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

## 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.

## Examples: cos(30°) + isin(30°) = √(3)/2 + 1/2i

cos(60°) + isin(60°) = 1/2 + √(3)/2i

[90 - 30 = 60]

### 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!

cos(30°) - isin(30°) = √(3)/2 - 1/2i

cos(60°) - isin(60°) = 1/2 - √(3)/2i

[90 - 30 = 60]

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

cos(15°) + isin(15°) = 0.9659258263... + 0.2588190451...i

cos(75°) + isin(75°) = 0.2588190451... + 0.9659258263...i

[90 - 15 = 75]

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°.

cos(45°) + isin(45°) = √(2)/2 + √(2)/2i

cos(45°) + isin(45°) = √(2)/2 + √(2)/2i

[90 - 45 = 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. 