Conversion from Cartesian coordinates \((x, y)\) to Polar coordinates \((r, \theta)\) is a fundamental concept in mathematics.
What are the two systems?
- Cartesian System → A point is represented as \((x, y)\)
- Polar System → A point is represented as \((r, \theta)\)
Where:
- \(r\) = distance from origin
- \(\theta\) = angle with the positive \(x\)-axis (in radians or degrees)
Conversion Formulas
To convert from Cartesian \((x, y)\) to Polar \((r, \theta)\):
Radius (r)
\[
r = \sqrt{x^2 + y^2}
\]
Angle (θ)
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
The formula \(\theta = \tan^{-1}(y/x)\) gives the correct angle only in certain quadrants. So:
- If \(x > 0\), use directly
- If \(x < 0\), add \(180^\circ\) (or \(\pi\) radians)
- If \(x = 0\):
- \(y > 0\) → \(\theta = 90^\circ\)
- \(y < 0\) → \(\theta = 270^\circ\)
In calculators/programming, we use:
\[
\theta = \text{atan2}(y, x)
\]
which automatically handles quadrants.
Example
Convert \((x, y) = (3, 4)\) into polar form:
Step 1: Find r
\[
r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5
\]
Step 2: Find θ
\[
\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ
\]
Final Answer:
\[
(3, 4) \rightarrow (5, 53.13^\circ)
\]