Write the angle made by the tangent

Question:

Write the angle made by the tangent to the curve $x=e^{t} \cos t, y=e^{t} \sin t$ at $t=\frac{\pi}{4}$ with the $x$-axis.

Solution:

Given that the curve $x=e^{t} \cos t, y=e^{t} \sin t$

$\frac{d x}{d t}=e^{t} \cos t-e^{t} \sin t$ and $\frac{d y}{d t}=e^{t} \sin t+e^{t} \cos t$

$\Rightarrow \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{e^{t} \sin t+e^{t} \cos t}{e^{t} \cos t-e^{t} \sin t}=\frac{\sin t+\cos t}{\cos t-\sin t}$

Now, for $t=\frac{\pi}{4}$

$\frac{d y}{d x}=\frac{\sin \frac{\pi}{4}+\cos \frac{\pi}{4}}{\cos \frac{\pi}{4}-\sin \frac{\pi}{4}}=\frac{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}=\infty$

Let $\theta$ be the angle made by the tangent with the $x$-axis.

$\therefore \tan \theta=\infty$

$\Rightarrow \theta=\frac{\pi}{2}$

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