Prinicple value of both of them
For calculating principal value of given function. We know that principal value branch of sin thetha is -?/2 to ?/2,.
So convert 2?/3 in terms of ?/2
Sin ^-1 sin (2?/3)
Sin^-1 sin (?-?/3)
Sin^-1 sin (?/3)
Now sin inverse and sin cancle each other
Principal value =? /3
2)cos 60?= cos (?/3)= 1/2
Sin30= sin (?/6)= 1/2
Place these value in the eq
Cos^-1(cos ?/3)+2sin^-1(sin ?/6)
?/3+2?/6
?/3+?/3
=2?/3
So convert 2?/3 in terms of ?/2
Sin ^-1 sin (2?/3)
Sin^-1 sin (?-?/3)
Sin^-1 sin (?/3)
Now sin inverse and sin cancle each other
Principal value =? /3
2)cos 60?= cos (?/3)= 1/2
Sin30= sin (?/6)= 1/2
Place these value in the eq
Cos^-1(cos ?/3)+2sin^-1(sin ?/6)
?/3+2?/6
?/3+?/3
=2?/3