Final answer:
To find the slope of the tangent line to the polar curve r = 8/θ at θ = π, we can convert the polar equation to rectangular coordinates, differentiate the equations, and substitute the value of θ. The slope of the tangent line at θ = π is 0.
Step-by-step explanation:
To find the slope of the tangent line to a polar curve, we need to find the derivative of the polar equation with respect to θ, and then substitute the value of θ at the specified point. In this case, the polar equation is r = 8/θ, and we need to find the slope at θ = π.
To find the derivative, we can convert the polar equation to rectangular coordinates using the formulas x = r*cos(θ) and y = r*sin(θ). Differentiating with respect to θ, we get dx/dθ = -8*sin(θ)/θ^2 and dy/dθ = 8*cos(θ)/θ^2.
Next, we can find the slope at θ = π by substituting θ = π into the derivative equations. The slope of the tangent line to the polar curve at θ = π is dy/dx = (dy/dθ)/(dx/dθ) = (8*cos(π)/π^2) / (-8*sin(π)/π^2) = -cot(π) = 0.