How do you evaluate tan(arcsin(23))?

1 Answer
Dean R.
May 9, 2018

The principal value gives

tanArcsin(23)=2√5

Treating arcsin as a multivalued expression gives

tanarcsin(23)=±2√5

Explanation:

This question has a different answer depending on whether we interpret arcsin(23) as all the angles whose sine is 23 or just the one in the first quadrant. I prefer the former interpretation, reserving Arctan(23) for the principal value.

Let's answer first considering arcsin(23) to be a multivalued expression. That means

θ=arcsin(23)

is equivalent to

sinθ=23

which is an equation with multiple solutions.

Then there are two possible values for cosθ:

cos2θ+sin2θ=1

cosθ=±√1−sin2θ

In our case,

cosθ=±√1−(23)2=±√53

That's also apparent if we treat the right triangle in question as having opposite side 2 and hypotenuse 3 so other side √32−22=√5.

So we get

tanarcsin(23)=tanθ=sinθcosθ=23±√53=±2√5

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

tanArcsin(23)=2√5

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