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How transform sinx+cosx in product?

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greenbel | Honors

Posted July 25, 2013 at 2:44 PM via web

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How transform sinx+cosx in product?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 25, 2013 at 3:12 PM (Answer #1)

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You need to convert the summation in a product, and since only a summation of like trigonometric functions can be converted into a product, either, you convert `sin x` into `cos (pi/2 - x)` , or `cos x` in `sin(pi/2 - x)` , such that:

`sin x + cos x = sin x + sin(pi/2 - x)`

You should use the following identity, such that:

`sin a + sin b = 2 sin(a/2 + b/2)cos(a/2 - b/2)`

Reasoning by analogy, yields:

`sin x + sin(pi/2 - x) = 2 sin(x/2 + pi/4 - x/2)cos(x/2 - pi/4 + x/2)`

`sin x + sin(pi/2 - x) = 2 sin(pi/4)cos(x - pi/4)`

Replacing `sqrt2/2` for `sin(pi/4)` yields:

`sin x + sin(pi/2 - x) = 2*(sqrt2/2)*cos(x - pi/4)`

Reducing duplicate factors, yields:

`sin x + sin(pi/2 - x) = sqrt2*cos(x - pi/4)`

 

Hence, converting the summation into a product, yields `sin x + cos x = sqrt2*cos(x - pi/4).`

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