Convert Polar to Rectangular Calculator with Steps
To convert polar coordinates to rectangular coordinates, you need to understand the relationship between the two coordinate systems and follow a simple two-step process. Let’s break it down and create a comprehensive table to illustrate the conversion.
Conversion Process
- Calculate x: x = r * cos(θ)
- Calculate y: y = r * sin(θ)
Where:
- r is the radius (distance from the origin)
- θ is the angle in radians (measured counterclockwise from the positive x-axis)
Conversion Table
Here’s a table showing the conversion of some common polar coordinates to rectangular coordinates:
| r (Radius) | θ (Degrees) | x (Rectangular) | y (Rectangular) |
|---|---|---|---|
| 1 | 0 | 1.000000 | 0.000000 |
| 1 | 90 | 0.000000 | 1.000000 |
| 1 | 180 | -1.000000 | 0.000000 |
| 1 | 270 | 0.000000 | -1.000000 |
| 2 | 30 | 1.732051 | 1.000000 |
| 2 | 45 | 1.414214 | 1.414214 |
| 2 | 60 | 1.000000 | 1.732051 |
Key Points to Remember
- Angle Conversion: If θ is given in degrees, convert it to radians before applying the formulas. To convert degrees to radians, use the formula: θ (in radians) = θ (in degrees) * (π / 180).
- Quadrant Awareness: The signs of x and y in the rectangular coordinates will depend on the quadrant in which the polar point lies.
- Special Angles: Memorizing the sine and cosine values for common angles (0°, 30°, 45°, 60°, 90°) can be helpful for quick mental conversions.
- Calculator Use: For more complex calculations, use a scientific calculator with trigonometric functions.
- Precision: The number of decimal places in the rectangular coordinates may vary depending on the required precision. The table above uses six decimal places for illustration.
By understanding these principles and using the conversion formulas, you can easily convert any polar coordinate to its rectangular equivalent.