Find the points of horizontal tangency to the polar curve.r = a sin(𝜃)      0 ≤ 𝜃 < 𝜋, a > 0

To find the points of horizontal tangency to the polar curve r = a sin(θ), we need to find the points where the derivative of the function is equal to zero.

Step 1: Convert the polar equation to Cartesian coordinates. In polar coordinates, x = r cos(θ) and y = r sin(θ). Substituting r = a sin(θ) into these equations gives x = a sin(θ) cos(θ) and y = a sin^2(θ).

Step 2: Differentiate y with respect to x. Using the chain rule, dy/dx = (dy/dθ) / (dx/dθ). Differentiating y = a sin^2(θ) with respect to θ gives dy/dθ = 2a sin(θ) cos(θ). Differentiating x = a sin(θ) cos(θ) with respect to θ gives dx/dθ = a cos^2(θ) - a sin^2(θ).

Step 3: Set dy/dx equal to zero and solve for θ. Setting dy/dx = 0 gives (2a sin(θ) cos(θ)) / (a cos^2(θ) - a sin^2(θ)) = 0. This simplifies to 2 sin(θ) cos(θ) = 0, which has solutions θ = 0, π/2, π.

Step 4: Substitute these values of θ back into the original polar equation to find the corresponding values of r. Substituting θ = 0 into r = a sin(θ) gives r = 0. Substituting θ = π/2 gives r = a. Substituting θ = π gives r = 0.

So the points of horizontal tangency to the polar curve r = a sin(θ) are (0, 0), (a, π/2), and (0, π).

To find the points of horizontal tangency to the polar curve r = a sin(θ), we need to find the points where the derivative of the function is equal to zero.

Step 1: Convert the polar equation to Cartesian coordinates. In polar coordinates, x = r cos(θ) and y = r sin(θ). Substituting r = a sin(θ) into these equations gives x = a sin(θ) cos(θ) and y = a sin^2(θ).

Step 2: Differentiate y with respect to x. Using the chain rule, dy/dx = (dy/dθ) / (dx/dθ). Differentiating y = a sin^2(θ) with respect to θ gives dy/dθ = 2a sin(θ) cos(θ). Differentiating x = a sin(θ) cos(θ) with respect to θ gives dx/dθ = a cos^2(θ) - a sin^2(θ).

Step 3: Set dy/dx equal to zero and solve for θ. Setting dy/dx = 0 gives 2a sin(θ) cos(θ) / (a cos^2(θ) - a sin^2(θ)) = 0. This simplifies to 2 sin(θ) cos(θ) / (cos^2(θ) - sin^2(θ)) = 0. This equation is satisfied when sin(θ) = 0, which gives θ = 0 and θ = π.

Step 4: Substitute θ = 0 and θ = π back into the original polar equation to find the corresponding r values. Substituting θ = 0 into r = a sin(θ) gives r = a sin(0) = 0. Substituting θ = π into r = a sin(θ) gives r = a sin(π) = 0.

So, the points of horizontal tangency to the polar curve r = a sin(θ) are (0, 0) and (π, 0).