# What is the equation of the line that is normal to the polar curve #f(theta)=-5theta- sin((3theta)/2-pi/3)+tan((theta)/2-pi/3) # at #theta = pi#?

The line is

This behemoth of an equation is derived through a somewhat lengthy process. I will first outline the steps by which the derivation will proceed and then perform those steps.

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To find the equation of the line that is normal to the polar curve f(θ) = -5θ - sin((3θ)/2 - π/3) + tan((θ)/2 - π/3) at θ = π, first, find the slope of the tangent line at θ = π by evaluating the derivative of the function f(θ) with respect to θ and then substitute θ = π into the derivative to find the slope. Then, find the negative reciprocal of this slope to get the slope of the normal line. Finally, use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency, to write the equation of the normal line.

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