Calculus II

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First we find polar slope by using the following formula

`tan mu=r/(r')`

where `mu` is angle between tangent line and polar radius.

`tan mu=(3+cos3theta)/(-3sin3theta)`  

at point `(3pi)/2` that is equal to

`tan mu=(3+cos((9pi)/2))/(-3sin((9pi)/2))=3/(-3)=-1`

`mu=arctan -1=-pi/4`  

Now to calculate the slope of tangent line we use the following formula

`k=tan(theta+mu)`

So our slope `k` is

`k=tan((3pi)/2-pi/4)=tan((5pi)/4)=1`  ` `

Slope of the tangent line is 1.

In the image below blue is your curve and red is the tangent line at `theta=(3pi)/2.`

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