Pythagorean Identities

We can change the symbols used for the sides of a Right Triangle. The use of {a,b,c} was convenient furthering the Pythagorean theorem, but we may wish to change the set of symbols to {h,a,o}. These are the symbols used when defining Trigonometric Functions.

We rewrite the Pythagorean Theorem equation as follows:

h^2 = a^2 + o^2

It is legal to divide every term by the same expression:

\dfrac {h^2}{h^2} =\dfrac {a^2}{h^2} + \dfrac {o^2}{h^2}

If a ratio equals a function then the square of that ratio equals the square of the function:

1 = cos^2(\theta) + sin^2(\theta)

where we used the following definitions:

sin (\theta) = \dfrac {o}{h}

cos (\theta) = \dfrac {a}{h}

The identity cos^2(x) + sin^2(x) = 1 is a golden egg. You will use it several times and you will feel joy as yoy watch a factor with trig functions in it disappear.

Of lesser importance are the two other Pythagorean Identities obtained using a and o:

  • \dfrac {h^2}{a^2} =\dfrac {a^2}{a^2} + \dfrac {o^2}{a^2}
  • sec^2(\theta) = 1 + tan^2(\theta)

and

  • \dfrac {h^2}{o^2} =\dfrac {a^2}{o^2} + \dfrac {o^2}{o^2}
  • csc^2(\theta) = cot^2(\theta) + 1