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

Marks : 6 M

Year : May 2013

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} \;=\; 2isinn\theta \; \; \; \; \ldots (i) \\ \; \\ \; \\ sin^7\theta \;=\; (sin\theta)^7 \;=\; \Bigg[ \dfrac{1}{2i} \bigg( x - \dfrac{1}{x} \bigg) \Bigg]^7 \; \; \; \; from \; (i) \;=\; \dfrac{1}{128 \times i^7} \bigg( x - \dfrac{1}{x} \bigg)^7 \\ \; \\ \; \\ \;=\; \dfrac{1}{128 i^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 i^7} \Bigg[ \bigg(x^7 - \dfrac{1}{x^7} \bigg) - 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) \Bigg] \\ \; \\ \; \\ = \dfrac{1}{128 \times i^7 } [ 2isin7\theta - 7(2isin5\theta) + 21(2isin3\theta) - 35(2isin\theta) ] \; \; \; from \; (i) \\ \; \\ \; \\ \;=\; \dfrac{1}{64 \times i^6} [ sin7\theta - 7sin5\theta + 21 sin3\theta - 35 sin\theta ] \\ \; \\ \; \\ \therefore sin^7\theta \;=\; \dfrac{-1}{64} [ sin7\theta - 7sin5\theta + 21 sin3\theta - 35 sin\theta ] $