Rotating one square within another square (not homework!)

[Jazzsammich]

Hi!

I'm not a student, but an artist working on something that requires some mathematical precision, and it's been over three decades since I've taken geometry and trig. Here's my problem:

I have two concentric squares: the larger one with side m and the smaller one with side n.

The smaller square is rotated by an angle θ such that (a) 0° < θ < 45° and (b) all four corners of the smaller square lie on the sides of the larger square, dividing m into smaller segments a and o.

I'm trying to figure out how to calculate θ given known values of m and n.

Thank you!

edit:

Obvious things I've got so far:

[imath]o^2 + a^2 = n^2[/imath]

[imath]m=a+o[/imath]

[math]sinθ=(o/n) ; cosθ = (a/n) ; tanθ=(o/a)[/math]

 

[mario99]

It is easy to show that:

[imath]\displaystyle \cos \theta = \frac{m - o}{n}[/imath]

sinθ=(o/n) ; cosθ = (a/n) ; tanθ=(o/a)

Now you can use your first formula to get rid of [imath]\displaystyle o[/imath].