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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.

Charlz97
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  • I don't understand your question. Are you asking why we can use trig ratios in quadrants II, III, and IV? Can't you just draw another triangle oriented differently? – Andrew Li May 07 '18 at 02:46
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    It really is as simple as "this is how we define sin and cos". Because of this, sin and cos are defined on all real numbers. – Kaynex May 07 '18 at 02:48
  • Historically, the trigonometric functions were defined on the circle first. Applications to right triangles came later. You should think of the trigonometric ratios for the right triangle as a special case of the trigonometric functions defined on the unit circle. – N. F. Taussig May 07 '18 at 09:56

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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.

Cedron Dawg
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  • You know from all the answers, I think you've best "gotten"my misunderstanding so to speak. So I'll ask - intuitively trigo ratios such as O/H, O/A, A/H are all using lengths isn't it? And aren't lengths always positive? Ratios are done using lengths!

    https://math.stackexchange.com/questions/2768581/why-is-cos135-circ-negative-when-length-is-always-positive?noredirect=1#comment5710209_2768581

    I laid this out in my initial question but no one seemed to get it.

     – Charlz97 May 07 '18 at 03:34
  • @Charlz97, Thank you. Lengths are always positive due to convention and the Pythagorean theorem, displacements are not. Think of the case of calculation a slope. If I only used lengths, all slopes would be positive. One step forward, two steps down is the same as one step backward, two steps up, and both are a slope of -2. The tangent function is the ration of sine to cosine, which each can be positive or negative, as can their ratio. The tangent is the ratio of y to x on the unit circle and is thus the same scale as slope. – Cedron Dawg May 07 '18 at 03:40
  • Okay, so digging deeper, why is it we use displacement (which then allows for negatives) instead of mere length/distance when calculating trigonometric ratios for angles in a quadrant other than the first? – Charlz97 May 07 '18 at 03:42
  • @Charlz97, Because the cosine is the x value, not the length along the x axis, similarly the sine is the y value, not the length along the y axis. Think of the graphs of sine and cosine that look like waves. They are the same graph, shifted by a quarter cycle (trip around the circle) and half the time they are each negative (below the horizontal). – Cedron Dawg May 07 '18 at 03:48
  • But cosine/sine are ratios. The formula TOA CAH SOH tells that - let's just use cosine - cosine of an angle = A/H. I look at how long A is - say 1 unit, I look at how long H is 1 unit, regardless of quadrant. Does TOA CAH SOH only apply for acute angles? Why again, in this context, am I wrong?

    Apologies if it seems difficult to explain here, clearly the misunderstanding is rather deep, so thank you very much in any case.

     – Charlz97 May 07 '18 at 03:52
  • @Charlz97, No, they aren't ratios. They become a ratio when they are divided by the constructed (added radius line segment) hypotenuse of one. You have the cart before the horse. Please reread all I have written. The "ratio definitions" are a special case of the "point on the circle definition". The former is the cart, the latter is the horse. If the ratio (of lengths) definition were the totality of the definition, the functions could never take on negative values. – Cedron Dawg May 07 '18 at 04:06
  • So it's more of...reframing how I see these concepts? – Charlz97 May 07 '18 at 04:07
  • @Charlz97, Yes. If you are not familiar with complex numbers, or the meaning of Euler's equation, now would be a good time to introduce them. They should help make your "reframing" easier. I recommend you read my first blog article: https://www.dsprelated.com/showarticle/754.php – Cedron Dawg May 07 '18 at 04:17
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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.

Mohammad Riazi-Kermani
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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

mallan
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