Teach an Old Dog New Trigs

If you studied your Trig tables in high school, you probably remember some facts like $$\begin{align*}
\cos(30) &= \sqrt{3}/2,\\
\cos(45) &= \sqrt{2}/2, \text{ and}\\
\cos(60) &= 1/2.
\end{align*}$$ With a few formulas you can get at a few more angles, e.g., $$\cos(15) = \cos(45 – 30) = \cos(45) \cos(30) + \sin(45) \sin(30) = \frac{ \sqrt{6} + \sqrt{2} }{4},$$ and likewise \(\cos(75) = (\sqrt{6} – \sqrt{2})/4\).

I now offer two more angles that are not as straightforward but nevertheless have surprisingly nice trig values: 18 and 36. Specifically, we have $$\cos(36) = \frac{1 + \sqrt{5}}{4} = \frac{\phi}{2}, \text{ and }
\sin(18) = \frac{-1 + \sqrt{5}}{4} = \frac{1}{2 \phi},$$ where \(\phi = (1 + \sqrt{5})/2\) is the golden ratio. Try to prove these! (Here’s a hint: Draw a 36-72-72 triangle, and bisect one of the large angles to form a 36-36-108 triangle and a smaller 36-72-72 triangle. Both are isosceles! And one of them is similar to the original triangle!)

You may ask for which other angles we can get “nice” answers. Well, this at least lets us get 18 – 15 = 3 degrees, and therefore any multiple of 3 degrees, as long as we don’t mind multiply nested square roots. But if we’re allowing this then there are even more! For example, Gauss showed that $$\begin{align*}\cos(360/17) &= \frac{1}{16}\bigg( -1 + \sqrt{17} + \sqrt{34-2 \sqrt{17}} \\
& \qquad {} + 2 \sqrt{ 17 + 3 \sqrt{17} – \sqrt{34-2 \sqrt{17}} – 2 \sqrt{34+2 \sqrt{17}} } \bigg),\end{align*}$$ and there are similar (lengthier!) formulas for \(\cos(360/257)\) and \(\cos(360/65537)\) using just nested square roots. Why these denominators? You’ll have to ask our friends Fermat and Galois.

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