MATHS INVERSE TRIGNOMETRY PRINCIPAL VALUES

Question

For finding the principal value,in this question:

Find the principal value of cot^-1(-1/√3)

The answer is cot(π-π/3) which is cot (2π/3)

why is it not just -π/3?

In a similar question, Find the principal value of sin^-1(-1/2), I tried doing sin(π+π/6) since sin is negative in 3rd quadrant.But the answer was just -(π/6).Why is there this difference?Someone please explain.Thank You

Answer 1

arccot is defined for all real values of the argument except 0, i.e. it does NOT have a principal value in the same sense as arcsin or arccos, but the cot(120°) = -(1/3)√3 = cot(300°) ,etc.

arccot(x) always has a value between 0 and π, unless specified otherwise.

In order to understand better you should look at the graphs of arcsin and arccot along with the graphs of cot and sin. (You may need to put the graph of y=x in the same graphs!)