Trigonometry Tricks 7: Swapping A & B in a Complex Number

When considering complex numbers, remember this formula below:

(To the right of the equal sign is the complex number in Polar form; to the left of the equal sign is the complex number in Rectangular Coordinates form.)

a + bi = R(cos(Θ) + isin(Θ))

a is the real part, b is the imaginary part, R is the absolute value of the complex number, Θ is the angle of the complex number in respect to the origin & finally, i is the imaginary unit equal to the square root of -1.

However, if you subtract Θ from a 90-degree angle, a & b swap places!

b + ai = R(cos(90° - Θ) + isin(90° - Θ))

Examples:

Note: You can also do this math trick on the conjugate of your chosen complex number, as I'll show you in the next example!

The conjugate of a + bi is simply a - bi!

Notice how the irrational numbers swapped places in this example. The triple periods mean that the digits after the decimal point never repeat or terminate, making the number irrational!

When a & b are equal, Θ = 45°.

Since 45 is half of 90, subtracting it from 90 leaves you with the same subtrahend, which is 45. The difference & the subtrahend are always the same if the subtrahend is half of the minuend! That is true for all numbers!

X - X/2 = X/2

Finally, even if either variable a or b is negative, the trick still works out! It'll also work out even if Θ > 90°, but the signs will change! In other words, if Θ > 90°, then the plus sign will become a minus sign or vice versa!

Whenever the subtrahend is greater than the minuend, the difference will be negative, which causes the sign to change. Don't forget that subtracting a number is the same as adding its additive inverse! For example, 90 - (-15) = 90 + 15 = 105.

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© Derek Cumberbatch