Question
sin2 7Ëš + sin2 9Ëš +
sin2 11Ëš + sin2 12Ëš + ……… + sin2 83Ëš = ?Solution
sin 2 7Ëš + sin 2 9Ëš + sin 2 11Ëš + sin 2 12Ëš + ……… + sin 2 83Ëš = (sin2 7Ëš + sin2 83Ëš)+(sin2 9Ëš + sin2 81Ëš) + .…. + (sin2 43Ëš + sin2 47Ëš) + sin2 45Ëš = (sin2 7Ëš + cos2 7Ëš) + (sin2 9Ëš + cos2 9Ëš) + .…. + (sin2 43Ëš + cos2 43Ëš) + sin2 45Ëš {As sin (90Ëš - θ) = cos θ} = (1)+(1) + .…. upto 19 times + sin2 45Ëš { As sin 2 θ + cos 2 θ= 1} = 19 + sin2 45Ëš {As there are 19 odd numbers from 7 to 43}. = 19 + (1/√2)2 {As sin 45Ëš =1/√2 } = 19 + 0.5 = 19.5 ` ` ` ` ` ` ` ` ` ` ` `
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