Final answer:
To find the points at which the polar curve has a horizontal or vertical tangent line, we need to set the derivative of r with respect to θ equal to zero. For horizontal tangent lines, set the derivative equal to zero and solve for θ. For vertical tangent lines, set the derivative equal to infinity. The corresponding points are (0, 0), (10, 0), (-5, -5), and (-5, 5) respectively.
Step-by-step explanation:
The given polar curve is r = 5 - 5sin(θ). To find the points at which the curve has a horizontal or a vertical tangent line, we need to find the values of θ that make the derivative of r with respect to θ equal to zero.
To find the horizontal tangent lines, we set the derivative equal to zero and solve for θ. Differentiating r = 5 - 5sin(θ) with respect to θ gives dr/dθ = -5cos(θ). Setting dr/dθ = 0, we get -5cos(θ) = 0, which gives θ = ½π or θ = &frac32;π. Substituting these values of θ into the equation for r, we find that the corresponding points are (0, 0) and (10, 0), respectively.
To find the vertical tangent lines, we set the derivative equal to infinity. We know that the slope of a tangent line is given by dy/dx = (dy/dθ) / (dx/dθ). For our given polar curve, dy/dθ = r sin(θ) + r' cos(θ) and dx/dθ = r cos(θ) - r' sin(θ). Setting dx/dθ = 0, we get r cos(θ) - r' sin(θ) = 0, which simplifies to 5 cos(θ) + 5sin(θ) sin(θ) = 0. Solving this equation yields θ = π/2 or 3π/2. Substituting these values of θ into the equation for r, we find that the corresponding points are (-5, -5) and (-5, 5), respectively.