The general solution of any trigonometric equation is given as:
$$ \[sin\text{ }x\text{ }=\text{ }sin\text{ }y\]
Or,
\[x\text{ }=\text{ }n\pi \text{ }+\text{ }{{\left( \text{ }1 \right)}^{n~}}y\] , where n ∈ Z.
\[cos\text{ }x\text{ }=\text{ }cos\text{ }y,\]
or,
\[x\text{ }=\text{ }2n\pi ~\pm ~y,\] where n ∈ Z.
\[tan\text{ }x\text{ }=\text{ }tan\text{ }y,\]
or,
\[x\text{ }=\text{ }n\pi ~+\text{ }y,\] where n ∈ Z.
\[\left( \mathbf{i} \right)~sin\text{ }x\text{ }=\text{ }1/2~~~~~~\]
As, \[sin\text{ }{{30}^{o}}~=\text{ }sin\text{ }\pi /6\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]
Hence,
\[Sin\text{ }x\text{ }=\text{ }sin\text{ }\pi /6\]
∴ the general solution is
\[x\text{ }=\text{ }n\pi \text{ }+\text{ }{{\left( \text{ }1 \right)}^{~n~}}\pi /6,\]
where n ∈ Z.
[as,
\[sin\text{ }x\text{ }=\text{ }sin\text{ }A\]
\[=>\text{ }x\text{ }=\text{ }n\pi \text{ }+\text{ }{{\left( \text{ }1 \right)}^{~n}}~A]\]
\[\left( \mathbf{ii} \right)~cos\text{ }x\text{ }=\text{ }\text{ }\surd 3/2\]
as, \[cos\text{ }{{150}^{o}}~=\text{ }\left( -\text{ }\surd 3/2 \right)\text{ }=\text{ }cos\text{ }5\pi /6\]
Hence,
\[Cos\text{ }x\text{ }=\text{ }cos\text{ }5\pi /6\]
∴ the general solution is
\[x\text{ }=\text{ }2n\pi ~\pm \text{ }5\pi /6\] , where n ϵ Z.