In this, a new mini-series about complementary knowledge for secondary maths teachers, I’m going to be looking at… the mighty Sine and Cosine.
I suspect a lot of Secondary maths teachers (i wouldn’t exclude all a-level teachers either) don’t really know where these come from (i’m also basing this on my experiences as a HoD, consultant and teacher trainer, so I’m hopefully not being THAT presumptuous). Knowledge of their origins isn’t critical for GCSE teaching, but have you ever asked what these mysterious things are or where they come from? Ask no more.
First a little history:
The Indian mathematician Aryabhata (5th Century) was the first to tabulate the values of half-chords (later known as… a table of sines) in his rather clever magnum opus, the Aryabhatiya which still exists today (!).
“Half-chords?” you say? Well yes. Before Aryabhata, mathematicians used a whole bunch of trigonometric tables as reference books for various triangles of differing lengths, angles etc, which were used throughout history to estimate distances that can not be measured directly (using properties of similar triangles). One of those books was a book of chords, which make handy triangles.
See? A lovely isosceles triangle is made from a chord and two radii.
Aryabhata went one step further with his table, based on the half-chord:
Now we have a trusty right-angled triangle (green) instead of an isosceles.
If you had absolutely nothing to do for the rest of your life, you could make your own table of chords, using this fancy Geogebra Applet. Note that at 900 the length of the half-chord is the length of the radius, and at 00 the length of the half-chord is zero. (Hence Sin90 = 1 and Sin0 = 0)
At this point you’ve probably noticed we’re using a particular circle for these tables. The unit circle which has a radius of 1 and is sometimes denoted as S1 . I’m sure you’ll agree, working with a unit circle is far easier than working with a circle of any other radius. And that’s kind of the point of it.
Now this value eventually became known as the Sine value (of the angle), even though it shouldn’t have (it was mistranslated somewhere along the way to Europe). And it’s important to think of it as a ‘reading’ , not as a constant.
Often students assume it’s a constant, because in algebra, ab = a x b, so Sin30 = Sin x 30 right? No. Sin30 means the value of Sine (the length of a half-chord on a unit circle) at 30 degrees (in other words, the length of the red line below).
*Note that the length is also the y-value if you plot the unit circle with centre (0,0), which is generally the ‘done’ thing.
Now, going back to the original table of chords, only one value changed. The chord. The other two sides of the triangle were both radii, so they stayed constant. But with the half-chord, there is a second triangle side length that is changing. Guess what that one is? The mighty Cosine. (*Note, that’s the x-coordinate on the unit circle plotted with centre (0,0)
With a bit of deduction, you can see that Cosine is in fact, the Sine value of the complementary angle (hence the name).
Here’s yet another useful Geogebra app that shows the relationship between the sine (length of half chord) and cosine (length of the adjacent side of the triangle made by a half-chord on a unit circle). Note the hypotenuse stays constant (at 1) because it’s on a unit circle (sadly the circle isn’t shown on that app). But the adjacent side changes, and is mapped onto a graph that makes… the sine and cosine waves. It’s a useful app to visualise what’s going on at Sin45 and Cos45 to see why their values are equal (you made an isosceles triangle).
Now if you haven’t figured it out yet, using a little Pythagoras gives you your trig functions (remember the hypotenuse = 1 on the unit circle)
In other words,
SinѲ = opp/1 (which is the half chord)
which you could interpret as ‘the sine value of Ѳ is equal to the opposite side of the triangle divided by the hypotenuse”
**for a unit circle ONLY, the sine value of Ѳ IS the opposite side, because the hyp of a unit circle = 1)
CosѲ = Adj / 1 (which is the adjacent triangle side length on the triangle made by the half chord)
Remember, with any right angled triangle, we’re simply scaling a triangle you can make using a half-chord on the unit circle.
So there you have it. Sine and Cosine. Not just ‘get the answer’ buttons on a calculator.
We’ll talk more about Tan next time. In the meantime, here’s a couple of Sine and Cosine Wave Gifs to help you visualise what the hell is going on.