A couple days ago I saw this tweet.
Pretty cool! Let’s see why
Two terms are easy to deal with:
But why is
One way to prove this is via the inverse trigonometric identity
In this case
Because and are between 0 and , .
I didn’t use this approach on my first attempt to solve the problem. (To be honest, I didn’t know this identity existed before finding it online.) I used geometry and trigonometry. We are interested in the angle in the diagram below.
The law of cosines tells us that
This implies that and hence
(I wonder if there is a year of tau coming up sooner than the year 112233.)
Update: I just saw that Cut-the-Knot has a page devoted to this topic too.
I think of this in terms of complex numbers: (1+2i)(1+3i) = -5 + 5i. Taking arguments of the complex numbers gives arg (1+2i) + arg(1+3i) = arg(-5 + 5i). Recalling that arg(a+bi) = arctan (b/a) gives what you’re looking for.
tan(A+B) = (tan(A) + tan(B)) / (1 – tan(A) * tan(B))
So, tan(arctan(2) + arctan(3)) = (2 + 3) / (1 – 2 * 3) = -1
=> arctan(2) + arctan(3) = (3/4) * pi
cos theta = – 1/ sqrt(2) and not 1/sqrt(2)
Cosine value should be negative to make the resulting angle 3pi/4.
Right—thanks! Good catch.
If tanA, tanB, and tanC are nonzero, then tanA + tanB + tanC = tanA*tanB*tanC if and only if A + B + C = Pi*K, where k is some integer. This together with the fact that 1 + 2 + 3 = 1*2*3 immediately gives arctan1 + arctan2 + arctan3 = Pi.
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