Question

How do you evaluate `arctan 133.3`?

Answer 1

`arctan(133.3) ~~ 1.56329459205`

Explanation:

Note that:

`arctan x = x - x^3/3+x^5/5-x^7/7+...`

This is practical to use for small values of `x`, but what about `133.3`?

The trick is that:

`arctan(x) = pi/2 - arctan (1/x)`

So we can evaluate `arctan(1/133.3)` and subtract it from `pi/2`.

`arctan (1/133.3) = arctan (10/1333)`

`color(white)(arctan (1/133.3)) ~~ 10/1333 - 10^3/(1333^3*3)`

`color(white)(arctan (1/133.3)) ~~ 10/1333 - 1000/7105779111`

`color(white)(arctan (1/133.3)) ~~ 0.00750173474`

So

`arctan(133.3) ~~ pi/2 - 0.00750173474 ~~ 1.56329459205`