Value of sin[1/4arc sin((√63)/8)

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Solution

Let arcsin[√(63)/8] = x
=> sinx = √(63)/8
=> cos^2 x = √(1 - sin^2 x)
= √(1 - 63/64)
=> cosx = 1/8

2cos^2 (x/2) = 1 + cosx
= 1 + 1/8 = 9/8
=> cos^2 (x/2) = 9/16
=> cos(x/2) = 3/4
=> 2cos^2 (x/4)
= 1 + cos(x/2) = 1 + 3/4 = 7/4
=> cos^2 (x/4) = 7/8
=> sin^2 (x/4) = 1 - 7/8 = 1/8
=> sin(x/4) = 1/(2√2)
=> sin [(1/4) arcsin[√(63)/8]
= 1/(2√2).

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