How do you calculate cos1[cos(7π6)]?

2 Answers
May 24, 2018

The answer will be 5π6

Explanation:

You can write 7π6 as (π+π6)
Thus we can clearly see the angle falls in the third quadrant. And the cosine value in third quadrant is always negative.

Hence, cos(π+π6)=cos(π6)

coming back to the question
cos1[cos(7π6)]=cos1[cos(π6)]
=πcos1[cos(π6)]
=ππ6
=5π6

May 24, 2018

arccos(cos(7π6))=±7π6+2πk, integer k

Arccos(cos(7π6))=5π6

Explanation:

I treat arccos(cos(7π6)) as a multivalued expression, all the angles whose cosine equals cos(7π6).

In general cosx=cosa has solution x=±a+2πk, integer k

x=arccos(cos(7π6))

cosx=cos(7π6)

x=±7π6+2πk

arccos(cos(7π6))=±7π6+2πk, integer k

The principal value is in the second quadrant, given here by the minus sign and k=1.

Arccos(cos(7π6))=7π6+2π=5π6