In an introductory trignometry course, there are many options for introducing trigonometric functions:
- As ratios of sides of right triangles
- As coordinates (or ratios of coordinates) of intersections of the unit circle with rays from the origin
- 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?