# Why is this angle not $22.5^\circ$? And does it have an exact value?

Since the angle which splits a square in a half, starting from it's bottom left corner, is $$45^\circ$$, I intuitively thought that, if I put two squares to be horizontally adjacent, the angle between the bottom side of the resulting rectangle and the line going from it's bottom left vertex to it's top right vertex would be $$22.5^\circ$$. My (flawed) reasoning was that it does "half the vertical space a $$45^\circ$$ angle does". It looks like it's not though, as it's somewhere around $$27^\circ$$.

I'm sorry if it's a lame question, but why is that? And does that angle have an exact value that we can mathematically derive?

The core reason that it is not $$22.5^\circ$$, essentially, is that the trigonometric functions are not linear functions. For example, aside from particular values of $$x$$,

$$\sin(2x) \neq 2 \cdot \sin(x)$$ $$\sin(x+y) \neq \sin(x) + \sin(y)$$

(Similar truths hold for the other functions.) Thus, you cannot expect to get half the angle just by doubling the length of a side of a triangle.

As for the angle in question, it can shown to be given by $$\tan^{-1}(1/2)$$:

For all my looking, this expression doesn't seem to have an exact value, and is probably irrational. A variety of representations (e.g. continued fractions, integrals, infinite sums) can be found through Wolfram Alpha.

As for approximations,

$$\tan^{-1}(1/2) \approx 26.57^\circ \approx 0.4636476 \text{ radians}$$

• Great answer! Only one detail I'd like to understand better: what kind of reasoning do we employ to know that the value is arctan(1/2)? Does it have to do with the fact that the angle leads the line defining it to intersect the opposite side of the square in the 1/2 position? Sorry for my awful mathematical terminology. – Eärendil Baggins Jan 6 at 9:53
• It's basically because that diagonal - the line you make through the two squares - essentially divides the rectangle into two right triangles. Opposite the angle you're concerned with, the side length is $1$, and the adjacent side is $2$, so per definition of the tangent function, when $x$ is the angle in question, we say $\tan(x) = 1/2$. Then, taking the inverse tangent of both sides, $x = \tan^{-1}(1/2)$. – Eevee Trainer Jan 6 at 9:56
• You have a right-angled triangle with legs of 1 and 2 units. By definition of the tangent function, the pointiest angle has tangent $1/2$. @EärendilBaggins – Lord Shark the Unknown Jan 6 at 9:57

The angle is $$\tan^{-1}(1/2)$$. The tangent of $$22.5$$ degrees is $$\sqrt2-1$$ which is about $$0.414$$ so the actual angle is a bit more than $$22.5$$ degrees.