Sungjai's Recreational Mathematics

Last time, we explored Essence of Power Functions. This time, we will try to calculate for Essence of Exponentials in a similar way. $$\sum_{i=n}^m r^i = \int_{n-1}^mȢ\{r^x\}dx$$ Sequence Flatlining We were able to find Essence of Power functions using Sequence Flatlining. The basic idea behind Sequence Flatlining is to differentiate the equation for sequence until you reach 0 (Check out Sequence Curve: Essence of Power Function  for more). This method, however, will not work for Exponential Functions because they will never differentiate to 0. We will need to find another way to approach Essences. Essence Formula We start similarly to beginning of Sequence Flatlining: $$\sum_{i=n}^m f(i) = \int_{n-1}^mȢ\{f(x)\}dx \\ \sum_{i=n}^n f (i) = f(n) = \int_{n-1}^nȢ\{f(x)\}dx \\ \therefore \frac{d}{dn}f(n) = Ȣ\{f(n)\} - Ȣ\{f(n-1)\}$$ Rather than continuing to differentiate this like in Sequence Flatlining, we stop here. By Linearity of Essence (see Sequence Curve: Essential Pr

I'm back at it again with another fun math inquiry. We learn double angle formula for $\sin$ and $\cos$ but we usually do not go any further. Of course, we can build triple and quadruple angle formula with Angle Sum Identities, but why do we not discuss patterns of $\sin(n\theta)$ or $\cos(n\theta)$? Today, we will discuss intriguing patterns that arrises from n-Angle Trigs. Basics Let's first go over the basic identities: $$\begin{align*} \sin(\theta + \phi) &= \sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi) \\ \sin(\theta - \phi) &= \sin(\theta)\cos(\phi) - \cos(\theta)\sin(\phi) \\ \cos(\theta + \phi) &= \cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi) \\ \cos(\theta - \phi) &= \cos(\theta)\cos(\phi) + \sin(\theta)\sin(\phi) \\  \sin(2\theta) &= 2\sin(\theta)\cos(\theta) \\  \cos(2\theta) &= \cos^2 (\theta) - \sin^2(\theta) \end{align*}$$ From these simple identities we can build all n-angled trigs. But why aren't we taught them? Well,