Trig Functions: Sine and Cosine Definition

Definition: An algebraic approach

From defining a few general properties of the sine and cosine functions, we can algebraically derive the sine and cosine functions themselves.

First, define the sine and cosine functions to have these properties:

sin x = cos x

cos x = - sin x

sin 0 = 0

cos 0 = 1

Now, write the sine function as an arbitrary power series, in that let sin x = a₀x⁰ + a₁x¹ + a₂x² + a₃x³ + a₄x⁴ + ...
    sin 0 = 0 = a₀
          a₀ = 0

By differentiating the power series and equating it with the cosine by the original properties, sin x = cos x = 0 + (1)a₁x⁰ + (2)a₂x¹ + (3)a₃x² + (4)a₄x³ + ...
    cos 0 = 1 = (1)a₁
            a₁ = 1

Continuing, cos x = -sin x = -(0 + 0 + (2)(1)a₂x⁰ + (3)(2)a₃x¹ + (4)(3)a₄x² + ...)
    -sin 0 = 0 = -(2)a₂
             a₂ = 0

-sin x = cos x = -(0 + 0 + 0 + (3)(2)(1)a₃x⁰ + (4)(3)(2)a₄x¹ + ...)
     cos 0 = -1 = (3)(2)(1)a₃
              a₃ = -1/3!

cos x = sin x = 0 + 0 + 0 + 0 + (4)(3)(2)(1)a₄x⁰ + ...
    sin 0 = 0 = (4)(3)(2)(1)a₄
            a₄ = 0

Continue on and you get values of all a_n
   if n is even then a_n = 0
   if n is odd then a_n = 1/n! alternating positive and negative.
Or stated as:
   a_2n = 0
   a_2n+1 = (-1)ⁿ/(2n+1)!


Plugging these values into the equation for sine:
    sin x = x¹/1! - x³/3! + x⁵/5! - x⁷/7! + x⁹/9! - ... + (-1)ⁿx²ⁿ⁺¹/(2n+1)! + ...

          = (-1)ⁿx²ⁿ⁺¹ / (2n+1)!

and for cosine:
    cos x = 1 - x²/2! + x⁴/4! - x⁶/6! + x⁸/8! - ... + (-1)ⁿx²ⁿ/(2n)! + ...

          = (-1)ⁿx²ⁿ / (2n)!

These series converge for all x ÎÂ.

Source: Jeff Yates.