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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?

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Related but different matheducators.stackexchange.com/questions/1330/… –  quid Apr 27 '14 at 18:44

1 Answer 1

up vote 5 down vote accepted

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.

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