Score: \( 2 / 3 \) Penalty: none Question Write the equation of the line tangent to the graph of \( r=5-5 \sin \theta \) when \( \theta=\frac{5 \pi}{6} \). Answer Attempt 1 out of 2

To find the equation of the tangent line to the graph of \( r = 5 - 5 \sin \theta \) at \( \theta = \frac{5\pi}{6} \), we first need to determine the corresponding Cartesian coordinates. Converting to Cartesian form using \( x = r \cos \theta \) and \( y = r \sin \theta \): 1. Calculate \( r \) at \( \theta = \frac{5\pi}{6} \): \[ r = 5 - 5 \sin\left(\frac{5\pi}{6}\right) = 5 - 5 \cdot \frac{1}{2} = 5 - \frac{5}{2} = \frac{5}{2}. \] 2. Now convert to Cartesian coordinates using \( x = r \cos \theta \) and \( y = r \sin \theta \): \[ x = \frac{5}{2} \cos\left(\frac{5\pi}{6}\right) = \frac{5}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{5\sqrt{3}}{4}, \] \[ y = \frac{5}{2} \sin\left(\frac{5\pi}{6}\right) = \frac{5}{2} \cdot \frac{1}{2} = \frac{5}{4}. \] 3. Thus, the point on the curve is \( \left(-\frac{5\sqrt{3}}{4}, \frac{5}{4}\right) \). Next, we find the slope of the tangent line. The polar form \( r = 5 - 5\sin\theta \) gives derivatives in polar coordinates. The slope is found from: \[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}. \] Calculate \( \frac{dr}{d\theta} \): \[ \frac{dr}{d\theta} = -5 \cos \theta. \] Evaluating at \( \theta = \frac{5\pi}{6} \): \[ r = \frac{5}{2}, \quad \frac{dr}{d\theta} = -5 \cdot \left(-\frac{\sqrt{3}}{2}\right) = \frac{5\sqrt{3}}{2}. \] Now plug in the values to compute the slope: \[ \frac{dy}{dx} = \frac{\frac{5\sqrt{3}}{2} \cdot \frac{1}{2} + \frac{5}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right)}{\frac{5\sqrt{3}}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right) - \frac{5}{2} \cdot \frac{1}{2}}. \] Calculating the numerator: \[ \frac{dy}{dx} = \frac{\frac{5\sqrt{3}}{4} - \frac{5\sqrt{3}}{4}}{\frac{-5\cdot \frac{3}{2}}{2} - \frac{5}{4}} = \frac{0}{-\frac{15}{4} - \frac{5}{4}} = 0. \] The slope at that angle is 0, which means the tangent line is horizontal. Finally, the equation of the tangent line at the point \( \left(-\frac{5\sqrt{3}}{4}, \frac{5}{4}\right) \) is: \[ y = \frac{5}{4}. \]