Question

How do you evaluate `arctan(1)`?

Answer 1

`=45^@`
`=pi/4`

Explanation:

`arctan(1)`
`=tan^-1(1)`
`=45^@`
`=pi/4`

Answer 2

`arctan(1) = pi/4`

Explanation:

`color(white)()`
`theta = arctan(1)` is the angle `theta in (-pi/2, pi/2)` satisfying `tan(theta) = 1`

`color(white)()`
Note that the triangle formed by bisecting a unit square diagonally is a right angled triangle with sides `1`, `1`, `sqrt(2)` and angles `pi/4`, `pi/4` and `pi/2`.

So we find:

`tan(pi/4) = "opposite"/"adjacent" = 1/1 = 1`

So `theta = pi/4` satisfies `tan(theta) = 1` and is in the required range.

`color(white)()`
So:

`arctan(1) = pi/4`