Let’s then find the limit of p as n approches positive infinity (to resolve the indeterminate form, we define N as 1/n and we use the definition of derivatives.)ĭepending on the unit we use, the limit l we find is different: Let the natural number n denote the number of sides of the polygon and the positive real number r the radius of the circle (angles are taken in radians) the perimeter of the polygon is then given by the formula : So when the number of sides tends towards infinity, the perimeter of the polygon tends towards that of the circle. The more sides this polygon had, the closer to pi this value was. We were told at school that the value of pi was firstĪpproximated by drawing a regular polygon inscribed in a circle and dividing its perimeter by the radius. Radians are not only convenient, in some cases they are the only correct choice. When you measure the angles between the whole number intervals on the spiral you will find that the angles between them approach 2 Pi radians. The next one is just as, if not more mysterious than the first. The further out you go on this spiral the closer this measurement gets to Pi. For instance when you begin this spiral and measure the distance between the coils you start out with a measurement that is very close to Pi. This spiral is easy to construct but it reveals some relationships that are very hard to explain. The first thing to look at to get a closer view of this is a math object called the “Spiral of Squares” There are a lot of things about this relationship that aren’t apparent at first. One radian measures the angle when the radius of a circle is traced out on its edge. Radians connect algebra to trigonometry to geometry like nothing else can. Why radians? I understand that the first commentator tried to explain but most probably lost you in the first paragraph if not sooner. It’s a lot easier to understand why if you look at a graph. Same thing happens when you differentiate cosine or tangent or whatever. It’s not the end of the world if you try to do calculus with degrees (it’s close), it’s just that the result is multiplied by an inconvenient constant. If it weren’t for the fact that (when using radians) we wouldn’t have. That doesn’t look like a big deal, but keep in mind that all of trigonometry is just a rehashing of sine. For example, one of the more important things in the world (that’s not quite sarcasm) is the fact that. All of the calculus around trig functions can be based on the fact that. When someone says “in the limit as _ approaches _” it means they’re about to talk about calculus (and true to form…). In fact, in the limit as the angle approaches zero they are equal, or in mathspeak. For small angles sin(θ)≈θ, but only when that angle is described in radians.
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