Why do we choose a unit circle with center at origin to define trigonometric ratios?

Question

Alright I understand the if the circle is not a unit circle, it's still fine because we will get $r \cos \theta$ and $r \sin \theta$ and that way it won't change the definition of trig ratios, what I don't understand is, why does the circle have to be at origin. What if my circle is at some point $(h,k)$ how will I define trig ratios then?

Second question, they taught us Trig using a triangle in school. Now suddenly we are being taught this on Cartesian coordinates system which has negative angles and lengths (which isn't possible) what is going on? I can't even make a connection here anymore. Help!

EDIT : What does direction got anything to do with length? Basically it's a ratio of $2$ lengths, why would I want to "consider" it a vector quantity when it's not. Just like that? I'm having a hard time wrapping my hand around this. Help!

Answer 1

You ask what if you shift the centre of your circle to some other point. Well, you still have the usual trig ratios since they're defined relative to the centre of that circle (as opposed to the origin); so a displacement of the unit circle from the origin affects nothing -- in other words, the definition does not depend on the origin in particular; it is only chosen because, well, why do we need to overcomplicate matters when they don't matter?

Your second question is more eclectic; I hope the following sufficiently addresses it. It seems to me that you have not yet appreciated the rationale for studying trigonometry (or coordinatising the plane, for that matter). Well, if trig were not that important then it wouldn't even be a separate part of mathematics at the secondary school level, but just a topic or two. It turns out we need a knowledge of trig in a host of situations different from just the solution of triangles (at least such triangles are only usually in the background). Also, you need to be able to change your views to accommodate more generality as you advance in your studies -- this openness and laterality in thought will help you a lot not to get confused over things that don't matter. Trig is usually first introduced synthetically because at such stages the pupils know nothing of coordinates (consequently, their trig is limited to straight angles).

Historically, trig started in astronomy where a way of describing directions was sought. Today angles pop up everywhere -- they're involved with geometric vectors and even vector-like objects can get a notion of angle (cf. orthogonality of functions, e.g.). Much more than this, the study of trig leads to those functions that are the basis of the study of all oscillatory phenomena -- from circadian rhythms to the motion of the planets... Coordinates help us generalise the functions from just the set of numbers in the interval $[0,180]$ to all possible real numbers. But you seem to be averse to the notion of a negative number. First, I should assure you that you are in good company -- for many years even the best mathematical minds treated them as less real than the positive numbers, but eventually they got over this and realised how to think about them. Again, such flexibility of thought is important to comprehending higher mathematics -- if we co-ordinate the plane Cartesian style, every point can be referred to unambiguously. Without the negatives we couldn't do this (we only have a quarter-plane, which, I hope you'll agree, is unsatisfactory).

In any case come back to trig. There is a beautiful relationship between circles and triangles in general. It turns out one way to extend the trig functions to a larger domain is to measure angles by lengths of arcs of a circle. Draw a line through the centre of this circle and start measuring from one point of intersection of line and circle (which point and starting orientation you pick is irrelevant, but for most people it is convenient to imagine a horizontal line and measure counterclockwise from the right intersection). Now notice that the circle has been split right in half -- everything up is down and vice versa -- so think of this line as a mirror (again, which semicircle you wish to think of as 'real' and which to think of as the reflection is immaterial, but most people think of the upper one as 'real' and thus the lower as its reflection; we indicate this by labelling all perpendicular distances (ordinates) measured downwards from the line as negative). It is clear that we may now define the trig functions for all numbers in $[0,360],$ and the natural way to define anything beyond $180$ is to use reflection (we need another mirror through the centre of the circle and perpendicular to the first mirror to track cosines). Indeed we may extend beyond these numbers by thinking of a ray emanating from the centre of the circle. If it sweeps continuously counterclockwise, then it is clear that it traverses every positive real number; consequently, it sweeps through the negative real numbers in the clockwise direction. Thus we have extended the functions to the real line. This extension is beneficial for the study of periodic phenomena, and is the basis of Fourier theory and related theories.

Do not despise trigonometry -- love it, understand it, and never cease to look at it from different perspectives. Good luck in your studies.

Answer 2

The utility of the unit circle is to introduce the more generalised trig functions as distinct from the simple ratios that allow us to solve simple triangles. This allows us to extend the definition of trig function to all angles, not just the first quadrant. This, in turn, allows more advanced concepts that involve complex number and the use of exponential functions and important equations such as, $ e^{i\theta}=\cos(\theta)+i \sin(\theta)= cis(\theta) $. This then permits calculus operations on these functions and leads naturally to their Taylor-McLaurin series, etc.

There is another way to think about the trig functions. Imagine the unit vector rotating (anticlockwise) about the origin: the sine function is its shadow or projection against the vertical axis while cosine is its shadow or projection on the horizontal axis.

The utility of the unit circle at the origin is to simplify the equations, specifically to omit the need to constantly subtract the coordinates of the centre; that is, it purely for simplicity.