Mumbai University > First Year Engineering > Sem 1 > Applied Maths 1

Marks : 6 M

Year : Dec 2014

Let, $x=cos⁡θ+i sin⁡θ \; \; \; \therefore \dfrac{1}{x}=cos⁡θ-i sin⁡θ \\ \; \\ \; \\ \therefore x^m=(cos⁡θ+i sin⁡θ )^m \;=\; cos⁡mθ+i sin⁡mθ \; \; \; \; \ldots By \; De \; Moivre's \; Theorem \\ \; \\ \; \\ Also,\dfrac{1}{x^m} =(cos⁡θ-i sin⁡θ )^m \;=\;cos⁡mθ-i sin⁡mθ\; \; \; \; \ldots By \; De \; Moivre's \; Theorem \\ \; \\ \; \\ \therefore x+\dfrac{1}{x}=2 cos⁡θ \; \; and \; \; x^m+ \dfrac{1}{x^m} \;=\; 2 cos⁡mθ \\ \; \\ \; \\ \therefore x+\dfrac{1}{x} \;=\; 2 cos⁡θ \; \; and \; \; x^m+ \dfrac{1}{x^m} =2 cos⁡mθ \\ \; \\ \; \\ \therefore \bigg( x+\dfrac{1}{x} \bigg)^7 \;=\; 2^7 cos^7 θ \\ \; \\ \; \\ \therefore x^7+7x^6 \dfrac{1}{x}+21x^5 \dfrac{1}{x^2} +35x^4 \dfrac{1}{x^3} +35x^3 \dfrac{1}{x^4} +21x^2 \dfrac{1}{x^5} +7x \dfrac{1}{x^6} +\dfrac{1}{x^7} =2^7 cos^7 θ \\ \; \\ \; \\ \therefore (x^7+\dfrac{1}{x^7 })+7(x^5+\dfrac{1}{x^5 } )+21(x^3+\dfrac{1}{x^3 } )+35(x+\dfrac{1}{x })=128cos^7 θ \\ \; \\ \; \\ \therefore (2 cos⁡7θ )+7(2 cos⁡5θ )+21(2 cos⁡3θ )+35(2 cos⁡θ )=128cos^7 θ \\ \; \\ \; \\ \; \\ \therefore cos^7 θ=\dfrac{1}{64}[cos⁡7θ+7 cos⁡5θ+21 cos⁡3θ+cos⁡θ] $