In case you haven't noticed, the two series [tex]C(\theta)[/tex] and [tex]C(\theta)[/tex] in robphy's post are:fliptomato said:Ah--okay. So we can define [tex]\sin[/tex] and [tex]\cos[/tex] as ratios of a right triangle, or equivalently cartesian coordinates of a point on a unit circle. This has a nice geometric interpretation. However, at some point, we also prefer to define these trigonometric functions in a more general way to accomodate complex numbers:
[tex]\sin(x) = \frac{1}{2}(e^{ix}-e^{-ix})[/tex]
[tex]\cos(x) = \frac{1}{2}(e^{ix}+e^{-ix})[/tex]
One of the advantages of this formalism (aside from giving a natural definition for the hyperbolic trig functions) is that we can recover Euler's formula trivially:
[tex]\cos(x) + i \sin(x) = e^{ix}[/tex]
However, now I'm curious how we can relate this to the previous geometric interpretation. Instinctively, perhaps, we could recite what is commonly taught and "interpret" the imaginary exponential [tex]e^{i\theta}[/tex] as a point on the Cartesian plane with length one from the origin and angle [tex]\theta[/tex] from the positive [tex]x[/tex]-axis.
However, this is just an awkward definition--why do we interpret this complex exponential this way? I have an algebraic understanding of the exponential of real numbers.
[tex]e^3= e\cdot e\cdot e[/tex]
And by the laws for multiplication of exponentials, we can figure out how to appropriately define the exponential of a negative number, of zero, and even of rational and irrational numbers. However, this is all based on an understanding of the real numbers as things that are closely related to counting numbers, and in a sense we're counting how many times the base of the exponential is being multiplied to itself.
How then, can we understand the exponential of an imaginary number as a polar coordinate, therefore allowing us to connect our generalized definition of the trigonometric functions to our geometric intuition about them?
Perhaps put in another way, why is it that robphy's comment is true? (that the sine is the imaginary component of a complex exponential while the cosine is the real component). We can set this true by definition, but then how do we connect this to a geometric interpretation? (then you say that we can graph the complex exponential as polar coordinates--to which i ask why is a complex exponential supposed to be interpreted as polar coordinates?)
-Flip
Great question.Cheman said:My question has always been why is the tangent function called thwe tangent function?
Agreed.mathwonk said:i recommend reading my post again and thinking about it. it is much more elementary than the other posts here.