How To Calculate a Tangent In Degrees Without a Calculator

Question

So the other day in my Geometry class, I was bored so I decided to try and calculate pi (which is one of many things I do when I am bored). During that class, I finally developed an equation to calculate pi. I did this by using a polygon with an apothem of a and n sides and I found the perimeter before dividing that by 2 times the apothem in order to estimate pi by using a polygon with a large number of sides to represent a circle. pi = (2anTan(360/(2n)))/(2a) which I then simplified to pi = nTan(360/(2n)) where the value of n can be set to any value greater than zero. But the higher the value of m is, the more accurate the equation is. I set the n variable on my graphing calculator to 9,999,999,999 and typed in the equation. Sure enough, my calculator outputed 3.141592654. I also figured out that i can calculate pi indefinitely by setting n to an infinite number. For example, I set n to 4 and then do my long multiplication and division, then i add a 4 to the end of my multiplication and division problems and complete another digit of calculations. I can then repeat this process to set n as an infinite value.

But now I want to calculate pi even further. The only problem is, my calculator has a maximum output display of 10 digits. So, how can I calculate a tangent in degrees without a calculator? I have found plenty of infinite equations to calculate a tangent in radians. However, considering that I am only 14 and I haven't exactly learned what a radian is yet, this doesn't help. Can someone please tell me how to calculate a tangent in degrees? Or how to convert radians to degrees (if that's possible, again I don't know what a radian is)? And if necessary what is a radian?

Answer 1

If you take a circular arc of radius $r$ and arc length $l$, then the corresponding angle will be $l/r$ radians. In particular, a radian is just $180/\pi$ degrees. This would convert an expression like $\tan(360/2n)$ working with degrees to one like $\tan(\pi/n)$ working with radians. This is somewhat unfortunate, since you can't very well plug in $\pi$ if that's the value you're trying to calculate. Note that your method is basically equivalent to saying that the derivative of the tangent function at $x=0$ is equal to one.

My take is that the inverse tangent function provides a reasonable method for approximating $\pi$ by hand. Machin's formula gives the following expansion:

$\begin{eqnarray} \pi & = & 16\arctan(1/5) - 4\arctan(1/239) \\ & = & \left(\frac{4}{5} - \frac{4}{3*5^3} + \frac{4}{5*5^5} - \frac{4}{7*5^3} + \ldots \right) + \left(- \frac{4}{239} + \frac{4}{3*239^3} - \frac{4}{5*239^5} + \frac{4}{7*239^7} - \ldots \right) \end{eqnarray}$ The justification that this works requires Taylor series, but computing approximations this way is pure algebra.

The good news is that dividing by $5$ is easy by hand, and the convergence is fairly rapid. The bad news is that dividing by $239$ by hand is pretty painful. However, it's not hard at all to get beyond 10 digits. Historically, this method was used by William Shanks to correctly compute 527 digits of pi before the existence of computers.