Find the principal values of the following:

(i)  (ii)  (iii) 

(iv)  (v)  (vi)  

(vii)  (viii)  (ix) 

(x) 

(i) Let sin^-1 (-1/2) = y . then siny=-1/2 = -sin(Π/6) = sin (-Π/6)

Implies the range of the principal value branch of sin^-1 is [-Π/2,Π/2] 

and sin (-Π/6) = -1/2 implies the principal value of sin^-1(-1/2)= -Π/6

(ii) 

Let cos^-1(√3/2) = y . then cosy=√3/2 = cos(Π/6) 

Implies the range of the principal value branch of  cos^-1 is [0,Π] 

and cos(Π/6) = √3/2 implies the principal value of cos^-1(√3/2)= Π/6

(iii)

Let cosec^-1(2) = y . then cosecy=2 = cosec(Π/6) 

Implies the range of the principal value branch of  cosec^-1 is  [-Π/2,Π/2]-{0} 

and cosec(Π/6) = 2 implies the principal value of cosec^-1(2)= Π/6

(iv)

Let tan^-1(-√3) = y . then tany=-√3 = -tan(Π/3) =tan(-Π/3)

Implies the range of the principal value branch of  tan^-1 is  (-Π/2,Π/2)

and tan(-Π/3) = -√3 implies the principal value of tan^-1(-√3)= -Π/3

(v)

Let cos^-1(-1/2) = y . then cosy=-1/2 = -cos(Π/3) =cos(Π-Π/3) =cos(2Π/3)

Implies the range of the principal value branch of  cos^-1 is [0,Π] 

and cos(2Π/3) = -1/2 implies the principal value of cos^-1(-1/2)= Π/6

(vi)

Let tan^-1(-1) = y . then tany=-1 = -tan(Π/4) =tan(-Π/4)

Implies the range of the principal value branch of  tan^-1 is  (-Π/2,Π/2)

and tan(-Π/4) = -1 implies the principal value of tan^-1(-1)= -Π/4

(vii)

Let sec^-1(2/√3) = y . then secy=2/√3 = sec(Π/6) 

Implies the range of the principal value branch of  cosec^-1 is  [-0,Π]-{Π/2} 

and sec(Π/6)  = 2/√3 implies the principal value of sec^-1(2/√3)= Π/6

(viii)

Let cot^-1(√3) = y . then coty=√3 = cot(Π/6) 

Implies the range of the principal value branch of  cot^-1 is  (0,Π)

and cot(Π/6) = √3 implies the principal value of cot^-1(√3)= Π/6

(ix)

Let cos^-1(-1/√2) = y . then cosy=-1/√2 = -cos(Π/4) =cos(Π-Π/4) =cos(3Π/4)

Implies the range of the principal value branch of  cos^-1 is [0,Π] 

and cos(3Π/4) = -1/√2 implies the principal value of cos^-1(-1/√2)= 3Π/4

(x) 

Let cosec^-1(-√2) = y . then cosecy=-√2 = -cosec(Π/4) =cosec(-Π/4) 

Implies the range of the principal value branch of  cosec^-1 is  [-Π/2,Π/2]-{0} 

and cosec(-Π/4) = -√2 implies the principal value of cosec^-1(-√2)= -Π/4