Editing Trigonometric functions (section)

===Definition by differential equations===

Sine and cosine can be defined as the unique solution to the [[initial value problem]]:{{sfn|Bartle|Sherbert|1999|p=247}}
:<math>\frac{d}{dx}\sin x= \cos x,\ \frac{d}{dx}\cos x= -\sin x,\ \sin(0)=0,\ \cos(0)=1. </math>

Differentiating again, <math display="inline">\frac{d^2}{dx^2}\sin x = \frac{d}{dx}\cos x = -\sin x</math> and <math display="inline">\frac{d^2}{dx^2}\cos x = -\frac{d}{dx}\sin x = -\cos x</math>, so both sine and cosine are solutions of the same [[ordinary differential equation]]
:<math>y''+y=0\,.</math>
Sine is the unique solution with {{math|''y''(0) {{=}} 0}} and {{math|''y''′(0) {{=}} 1}}; cosine is the unique solution with {{math|''y''(0) {{=}} 1}} and {{math|''y''′(0) {{=}} 0}}.

One can then prove, as a theorem, that solutions <math>\cos,\sin</math> are periodic, having the same period. Writing this period as <math>2\pi</math> is then a definition of the real number <math>\pi</math> which is independent of geometry.

Applying the [[quotient rule]] to the tangent <math>\tan x = \sin x / \cos x</math>, 
:<math>\frac{d}{dx}\tan x = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = 1+\tan^2 x\,,</math>
so the tangent function satisfies the ordinary differential equation
:<math>y' = 1 + y^2\,.</math>
It is the unique solution with {{math|''y''(0) {{=}} 0}}.