Find the sum of 1 - - cos- - - cos 2- - cos3 - - - - cosn-1-

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Properties of Conjugate of a Complex Number

Find the sum ...

Question

Find the sum of $1 + | cos θ | + {cos}^{2} θ + | {cos}^{3} θ | + . . . . c o s (n - 1) θ$

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Solution

$\begin{matrix} M = cos θ + cos 2 θ + cos 3 θ + . . . . . . . . . . + cos (n - 1) θ M = \frac{2 sin \frac{θ}{2}}{2 sin \frac{θ}{2}} (cos θ + cos 2 θ + . . . . . . . . . .) = \frac{(sin \frac{3 θ}{2} - sin \frac{θ}{2}) + (sin \frac{5 θ}{2} - sin \frac{3 θ}{2}) + . . . . . . . .}{2 sin \frac{θ}{2}} M = \frac{- sin \frac{θ}{2} + sin (\frac{n + 1}{2}) θ}{2 sin \frac{θ}{2}} ∴ M + 1 = 1 + \frac{- sin \frac{θ}{2} + sin (\frac{n + 1}{2}) θ}{2 sin \frac{θ}{2}} M + 1 = \frac{sin \frac{θ}{2} + sin (\frac{n + 1}{2}) θ}{2 sin \frac{θ}{2}} \end{matrix}$

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Find the sum of 1+|cosθ|+cos2θ+|cos3θ|+....cos(n−1)θ

M=cosθ+cos2θ+cos3θ+..........+cos(n−1)θM=2sinθ22sinθ2(cosθ+cos2θ+..........)=(sin3θ2−sinθ2)+(sin5θ2−sin3θ2)+........2sinθ2M=−sinθ2+sin(n+12)θ2sinθ2∴M+1=1+−sinθ2+sin(n+12)θ2sinθ2M+1=sinθ2+sin(n+12)θ2sinθ2

Find the sum of $1 + | cos θ | + {cos}^{2} θ + | {cos}^{3} θ | + . . . . c o s (n - 1) θ$