# Note on the Formula for PI

May 25, 2009 4 Comments

In the previous post, we saw that PI = Lim (theta->0) 360 * sin(theta/2) / theta. (Here, theta is in degrees.)

Now, we all know that standard result: : Lim (**x** -> 0) sin **x** /** x** = 1. (Of course, **x** is in radians here.)

Important note: We’ll use **Boldface** for values in** radians**, and normal for degrees.

At first glance it may seem that Lim (theta->0) 360 * sin(theta/2) / theta = 1/2 * 360 = 180, and not PI.

So how can we verify that our formula for PI is actually correct?

The reason is rather simple. It is because: Lim (x->0) sin x / sin **x** = PI / 180. ( Numerator: x in degree; Denominator: x in radians).

Proof:

Lim (theta->0) sin theta/ sin **theta ** ( Numerator: theta in degree; Denominator: theta in radians).

= Lim (theta->0) sin ( PI * **theta** / 180) / sin **theta**. [ Now both numerator and denominator have value of the angle in radians.]

= Lim (theta->0) [sin (PI * **theta** / 180) / (PI * theta / 180)] * [ (PI * theta / 180) / sin **theta**]

Now, Lim (theta->0) [sin (PI *** theta **/ 180) / (PI * theta / 180)] = 1 ( We can use the standard limit formula here since the angle is in radians.)

and, Lim (theta->0) [ (PI * theta / 180) / sin **theta**] = PI / 180. (Again we can use the formula since angle is in radians.)

therefore, Lim (theta->0) sin theta / sin **theta** = PI / 180.

Using this, the formula for PI in the previous post can be verified.🙂

Very good posts daa.. keep it up..🙂

Thanks da.🙂

Perhaps you may consider using equation editor in MS Office followed by exporting it as an image for easier visualization

Yes da..It takes a bit longer… I’ll try n do that.