Part 2 in my series of ‘complementary maths’ – extra bits that complement the GCSE curriculum to hopefully aid student understanding : Tan.
A couple of weeks ago I explained the origins and meaning of Sine and Cosine, and how they relate to the unit circle, chords and right angled triangles.
Tan is a fairly obvious follow-up.
“Tan” is an abbreviation of “tangent”. Tangent stems from ‘Tangens’ (Latin) meaning “to touch”. With regards to our unit circle (yes, it’s back), we’re referring to a straight line touching the edge of the circle. Not just any straight line though. This one starts at the base of our half-chord right-angled triangle (with hypotenuse = radius).
Well, almost at the base. The base of said triangle is *inside* the circle, so we must extend the line. Time for a picture.
Ignore Sec, we’re sticking with GCSE here. Although now seems like a nice time to mention that Tan, Sec, Cosec and CoTan are all derivatives of Sine and Cosine. Sine and Cosine are the main ones, and you can do any trigonometric operation using only those two. In practice that’s annoying hence we use the others in A-Level and beyond.
Anyway, I digress. From the diagram above, you can see the tangential line that we use for ‘Tan’. It starts from an extension of our Cosine line (adjacent to the angle – CE in the second picture below, point E being the intersection with the tangent), and finishes where it meets the extension of our radial hypotenuse (BD below is the extension, D is the point of intersection).
So, in the unit circle, TanѲ means the length of DE (in the diagram above) at angle DAE (which is Ѳ). As with Sine and Cosine, these values change whenever we change the size of the angle.
So you’ll notice we now have two similar right-angled triangles.
We also know that using the smaller triangle ABC, the tangent of Ѳ is BC / AC, which in this unit circle is simply SinѲ / CosѲ
Hence TanѲ = SinѲ / CosѲ
Furthermore, if we use the larger triangle ADE, the tangent of Ѳ is DE / AE.
But we know AE is the radius, which in the unit circle = 1.
Hence TanѲ = DE,
Now, just like Sine and Cosine, if we plotted each value of Tan on a unit circle, you’d get the familiar corresponding graph. In the case of Tan, it looks like this:
So … why are there asymptotes?
Well, think back to how Sine and Cosine values are created using the unit circle. We need right-angled triangles. However, at 90 degrees, there is no triangle, there’s just a single line. Visualise that here. Or just look at the pictures below:
Now here’s where it gets a little weird. Technically, there’s no reason why Tan (90) can’t equal zero (*ducks bottle*), because, by the same reasoning(ish), Cosine (90) = 0, but that’s *allowed*. However, Tan 90 is considered ‘undefined’ because:
TanѲ = SinѲ/CosѲ and dividing by zero is world-endingly bad. So bad that we say it’s undefined.