# A trigonometric coincidence

“Hm,” I thought, “that’s odd.”

I don’t often work in degrees, but the student’s syllabus insisted. And $\sin(50º)$ came up. It’s 0.7660, to four decimal places. But… I know that $\sin\left(\frac 13 \pi\right)$, er, sorry, $\sin(60º)$ is 0.8660 – a difference that’s pretty close to $\frac 1{10}$.

Which got me wondering: is that something interesting, or just a coincidence?

Well… it’s a bit of both, I think.

There’s a nice trick for finding the difference between (say) two sines, two cosines or one of each, and it’s to use the formulas in your A-level book, the ones you never really look at, the ones after the compound angle formulas. ((Technically, they’re fair game, although I’m pretty sure I’ve only ever seen them in Solomon papers – with the recent trend towards more involved C3 questions, I wouldn’t be at all surprised to see them this summer.))

The one you’re interested in is $\sin(A) - \sin(B) = 2 \cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$.

In this case, it gives $\sin(60º) - \sin(50º) = 2 \cos(55º) \sin(5º)$ which, naturally, comes to just under 0.1. But why should it?

$\cos(55º)$, if you ask your calculator, is about 0.5736. Is that a familiar number? Our straw poll said… heehaw. However, it looks a bit familiar to me – 1 glorious radian is about 57.3 horrible degrees ($\frac {180}{\pi} \approx 57.296$ – roughly $100 \cos(55º)$.

Meanwhile, in radians, $\sin(\theta) \approx \theta$, which means in degrees, $\sin(xº) \approx \frac{\pi x}{180}$.

That means, $2 \cos(55º) \sin(5º) \approx 2 \times \frac{180}{100\pi} \times \frac {5\pi}{180}$. Cancel the 180s and the $\pi$s, obviously, to get $2 \times 5 \div 100 = 0.1$.

An upshot of this is that two angles the same distance either side of 55º will (for small differences) work out to roughly $\frac{1}{100}$ of the difference between them in degrees.

There’s another coincidental pair that I’ve found: $\sin(75º) - \sin(60º) \approx 0.1$ as well, although I can’t see any particular justification for that. However, those are both values that can be found exactly:

$\sin(75º) = \sin(45º + 30º) = \sin(45º)\cos(30º) + \cos(45º)\sin(30º) = \frac{\sqrt 2}{2} \frac{1}{2} + \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} = \frac{\sqrt{2} + \sqrt {6}}{4}$.

$\sin(60º) = \frac{\sqrt{3}}{2}$

So the difference between them is $\frac{\sqrt{2} + \sqrt {6} - 2\sqrt{3}}{4} $, which means that $\sqrt{2} + \sqrt{6} - 2\sqrt{3} \approx 0.4$ – in fact, it’s 0.3996 to 4dp.

Last few: $\sin(75º) - \sin(50º) \approx 0.2$; $\sin(40º)-\sin(20º) \approx 0.3$; $\sin(35º)-\sin(10º)\approx 0.4$, and $\sin(80º)-\sin(29º) = 0.49998$, which is very close indeed to a half!