Question

Just a question regarding inputs in trigonometric functions; I might've forgotten, or it might've never been formalized, but what exactly do you call the type of input that isn't degrees or radians but representative of its position on a unit circle? (e.g. cos(1), sin(1), tan(0), sin(-1). Does this have a name as a "thing"?

Answer 1

Maybe you can give context...
What is cos(1) ?

Answer 2

WHOOPS

Answer 3

All of these are inverses, my bad, lol.

Answer 4

hmm?

Answer 5

Thought so too :D

Answer 6

But yeah, nonetheless.

Answer 7

although, I don't see how sin(1) = 1...

Answer 8

Whether or not it's \(\sin\) or \(\sin^{-1}\)

Answer 9

Yeah, i'm messing up/confusing my representation of the unit circle in radians and in a coordinate system, one minute.

Answer 10

Oh... so you want a representation of tangent on the unit circle? :)

Answer 11

oh wait, that was inverse tangent... :/

Answer 12

Ugh, how do I describe this...no, I mean, I could easily right now write up a unit circle, but where is that one coming from, what is it analagous to? It makes sense for sine and cosine, because at the extremes of their angles, you can have stuff like inverse cosine (0) = 1, because the input is just x = 0 which falls in the dead middle of a unit circle, and the cosine will obviously have a value of one, and the same works for extremities of sine like inverse sine(0) = 0, because it would fall squarely on either 0 or pi.

Answer 13

Well, that does induce problems :D
Take a deep breath, and concentrate your thoughts :)

Answer 14

Okay: As inputs for inverse functions, it seems like both sine and cosine rely on the x and y coordinate values at some given point to distinguish themselves, like so:

Because at (theta) = 0, the coordinate is (1,0), sin-1(0) where the input 0 is representative of the y-value of the unit circle put in a coordinate frame, sin-1(0) = 0.

Because at (theta) = pi/2, the coordinate is (0,1), sin-1(1) where the input 1 is representative of the y-value of the unit circle put in a coordinate frame, sin-1(1) = 1.

But this doesn't work for sin-1(0), where the coordinate is (-1,0) and is representative of pi. The analogy just stops working, because by itself, there's nothing to numerically distinguish itself from the exact same input and output of sin-1(0) where the coordinate is (1,0) and is representative of 0 radians.

The same thing could be said of the cosine function as soon as you take the cosine of pi/2 or 3pi/2; they're indistinguishable from an angle perspective IF their input isn't in radians but phrased by their coordinates.

Answer 15

And that obviously just doesn't work at all when you get to (tan), because you don't have a physical, coordinate allegory for it; it's just trig functions acting on other trig functions.

Answer 16

There is a representation for tangent with the unit circle, you know :)

Answer 17

^Not in coordinates.

Answer 18

(Yes, in values that you get by sine/cosine, but not in the same way you can describe an angle by its x/y value in a coordinate system where the origin is the center of the unit circle.)

Answer 19

Now draw the line TANGENT to the circle at the point (1,0)

Answer 20

And that, in fact, is why the function is called "tangent" :)

Answer 21

Yeah, you're right; I meant it in a different way, but you're right on that, lol. I mean that if you say, what's inverse sine(0), ugh, i'm going to go bury myself in Wikipedia and try to correctly ask what i'm trying to ask, because everybody is misunderstanding what i'm asking at the moment, and it's not your guys' fault. Brb.