Math, asked by shona5723, 8 months ago

If cos alpha + cos theta + kam Kama is equal to zero is equal to sin alpha + beta + sin Gama train show that cos 3 alpha + cos 3 theta + cos theta is equal to 3 cos alpha + beta + Gamma

Answers

Answered by MaheswariS
1

\underline{\textbf{Given:}}

\mathsf{cos\alpha+cos\beta+cos\gamma=sin\alpha+sin\beta+sin\gamma=0}

\underline{\textbf{To prove:}}

\mathsf{cos3\alpha+cos3\beta+cos3\gamma=3\,cos(\alpha+\beta+\gamma)}

\underline{\textbf{Solution:}}

\underline{\textbf{Concept used:}}

\boxed{\begin{minipage}{7cm}$\mathsf{If\;a+b+c=0,\;then\;a^3+b^3+c^3=3\,abc}$\end{minipage}}

\mathsf{Consider,}

\mathsf{cos\alpha+cos\beta+cos\gamma=0}..........(1)

\mathsf{sin\alpha+sin\beta+sin\gamma=0}........(2)

\mathsf{(1)+i(2)\implies}

\mathsf{(cos\alpha+cos\beta+cos\gamma)+i(sin\alpha+sin\beta+sin\gamma)=0+i0}

\mathsf{(cos\alpha+i\,sin\alpha)+(cos\beta+i\,sin\beta)+(cos\gamma+i\,sin\gamma)=0+i0}

\implies\mathsf{x+y+z=0}

\mathsf{Then}

\mathsf{x^3+y^3+z^3=3\,xyz}

\mathsf{(cos\alpha+i\,sin\alpha)^3+(cos\beta+i\,sin\beta)^3+(cos\gamma+i\,sin\gamma)^3=3(cos\alpha+i\,sin\alpha)(cos\beta+i\,sin\beta)(cos\gamma+i\,sin\gamma)}

\mathsf{Using\;demovire's\;theorem}\boxed{\bf\;(cos\theta+i\,sin\theta)^n=cos\,n\theta+i\,sin\,n\theta}

\mathsf{(cos3\alpha+i\,sin3\alpha)+(cos3\beta+i\,sin3\beta)+(cos3\gamma+i\,sin3\gamma)^3=3\,e^{i\alpha}e^{i\beta}e^{i\gamma}}

\mathsf{(cos3\alpha+cos3\beta+cos3\gamma)+i(sin3\alpha+sin3\beta+sin3\gamma)=3\,e^{i(\alpha+\beta+\gamma)}}

\mathsf{(cos3\alpha+cos3\beta+cos3\gamma)+i(sin3\alpha+sin3\beta+sin3\gamma)=3\,[cos(\alpha+\beta+\gamma)+i\,sin(\alpha+\beta+\gamma)]}

\textsf{Equating corresponding real parts on bothsides, we get}

\boxed{\mathsf{cos3\alpha+cos3\beta+cos3\gamma=3\,cos(\alpha+\beta+\gamma)}}

\underline{\textbf{Find more:}}

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