How do you evaluate #arcsin(0)# without a calculator?

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Answer 1 · 2017-03-02T19:01:44

#arcsin(0) = 0,pi,2pi,3pi...#

#arcsin# is the inverse of #sin#, so I prefer calling it #sin^-1#, which is what I'll use.

If we say that the solution (which we don't know yet) is #x#, then

#sin^-1(0)=x#

so, taking the #sin# of both sides,

#sin(sin^-1(0)) = 0 = sin(x)#

Effectively, we are finding the point where #sin(x)=0#. You can do this if you know what a #sin# graph looks like already, or by using a #sin# graph.

graph{sin(x) [-10, 10, -5, 5]}

Using this graph, we can see that the curve #sin(x)# is #0# at the points #(0,0)#, #(pi,0)#, #(2pi,0)#, etc.

Therefore,

#sin^-1(0)=0, pi, 2pi, 3pi, 4pi...#

or

#sin^-1(x) = npi#, where #n# is any integer (positive or negative, as the graph extends in both directions).