Given (cos(11pi/2)+i sin(11pi/2))/(cos(pi/4)+i sin(pi/4))=a+ i b, determine a, b?

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First, let's simplify the given expression by putting in the values of the trigonometric functions:

`cos((11pi)/2) = cos(5pi + pi/2) = 0`

`sin((11pi)/2) = sin(5pi+pi/2) = sin(3pi/2) = -1`

`cos(pi/4) = sqrt(2)/2`

`sin(pi/4) = sqrt(2)/2`

Now the given expression becomes

`(0 + i(-1))(sqrt(2)/2 + isqrt(2)/2 = -i(sqrt(2)/2 + isqrt(2)/2)`

Let's multiply this out, keeping in mind that `i^2 = -1`

`-i(sqrt(2)/2 + isqrt(2)/2) = -isqrt(2)/2 - sqrt(2)/2`

Rearranging the terms, we can write

`-sqrt(2)/2 -isqrt(2)/2 = a + ib`

**From here we see that**

**`a = -sqrt(2)/2` and `b = -sqrt(2)/2` .**

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