If I'm understanding correctly, the unit circle is based on ratios of special triangles and gives precise angles in degrees and radians for said triangles. However, if a triangle's hypotenuse isn't equal to 1 or the angles aren't exactly what the circle portrays, what's the use of it? My guess so far is that we can use it to apply ratios to the sides of the special triangles to find angles in ones with different measurements, but even if that's correct it just seems counter intuitive to use such a seemingly limiting template.
Why do we use the unit circle?
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Dr.Peterson
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I think you are misunderstanding what the unit circle really is. The unit circle isn't just about special angles; it applies to any angle. You're just thinking of the fact that the special angles are commonly marked on a drawing of the circle. Teachers often use such a diagram to help you learn the special angles, but the unit circle is really much more than that; it's a concept that removes limitations!
Here is an explanation of the unit circle, which ends with the picture you are thinking of:
But what's important is what comes before that picture! Note that the picture is not called "the unit circle", but "Points of Special Interest on the Unit Circle". The unit circle is the whole circle itself; it is not just the special angles. (Special angles are just a few angles for which we can give exact values, so teachers tend to use them in exercises. In real life, they aren't usually very important!)
I've seen many students with this same misunderstanding, because teachers focus their attention on the special angles rather than the concept; that is why when I have taught trig, I have not handed out that sheet showing the unit circle! There are easier ways to think about special angles, and about quadrants, than memorizing everything on the picture, and the concept is far more useful than that.
What the unit circle concept does, mainly, is to extend the trig functions to any angle (in any quadrant). It's a new definition of the sine and cosine based on the coordinates of a point on the unit circle, which is equivalent to the right triangle definition for acute angles. The cosine is the x-coordinate, and the sine is the y-coordinate -- for any point at any angle on the circle. That's the concept that I say is important.
As for triangles with a hypotenuse other than 1: Trig functions are ratios -- so it doesn't matter how long the sides of a triangle are. (In fact, that's how they can be functions of the angle -- there is one value for a given angle, regardless of how big the triangle is.) If the hypotenuse of a triangle isn't 1, you can multiply the sine or cosine by that hypotenuse to get the lengths of the legs. Scaling it to 1 helps focus our attention on the ratio rather than the triangle.
That makes much more sense, thank you!
In addition to Dr. Peterson’s excellent points, the trigonometric functions were originally developed for the angles in triangles with scientific use in astronomy and more practical use in surveying, cartography, and navigation. By extending the concept from triangles to the circle, the trigonometric functions now can be applied to any periodic phenomenon such as the study of waves.
Are we correct in saying the acute angled ratio definitions of sine, cosine are essentially special cases of the unit circle? So the unit circle was developed first? Or the other way around - ratios came first then the concept was extended or is it just that we teach it in this way?
Dr.Peterson
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Logically, we tend to think of the unit circle formulation as a generalization of the right triangle formulation, though it doesn't have to be done that way.
Historically, I think one could say that the concepts started with circles, in astronomy; the sine comes from what was originally a chord (of a circle in the sky). But when those ideas turned into trigonometry (which means "triangle measurement"), the focus turned to triangles (not necessarily right triangles).
But the unit circle formulation, I believe, came after formal definitions in terms of right triangles. So it's complicated. See
