Calculus II

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You need to evaluate the points of intersection between the curves `r = 1 + sin(theta/2)` and `r = 1 + cos(theta/2)` , hence, you need to solve the following system of equations, such that:

`{(r = 1 + sin(theta/2)),(r = 1 + cos(theta/2)):}`

`1 + sin(theta/2) = 1 + cos(theta/2) => sin(theta/2) = cos(theta/2)`

`sin(theta/2) - cos(theta/2) = 0`

You need to divide by `cos(theta/2)` such that:

`(sin(theta/2))/(cos(theta/2)) - 1 = 0`

Using the trigonometric identity `(sin(theta/2))/(cos(theta/2)) = tan(theta/2)` yields:

`tan(theta/2) - 1 = 0 => tan(theta/2) = 1 => theta/2 = tan^(-1) 1 + npi`

`theta/2 = pi/4 + npi`

`theta = pi/2 + 2npi => r = 1 + sin(pi/4) => r = 1 + sqrt2/2`

Hence, evaluating the point of intersection between the given curves, expressed in polar coordinates, yields `(r, theta) = (1 + sqrt2/2,pi/2).`

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