# What is the relation between trigonometry functions and circular functions?

In Calculus, by a function, $\;f$, we mean the rule which assigns to each element $x \in D$ (domain) an element $y \in \mathbb R$ (range). Where $x$ and $y$ are real numbers

According to the definition above, the trig function should be regarded as single-valued. But can trig functions operate on domains consisting of angles instead of real numbers? If not, how can we allow trig functions operate on domains consisting of real numbers? provided that we can't disregard traditional trigonometry because it exists beforehand. Whats unambiguous with traditional trigonometry that made mathematicians rethink and use real numbers rather than angles?

• Possible duplicate of what is sine of a real number – TRUSKI Sep 6 '18 at 7:16
• I believe my question is different than that of the possible duplicate because in this question I am not asking how the concept of differentiation would apply if the input to the trig function is an angle. Instead, I am asking about the reasons why we perceived differently the input to trig function machines when concepts like differentiation can still be understood if we think of the input as an angle and not as the length along the circumference of a circle? – AAMAIAK1998 Sep 6 '18 at 9:10

But can trig functions operate on domains consisting of angles instead of real numbers?

As far as I can tell one can describe angles by real numbers and vice versa. This includes negative numbers (turn in the other direction) and numbers greater or equal $360^\circ$ (multiple turns). $$(x, y) = r \, (\cos \phi, \sin \phi)$$

What difference do you perceive?

• In traditional trigonometry, we defined trig functions in terms of angles and not real numbers (which we visualize in terms of length). What compelled us to perceive the input as a real number? – AAMAIAK1998 Sep 6 '18 at 7:08
• I do not see the difference. E.g. angles specified in radians are given by the length of the resulting arc on the unit circle – mvw Sep 6 '18 at 8:00
• I am still confused. Before inventing radians measure, in traditional trigonometry, trig functions operated on angles formed by two rays, not on the length along the circumference of a circle. What made us switch our view of the input from angles to lengths When concepts like differentiability can still be understood how sin(angle in degrees) changes as the angle changes? – AAMAIAK1998 Sep 6 '18 at 8:47

In elementary geometry as taught and learned in high school the notion of "angle" is not grounded on unified axiomatic principles (which is o.k.). But you have learned that an angle can be measured as well in degrees as in radians. Radians seem more natural, insofar as degrees have to do with the fact that the Babylonians had a sexagesimal number system. Therefore it becomes reasonable to declare and tabulate the "official" trigonometric sine as a function of the angle measured in radians. So far our angles $\alpha$, real numbers all the while, belonged to the interval $\bigl[0,{\pi\over2}\bigr]$, say.

It then turns out (i) that there is a "natural" and simple extension of this sine function to $\alpha$-values outside of this interval, and that (ii) this same function, now in the form $\sin:\>{\mathbb R}\to[{-1},1]$, is of utmost imprtance in the description of phenomena that are periodic in time – an environment far away from the world of little triangles in school books.

But note that the "black box" denoted sin in your flow chart does not know what your intentions are: geometry, physics, what have you. It just computes for each given real input value $t$ the corresponding value $\sin(t)$.

• Yes, I have. I think I understand the resemblance of the sine function expressed in different angles of measure but I am afraid to have misinterpreted it. There are two ways of keeping track of the vertical distance from the x-axis if a point is traveling along the circumference of a unit circle. One is by taking the sin of the angle in degrees between the line joining the origin and where the point is along the circle, two is by taking the sin of the length the particle has traversed. These two should give the same resulting output. – AAMAIAK1998 Sep 7 '18 at 3:03
• Hence the two inputs should be equivalent. Input is dimensionless and because the sin machine is unaware of my intentions. s/r= θ. Is it more natural in the sense that we can understand things like sin(0) and sin(90) without resorting to a right angle triangle? The domain is not restricted as was in the case of angles in degree's mode – AAMAIAK1998 Sep 7 '18 at 3:11