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Unizor - Trigonometry - Series - Problems 2

Notes to a video lecture on http://www.unizor.com Math4Teens-Trigonometry-Trigonometric Series - Problem 2 Many (maybe, even most) problems related to trigonometric series can be reduced to the problem considered in the introductory lecture about this topic. Here are a couple of examples of such problems. Problem A Calculate the value of a trigonometric series: sin(x)·sin(2x)+sin(2x)·sin(3x)+... ...+sin(Nx)·sin((N+1)x) Hint Use the formula for a product of two sin we have used before and derived easily from a formula for a cosine of a sum of two angles: 2·sin(α)·sin(β) = cos(α−β)−cos(α+β) Answer: N·cos(x)/2−sin(Nx)·cos((N+2)x)/ /[2sin(x)] Problem B Calculate the value of a trigonometric series: sin³(x)+sin³(x+d)+... ...+sin³(x+(N−1)d) Hint You may express sin³(n·x) in terms of sin(3n·x) and sin(n·x) after which the problem is reduced to summation of trigonometric series we already considered before. Answer: 3·sin[x+(N−1)d/2]· ·sin(Nd/2)/[4·sin(d/2)] − − sin[3x+3(N−1)d/2]· ·sin(3Nd/2)/[4sin(3d/2)]