How this instrument works
A complex number can be named two ways: by its rectangular coordinates a + bi, or by its polar coordinates — a magnitude r and an angle θ measured counterclockwise from the positive real axis. This sheet converts the second description into the first. Drop a perpendicular from the point (a, b) down to the real axis and a right triangle appears, with hypotenuse r; its adjacent side is a = r·cos(θ) and its opposite side is b = r·sin(θ). Nothing about complex numbers demands new machinery here — it is the same right-triangle trigonometry used to turn plane-polar coordinates into Cartesian ones, with the second axis simply relabelled imaginary.
The shape of the formula falls straight out of Euler's identity. Because e^(iθ) = cos θ + i sin θ, multiplying through by r gives r·e^(iθ) = r·cos θ + i·r·sin θ — exactly a + bi. Reading the real and imaginary parts off that single equation is what the calculator does arithmetically: cosine supplies the real coordinate, sine supplies the imaginary one, and r scales both by the same factor because it sits outside the parentheses as a common multiplier, not as a separate rotation.
Two limits are worth holding onto. At θ = 0 the point sits on the positive real axis and b collapses to zero, so a + bi is just the real number r; at θ = 90° (π⁄2 radians) the roles swap and the number becomes purely imaginary, ri. Because cosine and sine repeat every 360°, infinitely many angles — θ, θ + 360°, θ + 720°, and so on — all land on the identical a + bi. This sheet's Magnitude field also only accepts r ≥ 0, matching the mathematical convention that a modulus is a nonnegative distance, never a signed quantity.
- Enter the modulus in Magnitude, r — the complex number's distance from the origin in the complex plane.
- Enter the angle in Angle, θ — the unit control beside the field accepts degrees, radians, or turns; the sheet converts internally to radians before multiplying.
- Read Real part, a and Imaginary part, b — together they give a + bi in rectangular form.
- Check the arithmetic if you like: a² + b² should equal r², since the modulus survives the conversion unchanged.
- Change r or θ and watch a and b update, tracing how the point moves around the circle of radius r.
Worked example — the 3-4-5 triangle's angle
Set Magnitude, r, to 5 and Angle, θ, to 53.13010235415598° — the sheet's default unit — which it converts internally to 0.9272952180016122 radians before multiplying, since arctan(4⁄3) is exactly the angle inside a 3-4-5 right triangle. It returns a = 5·cos(0.9272952180016122) = 3.0000000000000004 and b = 5·sin(0.9272952180016122) = 3.9999999999999996; the trailing digits are ordinary floating-point rounding, and the exact values underneath are the whole numbers 3 and 4. In rectangular form, the same complex number is 3 + 4i.
The check that matters here is a² + b² = r²: 3² + 4² = 9 + 16 = 25 = 5², so the modulus survives the round trip exactly, as it must — converting from polar to rectangular form never changes how far the point sits from the origin, only how that distance splits between the two axes. It is the same 3-4-5 triangle used to teach the Pythagorean theorem, walked in the opposite direction: instead of two legs producing a hypotenuse, one hypotenuse and one angle produce the two legs.
Questions
What do the magnitude r and angle θ actually represent?
r is the modulus — the complex number's straight-line distance from the origin, always taken as nonnegative — and θ is the argument, the angle measured counterclockwise from the positive real axis. Together (r, θ) is the polar form of the same number that a + bi describes in rectangular form; this calculator turns the first pair into the second.
How does this connect to Euler's formula?
a = r·cos θ and b = r·sin θ are exactly the real and imaginary parts of r·e^(iθ), because Euler's formula states e^(iθ) = cos θ + i sin θ. Multiply both sides by r and the real part becomes r cos θ while the imaginary part becomes r sin θ — precisely the pair of values this sheet computes.
Does the angle need to be entered in radians?
No — the unit control beside the Angle field accepts degrees, radians, or turns, and the sheet converts to radians internally before evaluating cosine and sine, since those functions are naturally defined on radian measure. Enter degrees directly if that is how the angle was given to you; there is no hand conversion required.
Can θ be negative or larger than 360°?
Yes — cosine and sine are periodic, so any real angle is valid input and the arithmetic works unchanged. The catch runs the other way: infinitely many angles, such as θ and θ + 360°, produce the identical a + bi, which is why converting rectangular form back to polar form requires choosing one principal angle, usually restricted to (−180°, 180°].
What mistake do people usually make converting polar form to a + bi?
The most common slip is swapping the two trigonometric functions — writing the real part as r·sin θ and the imaginary part as r·cos θ instead of the other way around. A close second is forgetting that b is the coefficient multiplying i, not a value that already contains i, which causes errors as soon as the number is used in further arithmetic.
How is this different from converting polar coordinates to Cartesian coordinates?
It isn't, mathematically — a = r cos θ and b = r sin θ are the identical formulas used for x = r cos θ and y = r sin θ in plane geometry. The only change is the label on the second axis: instead of a plain y-coordinate, b is read as the coefficient of i on the imaginary axis, because the two numbers being combined obey complex-number arithmetic rather than ordinary vector addition.