Where did the trigonometrical term "Arcsine" origninate from, and what is the meaning of it?

It's the inverse of a sine

Also known as 'asin', the inverse of the sine function, so that its value for any arguement is an angle in radians whose sine equals the given argument; that is, y=sin^-1x if and only if x = sin y. It is defined for arguments between -1 and 1, and its principal values (often written as Sin^-1y) are by convention taken to be those between -pi/2 and pi/2. (sigh, wish I had a pi sign...)

Don't worry about it. You've proved you know your acrsine from your hypotenuse....

In maths angles are traditionally measured anticlockwise around a unit circle starting from the rightmost point of the circle. The height of a point which traces angle A on the circle above the centre is the sine oof the angle (sin A), the horizontal distance is the cosine (cos A) and the gradient of the radius to that point is the tangent (tan A).
Given a sine of an angle between -1 and 1 there is an associated angle between -180 deg and 180 deg called the inverse sine (e.g. inv sin 0.5 is 30 deg).

But mathematicians now don't usually use a circle divided into 360 degrees for its unit of angle but a circle divided into 2 pi radians. This means that the angle turned through by our point is equal to th length of arc it traces on the unit circle.
The inverse sine in radians is not only an angle also a length of arc on the unit circle so is called the arcsine.
Arccosine and arctangent are similarly defined.

Pardon any typos but due to a browser error I had to type this twice.