Question

How do you find the exact value of `arctan(2)`?

Answer 1

This is not a rational number of degrees, nor a rational multiple of `pi` radians.

We can write:

`arctan 2 = pi/2 - sum_(k=0)^oo (-1)^k 1/(2^(2k+1)(2k+1))`

Explanation:

`arctan(2)` is an angle in a right angled triangle with sides `"adjacent" = 1`, `"opposite" = 2` and `"hypotenuse" = sqrt(5)`. It is not a rational multiple of `pi` radians nor a rational number of degrees.

We can represent it as the sum of an infinite series.

Note that:

`arctan x =sum_(k=0)^oo (-1)^k x^(2k+1)/(2k+1) = x - x^3/3+x^5/5-x^7/7+x^9/9-x^11/11+...`

However, this only converges for `abs(x) <= 1`.

To get a series that does converge, we can use:

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

So:

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

and hence:

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

`color(white)(arctan 2) = pi/2 - sum_(k=0)^oo (-1)^k 1/(2^(2k+1)(2k+1))`