InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
Find the principal value of each of the following: i) `cot^(-1)(-sqrt(3))` ii) `sec^(-1)(-sqrt(2))` iii) `"cosec"^(-1)(-1)`. |
|
Answer» i) We know that the range of the principal value of `cot^(-1)` is `(0,pi)`. Let `cot^(-)(-sqrt(3))=theta`. Then, `cottheta=-sqrt(3)=-cotpi/6=cot(pi-pi/6)=cot(5pi)/6`. `therefore theta=(5pi)/6 in (0, pi)`. Hence, the principal value of `cot^(-1)(-sqrt(3))` is `(5pi)/6`. ii) We know that the range of principal value of `sec^(-1)` is `[0,pi]-{pi/2}`. Let `sec^(-1)(-sqrt(2))=theta`. Then, `sec^(-1)(-sqrt(2))=theta`. Then, `sectheta=-sqrt(2)=-secpi/4 = sec(pi-pi/4)=sec(3pi)/4`. `therefore theta=(3pi)/4 in [0,pi]-{pi/2}`. Hence, the principal value of `sec^(-1)(-sqrt(2))` is `(3pi)/4`. iii) We know that the range of the princiapl value of `"cosec"^(-1)` is `[-pi/2,pi/2]-{0}`. Let `"cosec"^(-1)(-1)=theta`. Then, `"cosec"theta=-1`. `"cosec"theta=-1=-"cosec"pi/2="cosec"(-pi/2)`. `therefore theta=-pi/2 in [-pi/2, pi/2]-{0}`. Hence, the principal value of `"cosec"^(-1)(-1)` is `pi/2`. |
|