# 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 + b*i* = R(cos(Θ) + *i*sin(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 + a*i* = R(cos(90° - Θ) + *i*sin(90° - Θ))

__Examples:__

cos(30°) + *i*sin(30°) = √(3)/2 + 1/2*i*

cos(60°) + *i*sin(60°) = 1/2 + √(3)/2*i*

[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°) - *i*sin(30°) = √(3)/2 - 1/2*i*

cos(60°) - *i*sin(60°) = 1/2 - √(3)/2*i*

[90 - 30 = 60]

The conjugate of **a + b***i* is simply **a - b***i*!

cos(15°) + *i*sin(15°) = 0.9659258263... + 0.2588190451...*i*

cos(75°) + *i*sin(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°) + *i*sin(45°) = √(2)/2 + √(2)/2*i*

cos(45°) + *i*sin(45°) = √(2)/2 + √(2)/2*i*

[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.**

Back to Index Page Back to Math Trick Menu

© Derek Cumberbatch