The relation between the sine curve and the unit circle is illustrated in this animation. As the angle increases, the value of the sine traces the familiar sine curve, in red. Here, the angle in radians is represented by the blue arc around the circle and the blue line, both of which have the exact same length.
y = sin x (where x is the angle in radians)
How could we not repost this! Brilliant animation and brilliantly explained. Trigonometry is so much more than just finding angles in triangles - which it seems is all school wants to teach you. This is what Sine really looks like, beautiful hey?
Heres a derivation of Euler’s formula, I know the resulting identity is all over tumblr but not many people realize the derivation is relatively simple. All that is required is a knowledge of Taylor Series Expansions and it’s basic algebra from there.
I also included the fact that i^i is a real and gave its decimal approximation because its one of those things that twists the mind a little.