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

Marks : 6 M

Year : Dec 2012

Let, $ x\;=\; cos\theta+isin\theta \; \; \; \; \dfrac{1}{x} \;=\; cos\theta-isin\theta \\ \; \\ \; \\ \therefore x^n \;=\; (cos\theta+isin\theta)^n \;=\; cosn\theta+isinn\theta \\ \; \\ \; \\ Also \; \dfrac{1}{x^n} \;=\; (cos\theta-isin\theta)^n \;=\; cosn\theta-isinn\theta \\ \{ \because DeMoivre's \; Theorem \} \\ \; \\ \; \\ x^n + \dfrac{1}{x^n} \;=\; 2cosn\theta \; \; \; \; \; \ldots (i) \\ \; \\ \; \\ cos^7\theta \;=\; (cos\theta)^7 \;=\; \Bigg[ \dfrac{1}{2} \bigg( x + \dfrac{1}{x} \bigg) \Bigg]^7 \;=\; \dfrac{1}{2^7} \bigg( x + \dfrac{1}{x} \bigg)^7 \; \; \; \; from \; (i) \\ \; \\ \; \\ (cos\theta)^7 \;=\; \\ \dfrac{1}{2^7} \Bigg[ 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} \Bigg] \\ \ldots Expanding \; binomially \\ \; \\ \; \\ = \dfrac{1}{128} \Bigg[ x^7 + 7\bigg( x^5 + \dfrac{1}{x^5} \bigg) + 21\bigg( x^3 + \dfrac{1}{x^3} \bigg) + 35 \bigg( x + \dfrac{1}{x} \bigg) + \dfrac{1}{x^7} \Bigg] \\ \; \\ \; \\ = \dfrac{1}{128} [ 2cos7\theta + 7(2cos5\theta) + 21(2cos3\theta) + 35(2cos\theta) ] \; \; \; from \; (i) \\ \; \\ \; \\ \therefore cos^7\theta \;=\; \dfrac{1}{64} [ cos7\theta + 7cos5\theta + 21 cos3\theta + 35 cos\theta ] $