Find the general solutions of the following equations :
(i) sinθ=12(ii) cosθ=−√31(iii) cosecθ=−√2(iv) secθ=√2(v) tanθ=−1√3(vi) √3 secθ=2
We have,
(i) sinθ=12⇒sinθ=12 [∵sinπ6=12]⇒the general solution isθ=nπ+(−1)nπ6;n∈z[∵if sinθ=sinα⇒θ=nπ+(−1)nα](ii) cosθ=−√31We have,⇒cosθ=cos(π+π6)⇒cosθ=cos7π6 [∵cos7π6=−√32]∵the general solution isθ=2nπ=±7π6,n∈z(iii) cosecθ=−√2cosecθ=−√2⇒1sinθ=−√2⇒sinθ=−1√2⇒singθ=sin(π+π4)⇒sinθ=sin5π4or sinθ=sin(−π4)∵sin(−theta)=−sin]theta.∴θ=nπ+(−1)n+1π4.n∈z(iv) secθ=√2We have,secθ=√2⇒1cosθ=√2⇒cosθ=1√2⇒cosθ=cosπ4⇒θ=2nπ±π4,n∈z(v) tanθ=−1√3We have,tanθ=−1√3⇒tanθ=−tan(π6)⇒tanθ=tan(−π6)[∵tan(−θ)=−tanθ]⇒θ=nπ+(−π6),n∈zor θ=nπ−π6,n∈z(vi) √3 secθ=2We have,√3secθ=2⇒1cos θ=2√3⇒cos θ=√32⇒cos θ=cos(π6)⇒θ=2nπ±π6,n∈z