Find the slope of the tangent line to the polar curve at the given point.
B) 1
C) -1

Question

Find the slope of the tangent line to the polar curve at the given point. 
B) 1 
C) -1

Quizplus · Accepted Answer

To find the slope of the tangent line to the polar curve, we need to find the derivative of the polar equation with respect to $	heta$, then use the formula $\frac{dy}{dx}=\frac{\frac{dy}{d	heta}}{\frac{dx}{d	heta}}$ to get the slope.
The polar equation given cannot be directly differentiated with respect to $	heta$, we need to convert it to rectangular form first:
$r = \cos(	heta)$
$x = r\cos(	heta) = \cos^2(	heta)$
$y = r\sin(	heta) = \cos(	heta)\sin(	heta)$
Now we can differentiate with respect to $	heta$:
$\frac{dx}{d	heta} = -2\cos(	heta)\sin(	heta)$
$\frac{dy}{d	heta} = \sin^2(	heta) - \cos^2(	heta)$
And the slope at the point where $	heta=\frac{\pi}{4}$ is:
$\frac{dy}{dx}=\frac{\frac{dy}{d	heta}}{\frac{dx}{d	heta}} = \frac{\sin^2(	heta) - \cos^2(	heta)}{-2\cos(	heta)\sin(	heta)} = \frac{\sin^2\left(\frac{\pi}{4}ight) - \cos^2\left(\frac{\pi}{4}ight)}{-2\cos\left(\frac{\pi}{4}ight)\sin\left(\frac{\pi}{4}ight)} = 1$

Find the Slope of the Tangent Line to the Polar