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Last modified: January 26, 2018

Polar Coordinates
-----------------

                         P
                  ^ y   (x,y)           P = (r,θ) 
                  |    /                  ≡ (r,θ + 2nπ)
                  |   /                 x = rcosθ
                  |  /r                 y = rsinθ
                  | /
                  |/θ
    --------------+------------------>x
                 /|  
                / |
               /  |                    
              /   |
             Q    |   Q = (r,θ + π) 
          (-x,-y)       ≡ (-r,θ)
                        ≡ (-r,θ + (2n + 1)π)

Slope of Tangent Line at θ
---------------------------

        dy/dθ
dy/dx = -----  ... CHAIN RULE
        dx/dθ


Horizontal tangent line:  dy/dθ = 0

Vertical tangent line:  dx/dθ = 0

Example:

r = 2sinθ

x = rcosθ = 2sinθcosθ

y = rsinθ = 2sin2θ

dx/dθ = 2cos2θ - 2sin2θ = 0

∴ θ = π/4, 3π/4, 5π/4, 7π/4

dy/dθ = 4sinθcosθ = 0

∴ θ = 0, π/2, π, 3π/2

Not all of these are θs lines correspond to tangent 
lines.  To determine the valid ones it is necessary 
to look at the polar plot.



We can now see that:

π/4 and 3π/4 for vertical tangent lines.

0 and π/2 for horizontal tangent lines.

The r coordinate is found by solving the original polar
equation.  Therefore,

r = 2sin(π/4) = √3

r = 2sin(3π/4) = √3

r = 2sin(0) = 0

r = 2sin(π/2) = 2

Therefore, the polar coordinates of the tangent lines 
are:

(√3,π/4) (√3,3π/4) (0,0) (2,π/2)

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