Expert: Paul Klarreich - 12/6/2008

Question
arccos(cos7π/2)=  π/2. can you explain how do we get  π/2?

Answer
Questioner:   Steve
Category:  Calculus
Private:  Yes
 
Subject:  inverse properties
Question:  arccos(cos7pi/2)=  pi/2. can you explain how do we get  pi/2?
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Hi, Steve,

Two notes first:
1. Don't mark 'Private'. I change it anyway. If you don't want anyone on this site to see the question, send it somewhere else.
2. Don't use special math symbols. They get chewed up in the process.

Now then:

This is a matter of PRINCIPAL VALUE of a function.  For arccos(x), the principal value is between 0 and pi.  That means to find

y = arccos(x),

you do this:
1. Solve the equation x = cos y for the many values.
2. Pick the one that is between 0 and pi. (The P.V.)

Now:  cos (7 pi/2) = 0, so solve:

cos y = 0, which has many solutions, ONLY ONE OF WHICH, namely,  y = pi/2, is in the range  0 to pi.

That's it.