Different definitions of trigonometric functions?

Question

In school, we learn that sin is "opposite over hypotenuse" and cos is "adjacent over hypotenuse". Later on, we learn the power series definitions of sin and cos. How can one prove that these two definitions are equivalent?

Answer 1

Answer 2

Hi!

Well, the 'SOH CAH TOA' definitions are limited to 'positive acute angles'. In other words, this definition is in terms of an angle (not the right angle) of a right triangle. So, once the angle is selected then the side 'opposite' and 'adjacent' are determined.

The power series depend on the derivatives of these functions for the coefficients. And so, it is more natural to use the unit circle definitions, which is a generalization of the right triangle definition.

So, the power series definition is a generalization of the right triangle definition.

Answer 3

You are comparing two different things.

Within the context of a right triangle, YES, the sine is opposite over hypotenuse and

cosine is adhacent over hypotenuse.

However, with regard to the SINE FUNCTION, it can be expanded in an infinite series which

should agree with the angle measure for a specific value of theta.

The power series will dive an APPROXIMATION to the trig functions of the angle.

Answer 4

Good question!

The sticking point of your question is how to define the functions sin x and cos x.

Once you have defined those functions appropriately, then the power series is a natural result of the Taylor's series expansion and all the properties you want those function to have comes easily.

The most elegant way I have seen of defining the sine and cosine function and showing that the definitions make sense with what we started with in plane trigonometry is found is the textbook "Calculus" by Michael Spivak...not a easy read but worth the trouble if you really want the answer to your question.

Good luck