Question

What is `arctan(cospi)` ?

Answer 1

`"arctan"(cos(pi))=-pi/4`

Explanation:

`cos(pi)=-1`
`color(white)("XXX")`with an angle of `pi`
`color(white)("XXX")`the adjacent arm has the same magnitude as the hypotenuse
`color(white)("XXX")`but is negative.
`color(white)("XXX")`Therefore `cos(pi)= (-"hypotenuse")/"hypotenuse" =-1`

`"arctan"(-1)=pi/4`
`color(white)("XXX")`Note that as a function the range of `"arctan"`
`color(white)("XXX")`is limited to `[-pi/2,+pi/2]`

`color(white)("XXX")"arctan(-1)"` means that
`color(white)("XXX")`the `"opposite"/"adjacent"` for the angle must be `(-1)`
`color(white)("XXX")`or, in terms of `x and y` coordinates:
`color(white)("XXX")y/x = -1 rarr y=-x`
`color(white)("XXX")`and we have an equilateral right-angled triangle
`color(white)("XXX")`below the X-axis