How do you evaluate #tan(arcsin( 2/3)) #?
The principal value gives
This question has a different answer depending on whether we interpret
Let's answer first considering
is equivalent to
which is an equation with multiple solutions.
Then there are two possible values for
In our case,
That's also apparent if we treat the right triangle in question as having opposite side
So we get
In the case when we're talking about the principal value of the inverse sine, a positive sine ends us in the first quadrant, so a positive tangent as well. We'll write this