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I just recently did a project on the unit circle and the three main trig functions (sine, cosine, tangent) for my geometry class, and in it I was asked to provide an explanation for why sine is the y-coordinate and cosine is the x-coordinate. My initial response (which was demonstrated in the project) was that since the radius of the unit circle, or hypotenuse, is 1, opposite/hypotenuse=opposite and adjacent/hypotenuse=adjacent. Therefore, sine=opposite, or the y-coordinate, and cosine=adjacent, or the x-coordinate. This is pretty simple and makes sense. Then, I started thinking about how to derive the sine for a triangle in the first place, which requires knowing the measure of the opposite side. I looked at the following comment to a previous question I asked:

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from:

How would you describe a triangle with a sin 90 based on the definition of sine?

For example, if the side lengths of a 30-60-90 right triangle are 2, 1, and √3, you have to divide by the hypotenuse (2) in order to scale the triangle down to fit on the unit circle. By doing this, however, you divide each side of the triangle by hypotenuse, meaning that opposite becomes opposite/hypotenuse and adjacent becomes adjacent/hypotenuse. This already yields the result that in a triangle with radius 1, the opposite side is the sine and the adjacent side is the cosine, and we can come to the same conclusion as my initial explanation.

My question is, which would be more effective in explaining why these trig functions work in the unit circle? I get that the first explanation seems easier or more straight forward, at least for me, but the second one actually explains where the sine and cosine values come from for certain angles, making the first explanation feel almost feels unnecessary, for lack of a better term. I hope this makes sense and if not please let me know how I can clarify.

Note: The project has already been submitted, but I am wondering if this second explanation would have been better.

user386598
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2 Answers2

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Your question is ultimately about definitions--the important thing about having many definitions for the same concept is that they have to be equivalent (i.e., necessity and sufficiency). In your case, you just assume one is the definition, and can show the other follows.

You defined $\sin$ and $\cos$ using the SOHCAHTOA ratios on a right triangle, from pre-calc--this is a perfectly valid definition (similar to defining $\pi$ to be the ratio of the circumference to the diameter of an arbitrary circle). Assuming you know these to be the side-length ratios for $(\sin, \cos)$, you can apply them when the hypotenuse is $1$.

The comment defined $(\sin(\theta), \cos(\theta))$ as the point of intersection of the ray from the origin with angle $\theta$ with respect to the $x$-axis and the unit circle centered at the origin. Assuming you know these $(\sin, \cos)$ coordinate pairs on the unit circle, you can dilate or contract any right triangle to preserve the side-length ratios but with $1$ as the hypotenuse.

You can also define $\sin$ and $\cos$ as dot-product/cross-product ratios or as their power series or as solutions to certain differential equations--these also can be shown to be equivalent to the above. Certain definitions can be more useful for certain tasks (for example, if you are programming a game and calculating $\sin$ and $\cos$ many times per frame, a power series approximation and a lookup table could be better than calculating the ratio of the opposite and hypotenuse). There is no "more effective" definition--it's more like "here is the definition I chose, and here are the consequences of that choice."

Pavan C.
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    A teeny-tiny thing one most show before using the SOHCAHTOA definition is that you have to prove that the sides of similar triangles are proportional before you can assume the definition is consistent. That's pretty darned minor though. – fleablood Jun 05 '25 at 02:28
  • Very true! have to do some analytical or synthetic geometry :) – Pavan C. Jun 05 '25 at 05:20
  • No... It's basic Euclidean geometry stuff and one of the earlier propositions of Euclids elements. My point is one needs to be careful that ones definitions are valid before one states them. (Another example is $\pi = \frac {circumference}{diameter}$. And why does one know that's a constant? Actually that one is harder and it's possible we simply flubbed that one for a couple of millenia... Actually... off topic but does anyone know if Euclid ever used $\pi$ in any way?) – fleablood Jun 06 '25 at 21:46
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I suppose it depends on your audience but assuming you are presenting to peers, based on some classroom placements I've done, the first one might be better for groups who have recently been introduced to the basics of trig (GCSE Level in UK or about 13-14 year olds). They know SOH-CAH-TOA and the unit circle and this just provides a nice explanation that can be visualised easier. The second one as you said is a bit more rigorous, it gets into the WHY rather than just the what. Not sure if its within the scope of your project, but you could combine the second explanation with an animation/visualisation of an arbitrary triangle bigger than the unit circle, being scaled by dividing all sides by h, the hypotenuse, could also then factor in the concept of similar triangles/transformations.

Hope this is helpful but the tl;dr is, based on teaching a few levels of students:

First one is great for newer trig students:

  • Gives a simple explanation for what sine and cosine are.
  • Easy to show on a graph of the unit circle with a triangle drawn on.
  • Perhaps a bit more intuitive

Second is more suited for slightly more advanced students

  • Explains why we divide by the hypotenuse and why the unit circle is related to the trig functions
  • Can also show how multiplicative transformations retain angles as a corollary
  • Explains how $\frac{\text{Opposite}}{\text{Hypotenuse}}$ and $\frac{\text{Adjacent}}{\text{Hypotenuse}}$ lie on the x and y axes
  • A bit more freedom with showing this graphically

Hope this helps!

Charlie
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  • Thank you for the answer! Is there a way to perhaps blend these definitions together without being repetitive? They both seem important to understand to some degree. – user386598 Jun 05 '25 at 03:23
  • I tend to go for a visual approach with things like these, so start off with some triangle, either arbitrary or perhaps the 30-60-90 like you suggested, on the x-y plane, with hypotenuse>1 then have it "shrink" in an animation to the size of the unit circle, then show from our defn of SOHCAHTOA, the sine is the opposite over the hypotenuse but as we have scaled all our sides by 1/hyp the hypotenuse is simply 1 $\rightarrow$ sin = y component, x = cosine component. That way you get both the intuitive and more rigorous proof of where we get sine and cosine from. – Charlie Jun 05 '25 at 09:14
  • Could we also just say that since all the sides were already divided by hypotenuse they represent sine and cosine ratios (for all right triangles with one shared angle are similar by AA similarity)? And then we would not have to mention that opposite/1=hypotenuse and so on? ex. If the hypotenuse of a 30-60-90 triangle is 2 and if you scale the triangle by 1/2 each side is divided by two and then the sides become 1/2 and √3/2, which are already sine and cosine values depending on the given angle. Sorry if this is a lot I am just interested in mathematics and especially trigonometry. – user386598 Jun 05 '25 at 16:55
  • No need to apologise this is what this websites for! I don't see why not, its a great way to utilise the 30-60-90 triangle which is quite a familiar one and as students need to know certain exact values it helps with relating that. However, I think there is merit in using an arbitrary triangle with angles $\theta _1$, $\theta _2$, $\theta _3$ and side lengths $s_1, s_2, h$ as it allows us to prove the general case rather than just a specific, does that make sense or were you asking something slightly different? As another thought, you can also relate the graphs with changing $\theta$ as well. – Charlie Jun 05 '25 at 18:34
  • This makes sense and seems like a better way to explain it. However, I was more wondering if one can just say that since we scale a right triangle down to unit circle size by dividing by hypotenuse, each side becomes either sine or cosine (except for the hypotenuse). There is no need to then use the first definition in my question then, right? – user386598 Jun 09 '25 at 23:43