Note on the Formula for PI

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.🙂

4 Responses to Note on the Formula for PI

  1. Karthik says:

    Very good posts daa.. keep it up..🙂

  2. Karthik says:

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

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