Question

Why is it that sin^-1 has a principal value branch of [ -pi/2, pi/2]

and tan^-1 has a (-pi/2, pi/2)

how does one figure out the open/closed intervals from the graph? I know the graph and why the values are so but can't figure out why it has open and closed intervals

Answer 1

resticting the domain allows it to be a function....

Answer 2

break it down? I feel stupid :/

Answer 3

y=sin(x) is one to one on [-pi/2,pi/2]
y=tan(x) is one to one on (-pi,2pi/2)
we don't include the endpoints on the interval for y=tan(x) because we have vertical asymptotes at x=-pi/2 or x=pi/2 for y=tan(x), that is, the function doesn't exist there.
One to one means it passes a horizontal line test.
One to one means if we find a inverse it is going to be an inverse function on the intervals in which we restricted the domains to.