Find the points of horizontal tangency to the polar curve.r = 2 csc(𝜃) + 3      0 ≤ 𝜃 < 2𝜋

To find the points of horizontal tangency to the polar curve r = 2 csc(𝜃) + 3, we need to find the values of 𝜃 for which the derivative of r with respect to 𝜃 is zero.

Step 1: Express r in terms of sin(𝜃) r = 2/sin(𝜃) + 3

Step 2: Differentiate r with respect to 𝜃 dr/d𝜃 = -2cos(𝜃)/sin^2(𝜃)

Step 3: Set the derivative equal to zero and solve for 𝜃 -2cos(𝜃)/sin^2(𝜃) = 0 This implies that cos(𝜃) = 0

The solutions to cos(𝜃) = 0 in the interval 0 ≤ 𝜃 < 2𝜋 are 𝜃 = 𝜋/2 and 𝜃 = 3𝜋/2.

Step 4: Substitute these values back into the original equation to find the corresponding r values r = 2 csc(𝜋/2) + 3 = 2 + 3 = 5 r = 2 csc(3𝜋/2) + 3 = -2 + 3 = 1

So, the points of horizontal tangency to the polar curve are (5, 𝜋/2) and (1, 3𝜋/2).