In a unit circle, what allows the same trigonometric rules to be applied past the first quadrant?

Question

I had previously asked this question and was asked to learn more about the unit circle, which I've done. I now have further questions:

When the concept of trigonometric ratios for acute angles is being extended to angles beyond acute angles, the unit circle is used.

For teaching convenience, the teacher will begin in the first quadrant because that just "transfers" the initial right-angled triangle into the unit circle. The x-coordinate is shown to be $\cosθ$ whilst the y-coordinate is shown to be $\sinθ$.

This can be understood.

But what allows the same rules i.e. x-coordinate is always $\cosθ$ , y-coordinate is always $\sinθ$ to be extended beyond first quadrant in the first place? What are the assumption or rules underlying this? I understand it when it's in the first quadrant because I already understood the situation involving a plain right-angled triangle. I need a connecting explanation as to why we can extend the rules beyond the first quadrant, otherwise it seems to be circular reasoning (no pun intended) i.e. the rules apply past the first quadrant because we define it to be such.

I have been quoted definitions many time, I wish to understand why the definition is the way it is, not just accept the definition blindly. Any assistance would be greatly appreciated.

Answer 1

Part of mathematics is to extend the known concepts to the unknown territory.

we do this by definitions.

We started with natural numbers and defined negative numbers and zero to get to integers.

We started with integers and defined rational numbers,and finally real numbers and complex numbers.

To reach to the higher level of understanding we need new definitions and that is how we jump from a right triangle into a unit circle in order to define functions such as $\sin x$ and $\cos x$ for all real numbers.

Mathematics is an art and creativity is a vital part of it.

Accept the definitions and enter the beautiful world of trigonometry.

Answer 2

We can make a symmetry argument. If we can make a quad in the first quadrant, we can also make it in quadrants II, III, and IV by treating them as the quadrant I reflected about the y-axis, origin, and x-axis respectively. But this is messy and inconsistent, so to help that, $\theta$ is always measured ccw from the positive x-axis.

So each time $\theta$ goes into a new quadrant, we just flip the orientation of the right triangle we modeled in quadrant I to fit the demands of what signs its x and y coordinates should have in this new quadrant.

Once we complete one circle, we can begin another one where we initially started from, so both $\cos x$ and $\sin x$ have to be $2\pi$ periodic

Answer 3

Since you are asking again even though you have gotten a slew of good answers, I will try a different conceptual tack.

You've put the cart before the horse. Even though the word trigonometry means "trigon measurement", i.e. the study of triangles, the trigonometric functions are defined on the basis of the unit circle. Truthfully, though it is often first taught with degrees and triangle ratios, the student would be better served if trig were taught with radians and the unit circle to begin with. This would completely alleviate the misunderstanding that you seem to be having and make the transition to trig within the context of calculus much easier.

The study of the trig functions in relationships are truly most sensible only in the first quadrant because of the desire to use positive values for distances. Whereas, there is no conceptual hang up with thinking that the value of the x coordinate could be negative.

Simply put, the cosine returns the x value of a point a certain distance along the circumference, and the sine returns the y value of that same point. Drawing line segments and the corresponding ratios are merely an application of that definition.

Yeah, this is kind of an opinion piece, but it also a factual statement.