Why does f⁵⁰ = f² when f(x) = cos(2x)?

[dimpledur]

Homework Statement

if f(x)=cos2x, how do we know that f^50 = f^2 ??

In the solution it states that because 50 = (12)(4) + 2, then f^50 = f^2.

What the??

 

[Mark44]

We don't know that f⁽⁵⁰⁾ = f⁽²⁾ for the reason that it isn't true. If the function were f(x) = cos(x), it would be true, though, and this is easy enough to verify by taking a few derivatives and noticing the cyclical nature of these derivatives.

 

[dimpledur]

Okay, well how do they know that f^2 = f^50 for cosx simply by going 50 = (4)(12) + 2

 

[EstimatedEyes]

Because it repeats every four times
cosx
-sinx
-cosx
sinx
cosx