Tangent Lines to Polar Curves Question Write the equation of the line tangent to the graph of \( r=4 \cos \theta \) when \( \theta=\frac{\pi}{2} \). Answer Attempt 1 out of 2 \( \sqrt{~ S u b m i t ~ A n s w e r ~} \)

To find the tangent line to the polar curve \( r = 4 \cos \theta \) at \( \theta = \frac{\pi}{2} \), first calculate \( r \) at that angle: \( r = 4 \cos\left(\frac{\pi}{2}\right) = 0 \). This means the point is at the origin (0,0). Next, to find the slope of the tangent line, we need the derivative \( \frac{dy}{dx} \). In polar coordinates, this can be calculated using the formula: \[ \frac{dy}{dx} = \frac{r' \sin \theta + r \cos \theta}{r' \cos \theta - r \sin \theta} \] Calculating the derivative \( r' = -4 \sin\theta \) and substituting \( \theta = \frac{\pi}{2} \), we get: \[ \frac{dy}{dx} = \frac{(0) \sin\left(\frac{\pi}{2}\right) + (0) \cos\left(\frac{\pi}{2}\right)}{(-4) \cos\left(\frac{\pi}{2}\right) - (0) \sin\left(\frac{\pi}{2}\right)} = \frac{0}{0} \] Since we have an indeterminate form, we can conclude that the tangent line at the origin is vertical. Therefore, the equation of the tangent line is: \[ x = 0 \] Now let's add a bit of flavor! Did you know that the polar curve \( r = 4 \cos \theta \) represents a circle? Yep, it's centered at (2,0) with a radius of 2! It’s one of the classic examples of how polar coordinates can beautifully illustrate relationships that are a bit tricky in Cartesian forms. Also, here’s a pro tip: when dealing with polar curves, visualizing them helps tremendously! Sketch the curve to see how it behaves around the angles. This can be particularly useful when determining points, slopes, and tangent lines! Who knew math could be so creative?