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Sine

Definition​

Definition : By power series

Let x∈Rx \in \mathbb R, we define the sine function as

sin⁡(x)=x−x33!+x55!+⋯+(−1)kx2k+1(2k+1)!+⋯=∑n=0+∞(−1)nx2n+1(2n+1)!.\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} + \dots + (-1)^k \frac{x^{2k+1}}{(2k+1)!} + \dots = \sum_{n=0}^{+\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}.
Definition : By differential equation

The sine function is the unique solution of the following Cauchy problem

y′′=−y,y(0)=0,y′(0)=1y'' = -y, \quad y(0) = 0, \quad y'(0) = 1

Results​

Proposition : Addition formula

Let a,b∈Ra, b \in \mathbb R, we have

sin⁡(a+b)=cos⁡(a)sin⁡(b)+sin⁡(a)cos⁡(b).sin⁡(2a)=2cos⁡(a)sin⁡(a)sin⁡(π−a)=sin(a)sin⁡(π+a)=−sin(a)sin⁡(π2−a)=cos(a)\begin{align} \sin(a+b) & = \cos(a)\sin(b) + \sin(a)\cos(b). \\ \sin(2a) & = 2\cos(a)\sin(a) \\ \sin(\pi - a) & = sin(a) \\ \sin(\pi + a) & = - sin(a) \\ \sin(\frac{\pi}{2} - a) & = cos(a) \end{align}
Proposition : Derivative

Let x∈Rx \in \mathbb R, we have

sin⁡′(x)=cos⁡(x)\sin'(x) = \cos(x)
Proposition : Fundamental relation

Let x∈Rx \in \mathbb R, we have

sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1
Proposition : Outstanding values

Some outstanding values to keep in mind

sin⁡(0)=0,sin⁡(π6)=12,sin⁡(π4)=22,sin⁡(π3)=32,sin⁡(π2)=1.\sin(0) = 0, \quad \sin \left (\frac{\pi}{6} \right ) = \frac{1}{2}, \quad \sin \left (\frac{\pi}{4} \right ) = \frac{\sqrt 2}{2}, \quad \sin \left (\frac{\pi}{3} \right ) = \frac{\sqrt 3}{2}, \quad \sin \left ( \frac{\pi}{2} \right ) = 1.