Question

In each of the following , find the general value of x satisfying the equation :
(i) `sinx =(-sqrt(3))/(2)`
(ii) `cosx=(-1)/(2)`
(iii) `cotx=-sqrt(3)`

Answer 1

(i) `sinx=(-sqrt(3))/(2)=-"sin"(pi)/(3)= sin(pi+(pi)/(3))="sin"(4pi)/(3)`.
`thereforesinx " sin"(4pi)/(3)rArrx={npi+(-1)^(n)*(4pi)/(3))`, where `n in I`.
Hence , the general solution is x = `{npi+(-1)^(n)*(4pi)/(3))`, where `ninI`.
(ii) `cosx=(-1)/(2)=-"cos "(pi)/(3)=cos(pi-(pi)/(3))="cos "(2pi)/(3)`.
`therefore cos x="cos"(2pi)/(3)rArrx=(2npi+-(2pi)/(3))` , where `ninI`.
Hence , the general solution is x `=(2npi+-(2pi)/(3))` , where `nin I`.
(iii) `cotx=-sqrt(3)`
`rArrtanx =(-1)/(sqrt(3))=-"tan"(pi)/(6)="tan"(pi-(pi)/(6))="tan"(5pi)/(6)`
`rArr"tan " x = "tan " (5pi)/(6)`
`rArrx=(npi+(5pi)/(6))`, where `ninI`.