
Simplifying Polar Equations: Using reciprocal trigonometric identities (e.g., 1/sin(θ) = csc(θ), 1/cos(θ) = sec(θ)) to write equations in their most condensed form.
This video is perfect for pre-calculus and calculus students learning about coordinate system conversions.

Timestamps:

0:00 - Introduction: Converting Rectangular to Polar Equations
0:30 - Understanding the Relationship Between Rectangular and Polar Coordinates
1:00 - Key Conversion Formulas (x^2+y^2=r^2, tanθ=y/x, x=rcosθ, y=rsinθ)
2:30 - Example 1: x^2 + y^2 = 121 (Converting a Circle)
3:40 - Example 2: y = 7 (Converting a Horizontal Line)
5:20 - Example 3: y = √3x (Converting a Diagonal Line through the Origin)
7:20 - Example 4: x = 6 (Converting a Vertical Line)
8:50 - Conclusion & Next Steps: Converting Polar to Rectangular Equations
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