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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)