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**Βεν Γ. Κυθισ****Member**- Registered: 2018-10-09
- Posts: 20

So, I was browsing Wikipedia and came across the Right Triangle article's Altitude section.

Then I noticed that the altitude line splits the right triangle into two other right triangles.

Then I thought, "I wonder what would happen if I calculated the trig functions for one of these little triangles…"

So I came up with these steps:

Take a triangle ΔPAB where ∠A=90°, ∠P=θ, and line PB has length 1; then find the altitude and call it line f.

Mark the point where the altitude meets the hypotenuse as point F.

Take the new triangle ΔPAF and name line PA line b as it is the base of the triangle, and name line PF line e.

Take the other new triangle ΔBAF and name line BA line a, and name the line BF line d.

After those steps have been done, calculate the trigonometric functions for ΔPAF at ∠P.

You should get something that looks like this: https://www.desmos.com/calculator/j5wuwvtvk6 (the last three variables are capitalized because I had to capitalize E and I wanted it to be consistent).

For now I'm going to call these:

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1+1=|e^(π×i)-1|

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**Βεν Γ. Κυθισ****Member**- Registered: 2018-10-09
- Posts: 20

Also, if you make the hypotenuse have length √2, something interesting will happen to the graphs of these functions (you can do this by multiplying a and b by √2).

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1+1=|e^(π×i)-1|

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