# Unit circle vs. ratios of right triangles vs. wave functions for introducing trig functions

In an introductory trignometry course, there are many options for introducing trigonometric functions:

1. As ratios of sides of right triangles
2. As coordinates (or ratios of coordinates) of intersections of the unit circle with rays from the origin
3. As graphs that are periodic (and wavelike in the sin/cos case)

I was taught #1 first in high school, and then graphs. I saw #2 in college.

I feel that #1 is the traditional secondary-education method of introducing trigonometric functions, which has the benefit of coming a year or two after Euclidean geometry in the United States.

I feel that #2 is traditional in more rigorous textbooks used in University level courses, where radians are used.

Method #3 has the advantage that many students benefit from graphing calculators and visualize a function based on its graph.

Which of these three methods (or another unmentioned method like power series) would be best to introduce a college freshman with no math past algebra to trigonemetric functions with the goal of eventually covering the other two methods?

• Related but different matheducators.stackexchange.com/questions/1330/… – quid Apr 27 '14 at 18:44
• Why choose one? Demonstrate them all. Also, you forgot 4. As the real and imaginary parts of complex numbers $e^{i \theta}$. – Andrew May 4 '16 at 2:57
• 1. first but 2. is necessary before getting into sum angle formula. – user2139 May 4 '16 at 6:19

Physics approach (recommended, if the students have physics background or are simultaneously educated in the relevant physics):

Introduce the operations as the coordinates of a point ($\sin$ and $\cos$) resp. slope of the lines ($\tan$ and $\cot$) moving in a uniform motion along a circle of radius 1. Best example: Earth moves (nearly) uniformly along a (near) circle of radius 1 (AU). Give them other examples as well (car going in a curve, clock hand going around the clock).

Reason: Circular motions usually have more real life connection to the students than ratios of sides of triangles.

Afterwards, introduce the functions via harmonic mechanical oszillations. That way, you keep to the context of motions.

I teach (see my profile) at the high school level in the US. This is the sequence we use, and I think works.

• Triangles - students have already taken geometry, and are familiar with the 45/45/90 and 30/60/90 (degrees, of course) triangles. With the rules of similarity, we introduce the 3 basic trig functions and the initialism SOHCAHTOA which helps students remember the relationship for each. In my opinion, there's something concrete about the examples we offer here. The flagpole, tree, etc in sample problems.
• Unit circle - this seems a logical progression, as a way to get beyond the 30/45/60/90 angles, and introduce the cyclic nature of the trig functions. I've shown students how a rough sketch can easily remind them of the 30/45/60 sin/cos numbers with little to no memorizing.
• Periodic sinusoid graph - This seems the logical next step, graphing the sine or cosine value on the Y axis, vs degrees or radians on the X. It segues nicely to more complex equations where the period, amplitude, phase shift, etc, are introduced, and the use of the unit circle wouldn't be sufficient.
• Practice problems for the sine and cosine defined through a right-angled triangle seem very artificial although they technically come from "real life". So from the problem-based point of view, starting with the definition of the sine and cosine through a unit circle makes more sense because it can be immediately applied to important problems such as circular motion. – Pavlo Fesenko Apr 4 '18 at 20:28