How are numbers manipulated in a trigonometric function?

Question

Alright, I understand the definitions of the trigonometric functions, how it is a ratio between the lengths and hypotenuse. However, I wish to deepen my understanding how what goes on behind the scenes when I input sin(x degrees) into my calculator to get an output. How is the calculator able to get from this, e.g.sin(56), to that: 0.829 (3 s.f.)?

In: 11

Answer 1

If you want to deepen your understanding, then it’s good to lose focus on what happens in a calculator and to focus on learning properties of trig functions and how to compute special values.

* How do you compute sin(45)? Well, you gotta setup a triangle and use the Pythagorean theorem to help.

* How do you compute sin(30)? You have to use a different triangle, and you can do it.

* Can you compute sin(15) from this? Maybe the double angle formula can help.

These all give results involving square roots, so maybe you can think of them as roots to polynomials. For instance, since 30×3 = 90 then I can use the triple angle formula for cosine (cos(3x) = 4cos^(3)(x)-3cos(x)) to find cos(30). To do this, set y=cos(30) and note that cos(3×30) = 0 and with this substitution we get 0 =4y^(3)-3y. Solving for this gives the possible solutions y=0, y=sqrt(3)/2 and y=-sqrt(3)/2 which are the solutions.

Maybe we can solve for cos(10) from this same formula as well, but starting with cos(30)? This can then suggest more ways to compute values of trig functions by using these polynomials to do computations. If I know I can get to cos(20) by starting with 90, dividing by 3, dividing by 3, and multiplying by 2, and applying the polynomial results each time. It them seems that if you want to get cos(x), the all you need to do is find a safe starting point and a sequence of divisions and multiplications by whole numbers until you get close enough to x to call the final result “cos(x)”.

In this way, we’re less “doing stuff with x” and more “taking advantage of trigonometry” to do these computations. And this is the basic idea of many algorithms for the trig functions. But focusing on the geometric properties, the ties to algebra through various identities, can give you a more useful understanding from which the algorithms are a fun consequence.