sin and cos

What's the height of that tree, the teacher said, hoping I would be quiet for a bit longer. Obviously I had no clue, because I didn't understand trigonometric functions, the intricacies of infinite series, and didn't know what a radian was. That story may not be entirely true, however.

Triangles have three sides, so it's only logical there are three functions to define the angle ξ. sin (ξ) is the ratio of the opposite side and the hypothenuse, and cos (ξ) is the ratio of the adjacent side and the hypotenuse. tan(ξ) is the ratio of the two - but you probably already knew that. You probably also knew that you can use the inverse trig functions (arcsin, arccos and arctan) to work out what ξ is in the first place. And you probably already know about sec (ξ), cosec (ξ) and cot (ξ), which are more trig functions that are just the reciprocal of cos, sin and tan respectively. I won't bore you with that.

Defining ξ in radians, we can differentiate these ratios, and come up with some more peculiar outcomes. Differentiate sin (ξ) and you get cos (ξ). In fact, continue doing this, and you'll end up back at sin (ξ) - sin goes to cos goes to -sin goes to -cos goes to sin. And the same is true if you integrate every function, albeit in reverse. At this it's easier to recognise that the two functions are more identical than you may think - and that's not surprising, as sin (ξ) = cos (ξ - 90). You can also express sin and cos functions as the sums of the product of sin and cos functions. And the graphs of sin (ξ) and cos (ξ) are lovely at showing that. The sine wave especially is often seen in physics.

That's easy, I said, the height is obviously 4π radians.

Whilst sin (ξ) can be defined as the ratio of two sides of a triangle, it can also be expressed as an infinite Taylor series, which serves to approximate the sine wave. Specifically, it's this series:

ξ - (ξ³)/3! + (ξ⁵)/5! - (ξ⁷)/7! ... or in other words, the alternating adding and subtracting of odd powers of x divided by the factorials of those odd powers. Using the first four terms and letting x be 0.5, sin(0.5) ≈ 0.479; sin(0.5) = 0.479 as well (to 3sf). If we let x be a very small number, whatever that means, we get sin (ξ) ≈ ξ. 

We can do a similar thing with cos (x) - except this time, we use even numbers, starting from 0.

cos (ξ) = 1 - (ξ²)/2! + (ξ⁴)/4! - (ξ⁶)/6! ...; if we wanted to approximate it, though, we'd need to include the second term - for small angles, cos (ξ) ≈ 1 - (ξ²)/2. In fact, we can do this with all the trig functions, but they quickly get far more confusing. The expansion of tan (ξ) involves Bernoulli numbers, for example. There are also similar expansions for hyperbolic trig functions - however, the terms are all positive, there is no alternating in signs.

That's one reason why I suspect hyperbolic trig functions are called that -  they're based on exponentials, but can be expressed as a similar Taylor expansion. sinh (ξ), as it's called, is just (e^ξ - e^-ξ)/2, whilst cosh (ξ) is (e^ξ + e^-ξ)/2. You can link the two, though: sinh (ξ) = -isin (iξ), and cosh (ξ) = cos(iξ) - all can be found here. From here, the link between trig, complex numbers and exponentials has finally been realised. Euler's formula, e^iξ = cos (ξ) + isin(ξ), is probably the most iconic example of this link. From there, we can get to the famous equation e^iπ = -1, which occurs when ξ = π, or simply -1 + 0. And you don't even need complex knowledge of maths to know that - it's all in their graphs from the beginning.

If the story at the start was true, I never found out how tall the tree was. If only I knew about all these complex expansions.