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.
\[\left( \mathbf{i} \right)~sin\text{ }2x\text{ }=\text{ }\surd 3/2\]
or,
\[sin\text{ }2x\text{ }=\text{ }\surd 3/2\]
\[=\text{ }sin\text{ }\left( \pi /3 \right)\]
∴ the general solution is
\[2x\text{ }=\text{ }n\pi \text{ }+\text{ }{{\left( -1 \right)}^{n}}~\pi /3,\]
where n ϵ Z.
\[x\text{ }=\text{ }n\pi /2\text{ }+\text{ }{{\left( -1 \right)}^{n}}~\pi /6,\]
where n ϵ Z.
\[\left( \mathbf{ii} \right)~cos\text{ }3x\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]
or,
\[cos\text{ }3x\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]
\[=\text{ }cos\text{ }\left( \pi /3 \right)\]
∴ the general solution is
\[3x\text{ }=\text{ }2n\pi \text{ }\pm \text{ }\pi /3,\]
where n ϵ Z.
\[x\text{ }=\text{ }2n\pi /3\text{ }\pm \text{ }\pi /9,\]
where n ϵ Z.