Question

what is arcsin?

Answer 1

is it the inverse function of sin?

Answer 2

Can you explain that?

Answer 3

yes,I think it's about function! :)

Answer 4

but I don't know what exactly is it!

Answer 5

It's hard to say that it's anything more than the inverse of \(\sin(x)\), since that's what it's defined to be. So it is the function that makes this true
\[\arcsin(\sin(x))=x.\]

Answer 6

But we have to restrict which \(x\)'s it works for, since if we put in \(x=\pi\) say, we have
\[\arcsin(\sin(\pi))=\arcsin(0)=\pi.\]
But if we put in \(2\pi\) we also have that \(\arcsin(0)=2\pi\).

Answer 7

So if it should be a function we must restrict it's domain, so \(\arcsin(x)\) is only defined for \(-1≤x≤1\). It's graph is just \(\sin(x)\) tipped on it's side and cut off, so that it becomes a function.

Answer 8

I think that you may know that arcsin is abbreviation of arcsine which is an invers function arcsin = (opp/ hyp)= q

Answer 9

I havn't study about function so can I understand arcsin?

Answer 10

arcsin is part of the inverse function in calculus

Answer 11

I have been seen some of it, but not all so that is what I can tell you about it

Answer 12

if arcsin is part of the inverse function in calculus,and I don't know anything about function so I can't understand this?

Answer 13

The Sine of an angle is equal to, the ratio of the opposite side to the hypotenuse side, of a right angled triangle. \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenues}}\)

As theta goes from 0 to 2π ,(0° to 360°),the function returns to its starting value and repeats, \(\sin(\theta)=\sin(\theta+2\pi n)\)



The Arcsine of the ratio of the opposite side to the hypotenues side of the right angled triangle, is equal to the angle. \(\arcsin\left(\frac{\text{opposite}}{\text{hypotenues}}\right)=\theta\)

This time there is a restriction on the domain, the hypotenuse will always be larger than the opposite side and hence the ratio will never be larger in magnitude than 1.
This means that \(\arcsin \theta\) will only make sense / be defined for x between -1 and 1

Answer 14

@UnkleRhaukus , Thank you :)

Answer 15

I am pretty sure that you will understand it , if you put on your part ok don't worry too much. take care ok I got to go

Answer 16

But that definition of \(\sin(\theta)\) breaks down when \(\pi/2≤\theta≤0\). It's more practical to define it in terms of the unit circle. In this context \(\arcsin(y)\) is just the angle \(\theta\) for which \(\sin(\theta)=y\), and can be phrased as "for which angle \(\theta\) is the y-coordinate of a point on the unit circle equal to \(y\)?"

Answer 17

And \(\sin(\theta)\) could be phrased "what is the y-coordinate at angle \(\theta\) on the unit circle?"

Answer 18

@dape , Thank you very much :)