What this polar to rectangular calculator does
This calculator converts a number from polar form to rectangular form. You enter the magnitude r and angle θ, and the tool computes:
- x = r cos(θ) (real part)
- y = r sin(θ) (imaginary part)
The rectangular result is then shown as x + yi. This is useful for complex numbers, AC circuit analysis, vectors, and trigonometry homework.
Polar form vs rectangular form
Polar form
Polar form expresses a value using size and direction. For complex numbers, you often see: r∠θ or r(cos θ + i sin θ).
Rectangular form
Rectangular form expresses the same value as: a + bi, where a is the real component and b is the imaginary component.
Conversion formula
Use the standard relationships below:
y = r · sin(θ)
rectangular form = x + yi
If your angle is in degrees, convert to radians for computation:
How to use the calculator
- Enter the magnitude r.
- Enter the angle θ.
- Select Degrees or Radians.
- Choose decimal precision.
- Click Convert.
The calculator instantly displays the rectangular pair (x, y) and the complex number in standard form x + yi.
Worked examples
Example 1: 10∠53.13°
cos(53.13°) ≈ 0.6 and sin(53.13°) ≈ 0.8
- x = 10 × 0.6 = 6
- y = 10 × 0.8 = 8
Rectangular form: 6 + 8i
Example 2: 5∠π/2 radians
cos(π/2) = 0 and sin(π/2) = 1
- x = 5 × 0 = 0
- y = 5 × 1 = 5
Rectangular form: 0 + 5i
Common mistakes to avoid
- Using the wrong angle unit: degrees and radians are not interchangeable.
- Rounding too early: keep more digits until the final answer.
- Sign errors: angles in Quadrants II, III, and IV change the signs of x and y.
- Forgetting special angles: values like 0°, 90°, 180° can simplify quickly.
Where this is used
Polar-to-rectangular conversion appears in many practical settings:
- Electrical engineering (impedance and phasors)
- Signal processing and Fourier methods
- Physics vectors and force decomposition
- Computer graphics and rotation transforms
- Math classes involving complex arithmetic
Quick takeaway
To convert polar form to rectangular form, multiply the magnitude by cosine for the real part and by sine for the imaginary part. Use this calculator to get fast, accurate results with clear formatting.