Respuesta :
The equation in x and y for the line tangent to the curve is x - √3y - √3 = 0
A given curve can be defined by both its cartesian coordinates that is in x-y form or in polar coordinates that is in terms of r and θ there exists a relationship between two sets of coordinates
x = r cosθ
y = r sinθ
The tangent to any curve f(x) is defined by f'(x) . In order to find tangent to a given terms of r and θ , we can calculate from the above to expressions f'(x)
f'(x) = dy/dx = dx/dθ ÷ dy/dθ
It is given in the question that
r = 2 sinθ
Replacing the value of r to the standard equation,
x = (2 sinθ )cosθ = sin2θ
At θ = π/3 , x = √3 / 2
y = (2 sinθ)sinθ = 2sin²θ
At θ = π/3 , y = 3/2
Differentiating both w.r.t θ
=> dx / dθ = 2cosθ
=> dy/dθ = 4sinθcosθ
So, dy/dx = 2cosθ / 4sinθcosθ
=> 1 / 2sinθ
Therefore , slope of tangent : m = 1/2sinθ ( θ = π/3)
=> m = 1 / 2sin(π/3)
=> m = 2 / 2√3
=> m = 1 /√3
Hence , the equation of line is
( y - 3/2 ) = 1 /√3 (x - √3 / 2)
=> √3y - 3√3/2 = x - √3/2
=> x - √3y - √3 = 0
To know more about polar coordinates here
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