Follow questiong regarding trig inverse functions.?

Question

If sin θ = y/r, cos θ = x/r, tan θ = y/x, then: arcsin y/r = θ, arccos θ = x/r, arctan θ = y/x and the same holds true for the rest of the trig functions? Say csc θ = r/y, then arccsc r/y = θ??

Answer 1

sin θ = y / r -->θ = sin^(-1)(y / r)
cos θ = x / r-->θ = cos^(-1)(x / r)
tan θ = y / r-->θ = tan^(-1)(y / r)

cosec θ = 1 / cos θ = r / x-->θ = cosec^(-1) (r / x)
sec θ = 1 / sin θ = r / y -->θ= sec^(-1) (r / y)
cot θ = 1 / tan θ = x / y -->θ = cot^(-1) (x / y)

Answer 2

yes. They all hold true, provided none of their denominators become zero.

Note: arccos θ = x/r, arctan θ = y/x are incorrect.
arccos x/r = θ, arctan y/x = θ are the proper equations.

d:

Answer 3

Just remember: ARC means 'angle whose function is ...", e.g.

arcsin y/r = angle whose sine is y/r = theta.

arctan y/x = angle whose sine is y/x = theta.

Notice, arc * (any number) is an ANGLE, not a number.