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Definition by differential equations

Sine and cosine can be defined as the unique solution to the initial value problem:

Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x}, so both sine and cosine are solutions of the same ordinary differential equation

Sine is the unique solution with y (0) = 0 and y ′(0) = 1; cosine is the unique solution with y (0) = 1 and y ′(0) = 0.

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

Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x},

so the tangent function satisfies the ordinary differential equation

It is the unique solution with y (0) = 0.