Tangent Lines to Polar Curves Question Write the equation of the line tangent to the graph of \( r=4 \sin ^{2} \theta \) when \( \theta=\frac{\pi}{2} \). Answer Attempt 1 out of 2 \( \sqrt{ } \)

**Step 1. Convert the Polar Point to Cartesian Coordinates** The polar equation is given by \[ r = 4 \sin^{2} \theta. \] At \(\theta = \frac{\pi}{2}\), compute \(r\): \[ r = 4 \sin^{2}\left(\frac{\pi}{2}\right) = 4 \cdot 1^2 = 4. \] The Cartesian coordinates are given by: \[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta. \] For \(\theta = \frac{\pi}{2}\): \[ x = 4 \cos\left(\frac{\pi}{2}\right) = 4 \cdot 0 = 0, \quad y = 4 \sin\left(\frac{\pi}{2}\right) = 4 \cdot 1 = 4. \] Thus, the point of tangency is \((0, 4)\). --- **Step 2. Find the Slope of the Tangent Line** For a polar curve, the slope \(\frac{dy}{dx}\) is given by: \[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin \theta + r \cos \theta}{\frac{dr}{d\theta}\cos \theta - r \sin \theta}. \] First, find \(\frac{dr}{d\theta}\) for \(r = 4 \sin^{2}\theta\): \[ \frac{dr}{d\theta} = 4 \cdot 2 \sin \theta \cos \theta = 8 \sin \theta \cos \theta. \] Now, evaluate the numerator and denominator at \(\theta = \frac{\pi}{2}\). - **Numerator:** \[ \frac{dr}{d\theta}\sin \theta + r \cos \theta = (8 \sin \theta \cos \theta) \sin \theta + 4 \sin^2 \theta \cos \theta. \] At \(\theta = \frac{\pi}{2}\): \[ \sin\left(\frac{\pi}{2}\right) = 1, \quad \cos\left(\frac{\pi}{2}\right) = 0. \] So, \[ \text{Numerator} = 8 \cdot 1 \cdot 0 + 4 \cdot 1 \cdot 0 = 0. \] - **Denominator:** \[ \frac{dr}{d\theta}\cos \theta - r \sin \theta = (8 \sin \theta \cos \theta) \cos \theta - 4 \sin^2 \theta. \] At \(\theta = \frac{\pi}{2}\): \[ \text{Denominator} = 8 \cdot 1 \cdot 0 - 4 \cdot 1 = -4. \] Thus, the slope is: \[ \frac{dy}{dx} = \frac{0}{-4} = 0. \] --- **Step 3. Write the Equation of the Tangent Line** The tangent line with slope \(0\) is horizontal. Since it passes through \((0, 4)\), the equation is: \[ y = 4. \] --- **Final Answer** \[ y = 4. \]