To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), you use specific trigonometric equations. The direct method, as referenced, is to use the equations x = r cos θ , y = r sin θ.
Understanding the Conversion
Polar coordinates describe a point in a plane by its distance from a central point (the origin) and the angle measured from a reference direction (the positive x-axis).
- r: The radial distance from the origin (0,0) to the point.
- θ: The angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point.
Cartesian coordinates describe a point by its distances from two perpendicular axes, typically the x-axis and y-axis.
- x: The horizontal distance from the origin along the x-axis.
- y: The vertical distance from the origin along the y-axis.
The conversion equations bridge these two systems, utilizing trigonometry to find the horizontal (x) and vertical (y) components based on the distance (r) and angle (θ).
The Formulas
Based on the provided reference, the fundamental formulas for converting from Polar (r, θ) to Cartesian (x, y) are:
- x = r cos θ
- y = r sin θ
These equations essentially project the radial distance r onto the x-axis (using the cosine of the angle) and onto the y-axis (using the sine of the angle).
Step-by-Step Conversion Process
Converting a point from polar to Cartesian coordinates is straightforward:
- Identify r and θ: Determine the radial distance (
r) and the angle (θ) of the polar coordinate. - Apply the x-formula: Calculate the x-coordinate using the formula
x = r cos θ. - Apply the y-formula: Calculate the y-coordinate using the formula
y = r sin θ. - Write the Cartesian coordinate: Express the result as the ordered pair (x, y).
Example Conversion
Let's convert the polar coordinate (√2, −π/4) mentioned in the reference to Cartesian form.
-
Identify r and θ:
r= √2θ= −π/4 radians (or -45 degrees)
-
Apply the x-formula:
x = r cos θx = √2 * cos(−π/4)x = √2 * (√2 / 2)(Since cos(−π/4) = cos(π/4) = √2 / 2)x = 2 / 2x = 1
-
Apply the y-formula:
y = r sin θy = √2 * sin(−π/4)y = √2 * (−√2 / 2)(Since sin(−π/4) = −sin(π/4) = −√2 / 2)y = −2 / 2y = −1
-
Write the Cartesian coordinate:
- The Cartesian coordinate is (1, -1).
Therefore, the polar point (√2, −π/4) is equivalent to the Cartesian point (1, -1).
Summary Table: Polar vs. Cartesian Conversion
| Conversion Direction | Formulas Used | Notes |
|---|---|---|
| Polar (r, θ) to Cartesian (x, y) | x = r cos θ y = r sin θ |
Angle θ must be in radians for calculation using standard functions. |
| Cartesian (x, y) to Polar (r, θ) | r² = x² + y² tan θ = y/x |
Finding θ requires considering the quadrant of (x,y). |
This process allows you to translate the location of any point from its description based on distance and angle to its description based on horizontal and vertical displacements from the origin.
