Cunning Cosine!

√(A² + B² - 2ABcos(360°)) = |B - A| or |A - B|

In the Law of the Cosines Formula, if the angle is equal to 360 degrees, then the cosine of that angle is equal to 1, so you'll get the difference of A & B in absolute value!

You know that any number multiplied by 1 is still the same multiplicand, right? Because 1 is the number of multiplicative identity! (Whenever 1 is the multiplier, the product & the multiplicand are exactly the same!)

Examples:

√(3² + 4² - 2 • 12cos(360°)) = √(9 + 16 - 24cos(360°)) = √(25 - 24) = 1

(A = 3 & B = 4 in this 1st example; their product is 12; |3 - 4| = 1)

√(15² + 11² - 2 • 165cos(360°)) = √(225 + 121 - 330cos(360°)) = √(346 - 330) = √(16) = 4

(A = 15 & B = 11 in this 2nd example; their product is 165; |15 - 11| = 4)

√(13² + 8² - 2 • 104cos(360°)) = √(169 + 64 - 208cos(360°)) = √(233 - 208) = √(25) = 5

(A = 13 & B = 8 in this 3rd example; their product is 104; |13 - 8| = 5)

If the variables A & B have different signs, the minus sign in the formula turns into a plus sign because the product of the 2 variables will be negative. This number then gets multipied by 2. (Positive 2) Remember that subtracting negative numbers is the same as adding their positive additive inverses!

√(2² + (-6)² - 2 • -12cos(360°)) = √(4 + 36 + 24cos(360°)) = √(40 + 24) = √(64) = 8

(A = 2 & B = -6 in this 4th example; their product is -12; |2 - (-6)| = |2 + 6| = 8)

√((-2)² + (-6)² - 2 • 12cos(360°)) = √(4 + 36 - 24cos(360°)) = √(40 - 24) = √(16) = 4

(A = -2 & B = -6 in this 5th example; their product is +12; |-2 - (-6)| = |-2 + 6| = 4)

√((-2)² + 6² - 2 • -12cos(360°)) = √(4 + 36 + 24cos(360°)) = √(40 + 24) = √(64) = 8

(A = -2 & B = 6 in this 6th example; their product is -12; |-2 - 6| = |-8| = 8)

I'm sure that you already know that squaring negative numbers gives you a positive number instead of a negative number! It's because of how multiplication works for negative numbers. Please see the Web page entitled: "What Type of Number Will You Get?" for more details. Also, know that negative numbers have imaginary square roots.

Faye Pixie: "You can even use complex & imaginary numbers in this 'fairy'(very) special function!"

Note: If you plug imaginary or complex numbers into the variables A or B, then the result in the formula will sometimes be a complex number that matches their difference in absolute value!

In this example, A = B.

(The variables aren't capitalized in Brain's word balloon, but let's not be case-sensitive!)

√(5² + 5² - 2 • 25cos(360°)) = √(25 + 25 - 50cos(360°)) = √(50 - 50) = 0

(A = 5 & B = 5 in this 7th & final example; their product is 25; |5 - 5| = 0)

It's that simple! When A = B, the result in the formula is zero(0) since any number minus itself is always zero!

And remember this fact about the 2 idempotent numbers 1 & 0:

1 & 0 are both their own square roots!