![\[\left( \mathbf{ix} \right)~sin\text{ }2x\text{ }\text{ }sin\text{ }4x\text{ }+\text{ }sin\text{ }6x\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c193b9b72a5645ff4b50db775457ee5f_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[sin\text{ }2x\text{ }\text{ }sin\text{ }4x\text{ }+\text{ }sin\text{ }6x\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d1d09fe7f16237e99fe79e68cee1ba3d_l3.png "Rendered by QuickLaTeX.com")
using transformation formula,
![\[\text{ }sin\text{ }4x\text{ }+\text{ }sin\text{ }6x\text{ }+\text{ }sin\text{ }2x\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6522ac217b5124bfe69270f2feaf462a_l3.png "Rendered by QuickLaTeX.com")
By using the formula,
![\[sin\text{ }A\text{ }+\text{ }sin\text{ }B\text{ }=\text{ }2\text{ }sin\text{ }\left( A+B \right)/2\text{ }cos\text{ }\left( A-B \right)/2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-aa7f6758a867785e480ce64f7fd06caf_l3.png "Rendered by QuickLaTeX.com")
we get,
![\[\text{ }sin\text{ }4x\text{ }+\text{ }2\text{ }sin\text{ }\left( 6x+2x \right)/2\text{ }cos\text{ }\left( 6x-2x \right)/2\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c274af55c14a1397e8c8ed5702792ef1_l3.png "Rendered by QuickLaTeX.com")
![\[\text{ }sin\text{ }4x\text{ }+\text{ }2\text{ }sin\text{ }4x\text{ }cos\text{ }2x\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ddf440365d7f7c17a36ec2627f934939_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[Sin\text{ }4x\text{ }\left( 2\text{ }cos\text{ }2x\text{ }\text{ }1 \right)\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9d78c61a7e5761c20d81553d7e2f4946_l3.png "Rendered by QuickLaTeX.com")
So,
![\[Sin\text{ }4x\text{ }=\text{ }0\text{ }or\text{ }2\text{ }cos\text{ }2x\text{ }\text{ }1\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-90b880edc84f0bb5c6c4c00f2de4ab9c_l3.png "Rendered by QuickLaTeX.com")
![\[Sin\text{ }4x\text{ }=\text{ }sin\text{ }0\text{ }or\text{ }cos\text{ }2x\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-87553343d8872299feb0d4ab68375073_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[Sin\text{ }4x\text{ }=\text{ }sin\text{ }0\text{ }or\text{ }cos\text{ }2x\text{ }=\text{ }\pi /3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fc0976debe3ff556ea5efd4d7015038a_l3.png "Rendered by QuickLaTeX.com")
![\[4x\text{ }=\text{ }n\pi \text{ }or\text{ }2x\text{ }=\text{ }2m\pi \text{ }\pm \text{ }\pi /3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a5d8cc2eca2a08e11ef49adc3f42abd2_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[x\text{ }=\text{ }n\pi /4\text{ }or\text{ }x\text{ }=\text{ }m\pi \text{ }\pm \text{ }\pi /6\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9fac4525a404abacd1fed677c7ae7c38_l3.png "Rendered by QuickLaTeX.com")
∴ the general solution is
![\[x\text{ }=\text{ }n\pi /4\text{ }or\text{ }m\pi \text{ }\pm \text{ }\pi /6,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9a0468a095f756dbdae3a94c21038853_l3.png "Rendered by QuickLaTeX.com")
where m, n ϵ Z.
Or,
using transformation formula,
By using the formula,
we get,
Or,
So,
Or,
Or,
∴ the general solution is
where m, n ϵ Z.