Skip to main content

Cosine

Definition​

Definition : By power series

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

cos⁡(x)=1−x22!+x44!+⋯+(−1)nx2n(2n)!+⋯=∑n=0+∞(−1)nx2n(2n)!\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots + (-1)^n \frac{x^{2n}}{(2n)!} + \dots = \sum_{n=0}^{+\infty}(-1)^n\frac{x^{2n}}{(2n)!}
Definition : By differential equation

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

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

Results​

Proposition : Addition formula

Let a,b∈Ra, b \in \mathbb R, we have

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

Let x∈Rx \in \mathbb R, we have

cos⁡′(x)=−sin⁡(x)\cos'(x) = -\sin(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

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