Final answer:
To find the points of horizontal tangency to the polar curve r = 3 csc(5θ), differentiate the equation and set the derivative equal to zero. Solve for θ to find the values where the tangent line is horizontal.
Step-by-step explanation:
To find the points of horizontal tangency to the polar curve r = 3 csc(5θ), we need to first differentiate the equation with respect to θ to find the slope of the curve. The slope will be zero when the tangent line is horizontal.
Step 1: Differentiate the equation.
Given: r = 3 csc(5θ)
Differentiating both sides with respect to θ:
dr/dθ = 3d(csc(5θ))/dθ
Using the chain rule: dr/dθ = 3(−5csc(5θ)cot(5θ))
Step 2: Set the derivative equal to zero and solve for θ.
3(−5csc(5θ)cot(5θ)) = 0
Simplifying:
−5csc(5θ)cot(5θ) = 0
Step 3: Find the values of θ that satisfy the equation.
Since we know that cot(5θ) is equal to zero when θ is of the form nπ/5, where n is an integer, we can set θ = nπ/5.
Therefore, the points of horizontal tangency to the polar curve r = 3 csc(5θ) occur at θ = nπ/5 for n = 0, 1, 2, 3, 4.