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Question

# Find the polar equation of the tine tangent to the polar curve $r=\cos \theta+\sin \theta$ at the origin, and then find the slope of this tangent line.

Solution

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To find the equation of the line tangent to the polar curve $r=\cos \theta+\sin \theta$ at the origin we need to solve the equation $r=0$ for $\theta$:

\begin{align*} r&=0\Leftrightarrow\\ \cos \theta&=-\sin \theta\Leftrightarrow\\ \cos (-\theta)&=\sin (-\theta)\Leftrightarrow\\ -\theta&=\frac{\pi}{4}\Leftrightarrow \pmb{\theta=-\frac{\pi}{4}\,\,\lor \,\, \theta=\frac{3\pi}{4}} \end{align*}

Now that we know that the equation of the tangent line at the origin is $-\frac{\pi}{4}$ we see that the slope of the line is equal to

$\boxed{\tan \left( -\frac{\pi}{4}\right)=-1}$

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