The Mystery of the Sine of Nines: sin 999=sin 9999 ?

Today while playing with my calculator, I discovered a curious fact:

sin(999)=-0.987688
sin(9999)=-0.987688
sin(99999)= -0.987688

Note: All angles in degrees.

In fact, the sine of any number of nines (more than 3), always led to the same number!

This may not work with other digits, for example "8":
sin(888)= 0.20791
sin(8888)=-0.92718
sin(88888)=-0.52992

As a math tutor, definitely I was curious about the mathematics behind this phenomenon. If you want to try to unravel the mystery, do give it a try before reading the answer!

The Mystery of the Sine of Nines

The mystery is pretty straightforward once we notice the following:

$\sin (9999^\circ)=\sin (999^\circ+ 25(360^\circ))$

As we know, adding 360 degrees to an angle doesn't affect the result of its sine, since $\sin (x+360^\circ)=\sin (x)$. sin(9999) is actually sine of 25 times of 360 added to 999, hence they are essentially the same value!

Thus, sin(9999)=sin(999).

We can then proceed to show sin(99999)=sin(9999) in a similar way. This will keep on working since 9000=25x360 is already a multiple of 360, hence 9000...000 (more than 3 zeroes) will also be a multiple of 360!

This concludes the mysterious case of the Sine of Nines (it rhymes!).

Trigonometry is a really fun subject. But could it be taught better? Trigonometry often leads to nasty irrational numbers, for example sin(60) is already an irrational number ($\sqrt{3}/2$). Professor Wildberger, author of Divine Proportions: Rational Trigonometry to Universal Geometry argues that there is a better way to present Trigonometry, via the very novel (most people haven't heard of it, let alone seen it) Rational Trigonometry. I have followed his videos on YouTube, and personally it is an interesting idea. With Rational Trigonometry, irrational numbers (which are highly problematic if one thinks about them deeply) are banished, and we can only work with rational numbers.

 Check out the book here: