Question

Find principal solutions of the following equations :
(i) `sinx=(-sqrt(3))/(2)`
(ii) `cos x=(-1)/(2)`
(iii) `cotx=-sqrt(3)`

Answer 1

(i) The given equation is sin `x=(-sqrt(3))/(2)`.
We know that `"sin"(pi)/(3)=(sqrt(3))/(2)`.
`thereforesin(pi+(pi)/(3))=-"sin"(pi)/(3)=(-sqrt(3))/(2), and (2pi-(pi)/(3))=-"sin"(pi)/(3)=(-sqrt(3))/(2)`.
`therefore"sin"(4pi)/(3)=(-sqrt(3))/(2)and "sin"(5pi)/(3)=(-sqrt(3))/(2)`.
Hence , the principal solutions are x `=(4pi)/(3) and x=(5pi)/(3)`.
(ii) The given equation is `cosx =(-1)/(2)`.
We know that cos `(pi)/(3)=(1)/(2)`.
`thereforecos(pi-(pi)/(3))=-"cos"(pi)/(3)=-(1)/(2) , and cos (pi+(pi)/(3))=- "cos"(pi)/(3)=-(1)/(2)`*
`therefore "cos" (2pi)/(3)=(1)/(2) and "cos"(4pi)/(3)=-(1)/(2)`.
Hence , the principal solutions are `x=(2pi)/(3) and x =(4x)/(3)`.
(iii) The given equation is cot `x=-sqrt(3)hArrtanx=-(1)/(sqrt(3))`.
We know that tan `(pi)/(6)=(1)/(sqrt(3))`.
`therefore tan(pi-(pi)/(6))=-"tan"(pi)/(6)=(-1)/(sqrt(3)), and tan (2pi-(pi)/(6))=-"tan"(pi)/(6)=(-1)/(sqrt(3))`.
`therefore "tan"(5pi)/(6)=(-1)/(sqrt(3))and "tan"(11pi)/(6)=(-1)/(sqrt(3))`.
Hence , the principal solutions are `x=(5pi)/(6)andx=(11pi)/(6)`.