Complex Number Kookiness 9

|a + bi|sin(arg(a + bi)) = b

|a + bi|cos(arg(a + bi)) = a

Multiplying the absolute value of a complex number by the sine of its argument gives you the imaginary part & multiplying the absolute value of a complex number by the cosine of its argument gives you the real part!

NO, not that kind of argument! (Like how Burger & Sally argue in this cartoon above!)

The argument of a complex number is its angle in respect to the origin point & the positive side of the real axis. If the angle is negative, then the rotation is going clockwise; otherwise, it’s going counterclockwise!

(Don't forget that the origin point in the complex number plane is 0 + 0i.)

Examples:

|3 + 4i| = 5; arg(3 + 4i) = 53.13010235...°; sin(arg(3 + 4i)) = 4/5; cos(arg(3 + 4i)) = 3/5;

5sin(arg(3 + 4i)) = 4

5cos(arg(3 + 4i)) = 3

It figures, because:

X * N/X = N (X ≠ 0)

Notice how the sine & cosine of the angle are fractions with the absolute value (5) as their denominator!

Even if the real or imaginary parts of the complex number are negative, the math trick still works out!

|-1 - 3i| = √(10); arg(-1 - 3i) = -108.4349488...°; sin(arg(-1 - 3i)) = -0.9486832981...; cos(arg(-1 - 3i)) = -0.3162277656...;

√(10)sin(arg(-1 - 3i)) = -3

√(10)cos(arg(-1 - 3i)) = -1

This math trick can be used to calculate the height or the base of a triangle when you're trying to figure out the triangle's area! Complex numbers are very useful in trigonometry!

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