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calculate sinpie/16cospie/8cospie/4cospie/16

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mouhham | Student, Undergraduate | (Level 1) eNoter

Posted August 21, 2012 at 4:01 PM via web

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calculate sinpie/16cospie/8cospie/4cospie/16

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted August 21, 2012 at 4:15 PM (Answer #1)

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You should regroup the factors such that:

`(((sin(pi/16)*cos(pi/16))*cos(pi/8))*cos(pi/4))`

Notice that if you multiply `sin(pi/16)*cos(pi/16)`  by `2`  yields:

`2sin(pi/16)*cos(pi/16) = sin(2*pi/16) => sin(pi/16)*cos(pi/16) = (sin(pi/8))/2`

You need to substitute `(sin(pi/8))/2`  for `sin(pi/16)*cos(pi/16)`  such that:

`(sin(pi/8))/2*cos(pi/8)*cos(pi/4)`

Notice that if you multiply `sin(pi/8)*cos(pi/8)`  by `2`  yields:

`2sin(pi/8)*cos(pi/8) = sin(2*pi/8) =>sin(pi/8)*cos(pi/8) = (sin(pi/4))/2`

Hence `(sin(pi/8)*cos(pi/8))/2 = (sin(pi/4))/4`

You need to substitute `(sin(pi/4))/4`  for `((sin(pi/16)*cos(pi/16))*cos(pi/8))`  such that:

`(sin(pi/4))/4 * cos(pi/4)`

Notice that if you multiply `sin(pi/4)*cos(pi/4)`  by `2 `  yields:

`2sin(pi/4)*cos(pi/4) = sin(2*pi/4) =>sin(pi/4)*cos(pi/4) = (sin(pi/2))/2`

You need to remember that `sin(pi/2) = 1` , such that:

`sin(pi/4)*cos(pi/4) = 1/2 => (sin(pi/4)*cos(pi/4))/4 = 1/(4*2)`

`(sin(pi/4)*cos(pi/4))/4 = 1/8`

Hence, evaluating the given trigonometric product yields `(((sin(pi/16)*cos(pi/16))*cos(pi/8))*cos(pi/4)) = 1/8` .

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