I have made up a geometric definition for a function $\mathrm{polysin}(n, θ)$:
Construct a regular polygon of $n$ sides. Place the first vertex at $(1,0)$ and place the rest going counterclockwise.
Let a line through the origin intersect this "unit polygon," making an angle of θ with the positive half of the x-axis.
The y-coordinate of that point of intersection is equal to $\mathrm{polysin}(n, θ)$.
(You can play around with the function in this GeoGebra sketch.)
I would like to know a couple of things about this function, namely:
- How can I define $\mathrm{polysin}(n, θ)$ algebraically?
- How much error is there between $\mathrm{polysin}(8, θ)$ and $\sin(θ)$? Between $\mathrm{polysin}(n, θ)$ and $\sin(θ)$?
I am at a quite basic level of math education - I know how to do algebra and trigonometry, but I haven't taken any Calculus yet. I would really appreciate some insight into how one should approach math problems like this, and the broad strokes of what is involved in solving this problem. (As well as the answer, of course!)
I'll get you started. The first vertex, $v_1$, is at $(\cos(2\pi/n),\sin(2\pi/n))$. The line between $v_1$ and $v_0=(1,0)$ is $y= \frac{\sin(2\pi/n)}{1-\cos(2\pi/n)}(1-x)$; for $0\leq \theta \leq 2\pi/n$, you have defined $\text{polysin}(n,\theta)=\frac{\sin(2\pi/n)}{1-\cos(2\pi/n)}(1-\cos(\theta)) $. You can generalize this for any two consecutive vertices: find the equation of the line between them using point-slope form and evaluate at $\cos(\theta)$. If you're familiar with calculus, you can find the maximum error by taking the difference $\sin(x)-\text{polysin}(n,x)$, differentiating, and setting that equal to $0$. Neat idea, hope that helps!