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Show the number is natural  sin (a sin 1/2)+sin (a cos `sqrt(3)/2)`

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greenbel | (Level 2) Honors

Posted June 26, 2013 at 5:15 PM via web

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Show the number is natural 

sin (a sin 1/2)+sin (a cos `sqrt(3)/2)`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 26, 2013 at 5:26 PM (Answer #1)

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You need to use the following trigonometric identities, such that:

`sin(sin^(-1) x) = x`

`sin(cos^(-1) x) = sqrt(1 - x^2)`

Reasoning by analogy, yields:

`sin(sin^(-1) 1/2) = 1/2`

`sin(cos^(-1) sqrt3/2) = sqrt(1 - 3/4) => sin(cos^(-1) sqrt3/2) = 1/2`

Replacing `1/2` for `sin(sin^(-1) 1/2)` and `1/2` for `sin(cos^(-1) sqrt3/2)` yields:

`sin(sin^(-1) 1/2) + sin(cos^(-1) sqrt3/2) = 1/2 + 1/2 = 1`

Hence, testing if the given summation represents a natural number yields that `sin(sin^(-1) 1/2) + sin(cos^(-1) sqrt3/2) = 1 in N.`

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