Can Complex Numbers Rotate and Invert Sine Functions?

[zxh]

Sry, noob, but i didn't find this anywhere.

 

[Martin Rattigan]

No. Try Google.

 

[Mark44]

The two functions are inverses of one another. In general, the graphs of y = f(x) and x = f^-1(y) are identical, but the graphs of y = f(x) and y = f^-1(x) are reflections of each other in the line y = x.

The situation is a little more complicated with y = sin(x) and y = arcsin(x) = sin^-1(x) since the graph of the sine function isn't one-to-one (making the inverse not a function). The usual way around this is to restrict the domain of the sine function, defining y = Sin(x) = sin(x), with x restricted to the interval -pi/2 <= x <= pi/2.

 

[zxh]

So the title would be true for -sin, viewed as a curve?

 

[Mark44]

Are you asking whether y = arcsin(x) is the rotation by 90 deg of y = -sin(x)? If that's the question, then no.

If that isn't the question, then what are you asking?

 

[Martin Rattigan]

So the title would be true for -sin, viewed as a curve?

No it's only a segment of the curve, but the graph of y=arcsin(x) would fit over y=-sin(x) if rotated 90° either way about the origin.

 

[zxh]

Thanks, that's what i was looking for. I'm not too concerned about range definitions.
I came to this looking for a trig definition of a (half) circle (not the pythagorean Sqrt(r-x^2)).
At first i was wondering why Cos(Sin(x)) (given that the 2 functions for a circle in a parametric plot are sinx and cosx) didn't work but it turns out it's
Cos(ArcSin(x)).

 

[disregardthat]

You can show this algebraically. If you are familiar with complex numbers, this is easy.
1) multiply x+i (-sin x) with e^(i*pi/2) to rotate it by 90 degrees.
2) reflect x+i(-sin x) over the curve y=x to invert it.

 

[zxh]

You can show this algebraically. If you are familiar with complex numbers, this is easy.
1) multiply x+i (-sin x) with e^(i*pi/2) to rotate it by 90 degrees.
2) reflect x+i(-sin x) over the curve y=x to invert it.

thanks, good one.