The value of sin `18^(@)` is

Options

A. `(sqrt(5) +1)/(4)`
B. `(sqrt(5)-1)/(2)`
C. `(4)/(sqrt(5)+1)`
D. `(4)/(sqrt(5)-1)`

Answer

B

Explanation

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)`