Definitions Via Differential Equations
Both the sine and cosine functions satisfy the differential equation:
That is to say, each is the additive inverse of its own second derivative. Within the 2-dimensional function space V consisting of all solutions of this equation,
- the sine function is the unique solution satisfying the initial condition and
- the cosine function is the unique solution satisfying the initial condition .
Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions.
Further, the observation that sine and cosine satisfies y′′ = −y means that they are eigenfunctions of the second-derivative operator.
The tangent function is the unique solution of the nonlinear differential equation
satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation.
Read more about this topic: Trigonometric Functions
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