This equation is
incredibly important to physics and differential equations. Lemme see if I can remember the proof:
First off, this proof uses infinite series. Put basically, any differentiable function can be represented by the sum of an infinite series. No, I don't expect you to know what that means. Just accept that:
cos(x) = 1 - (x^2)/(2!) + (x^4)/(4!) - (x^6)/(6!) + . . .
sin(x) = x - (x^3)/(3!) + (x^5)/(5!) - (x^7)/(7!) + . . .
e^x = 1 + x + (x^2)/(2!) + (x^3)/(3!) + . . .
It follows that:
e^(i*x) = 1 + (i*x) - (x^2)/(2!) - (i*x^3)/(3!) + (x^4)/(4!) + (i*x^5)/(5!) - (x^6)/(6!) + . . .
i*sin(x) = (i*x) - (i*x^3)/(3!) + (i*x^5)/(5!) - (i*x^7)/(7!) + . . .
Now something interesting happens when you add cos(x) and i*sin(x) together. Lemme see if I can illustrate this clearly:
.....cos(x) = 1..........- (x^2)/(2!) ......................+ (x^4)/(4!) .................- (x^6)/(6!) + . . .
+ i*sin(x) = ..(i*x)..................... - (i*x^3)/(3!) .................+ (i*x^5)/(5!) - . . .
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....e^(i*x) = 1 + (i*x) - (x^2)/(2!) - (i*x^3)/(3!) + (x^4)/(4!) + (i*x^5)/(5!) - (x^6)/(6!) + . . .
Don't be discouraged if the idea of infinite series is unfamiliar to you - just take the series I gave for sin(x), cos(x), and e^x for granted, and everything else I show above follows from simple arithmetic on the series.
Anyway, this creates the general equation:
e^(i*x) = cos(x) + i*sin(x)
if x = PI, then:
e^(i*PI) = cos(PI) + i*sin(PI) = -1 + i*0 = -1
Oh, and shame on your math teacher for not knowing this proof
