How to solve this trigonometric mathematic function? (Details inside)

According to this wiki article, subtracting two arcsines can be rewritten as the arcsine of the first number times the square root of 1 minus the second number squared, minus the second number times the square root of 1 minus the first number squared:

arcsin(a) – arcsine(b) = arcsine(a*sqrt(1-b^2) – b*sqrt(1-a^2))

Does that point you in the right direction?

When you see the words “arcsin” Translate it in your mind to “the angle who’s sine is”

Also, you should know the sine of 5 basic angles in degrees. They are: 0, 30, 45, 60 and 90.
They are the most common and show up in most problems – even in life! The sine of those angles are 0, 0.5, 0.707, 0.866, and 1, respectively. An easy way to remember it is: 0, sqrt(1)/2 , sqrt(2)/2, Sqrt(3)/2 , and 1.

So, what’s arcsin(.707)? Say it out loud. The angle who’s sine is (0.707). That is 45 degrees
So what’s arcsin(0.5)?

Remember that arcsin results in an angle measurement.

Arcsin(.5) – or arcsin(½) – is the angle in a triangle where the opposite side is 1 and the hypotenuse is 2 (or any value with a 1:2 ratio, like 50 and 100). Like @worriedguy said, this is a particular case that you should know the properties of. Let me know if you need more help with that.

As for dealing with negative arcsin, do you understand reference angles? In this case, the angle made with the x axis will be the same, but your angle will be in a different quadrant. In the first quadrant, all trig functions are positive, in the second, only sine is, in the third, only tangent is, and in the fourth, only cosine. A mneumonic to remember that is “All Students Take Calculus.”

so arcsin is the inverse of sin?

Yup.

Yes. arcsin (x) is an angle. The angle who’s sine is x.

Thank you so much for your help :D

Make sure you note the two minus signs.