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J5.Find the general solution of the equationcos x + cos 2x + cos 3x = 0. |
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Answer» Cosx + cos2x + cos3x = 0 like cosx + cos3x + cos2x = 0. 2cos(x + 3x)/2. cos(3x - x)/2 + cos2x = 0 [ from cosC + cosD = 2cos(C + D)/2.cos(C - D)/2] 2cos2x.cosx + cos2x = 0 cos2x(2cosx + 1) = 0 cos2x = 0, cosx = -1/2 when cos2x = 0 = cosπ/2 2x = 2nπ ± π/2 , where n is integers x = nπ ± π/4 if n = 0, x = ± π/4 , but so, x = π/4 if n = 1 , x = π ± π/4, x = 5π/4 , 3π/4 but x ≠ 5π/4hence, x = π/4 and 3π/4 from cos2x = 0 when cosx = -1/2 = cos2π/3x = 2nπ ± 2π/3forX in 0 to πonly x = 2π/3 is solution hence, there are three solutions .e.g., x = π/4, 3π/4 and 2π/3 |
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