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How calculate cos pie/5cos pie/20 - sin pie/5sin pie/20? i dont know  sin...

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ruals | Salutatorian

Posted July 7, 2013 at 3:56 PM via web

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How calculate cos pie/5cos pie/20 - sin pie/5sin pie/20? i dont know  sin pie/5,sinpie/20...

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 7, 2013 at 4:15 PM (Answer #1)

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You do not need to know the values of the functions `sin(pi/5), cos(pi/5), sin(pi/20) or cos(pi,20)` , but you should use the following trigonometric identity, such that:

`cos a*cos b - sin a*sin b = cos(a + b)`

Reasoning by analogy yields:

`cos(pi/5)cos(pi/20) - sin(pi/5)sin(pi/20) = cos(pi/5 + pi/20)`

`cos(pi/5)cos(pi/20) - sin(pi/5)sin(pi/20) = cos((4pi + pi)/20)`

`cos(pi/5)cos(pi/20) - sin(pi/5)sin(pi/20) = cos((5pi)/20)`

Reducing duplicate factors yields:

`cos(pi/5)cos(pi/20) - sin(pi/5)sin(pi/20) = cos (pi/4)`

`cos(pi/5)cos(pi/20) - sin(pi/5)sin(pi/20) = sqrt2/2`

Hence, evaluating the given difference, using the indicated trigonometric identity, yields `cos(pi/5)cos(pi/20) - sin(pi/5)sin(pi/20) = sqrt2/2.`

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