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The value of sin `18^(@)` isA. `(sqrt(5) +1)/(4)`B. `(sqrt(5)-1)/(2)`C. `(4)/(sqrt(5)+1)`D. `(4)/(sqrt(5)-1)` |
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Answer» Correct Answer - B Let, `theta = 18^(@)` , then `2 theta = 36^(@) = 90^(@) - 54^(@) = 90^(@) - 3 theta` Now, `"sin" 2 theta = "sin" (90^(@) - 3 theta)` `implies "sin" theta cos theta = cos theta` `implies 2 "sin" theta cos theta = 4 cos^(3) theta - 3 cos theta` `implies 2 "sin" theta cos theta = cos theta (4 cos^(2) theta - 3)` `implies 2 "sin" theta = 4 cos^(2) theta - 3 (As cos theta != 0)` `implies 2 "sin" theta = 4 - 4 "sin"^(2) theta - 3` `implies "sin"^(2) + theta + 2 "sin" theta - 1 = 0` `implies "sin" theta = (-2 +- sqrt(4) + 16)/(2(4)) = (- 1 +- sqrt(5))/(4)` But as `"sin" theta gt 0` we have `"sin" theta gt 0` we have `"sin" = (sqrt(5) - 1)/(4)` `"sin" 18^(@) = (sqrt(5) - 1)/(4)` |
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