# What is the equation of the line that is normal to the polar curve #f(theta)= cos(pi-2theta)+thetasin(5theta-pi/2) # at #theta = pi/2#?

We want to figure out two things: the slope of the curve and the position of the point. Therefore, we can find a line that crosses perpendicular to that point.

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To find the equation of the line normal to the polar curve ( f(\theta) = \cos(\pi - 2\theta) + \theta \sin(5\theta - \frac{\pi}{2}) ) at ( \theta = \frac{\pi}{2} ), follow these steps:

- Compute the derivative ( f'(\theta) ).
- Evaluate ( f'(\theta) ) at ( \theta = \frac{\pi}{2} ) to find the slope of the tangent line.
- The slope of the normal line will be the negative reciprocal of the slope of the tangent line.
- Use the point ( (\theta, f(\theta)) ) where ( \theta = \frac{\pi}{2} ) to find the equation of the normal line using the point-slope form.

This process will yield the equation of the line normal to the polar curve at ( \theta = \frac{\pi}{2} ).

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