#### 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

\[\text { Here }, \]

\[x = e^t \cos t \text { and } y = e^t \sin t\]

\[\frac{dx}{dt} = e^t cos t - e^t \sin t \text { and }\frac{dy}{dt} = e^t \sin t + e^t \cos t\]

\[ \therefore \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{e^t \sin t + e^t \cos t}{e^t cos t - e^t \sin t} = \frac{\sin t + \cos t}{\cos t - \sin t}\]

\[\text { Now, } \]

\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_{t = \frac{\pi}{4}} =\frac{\sin \frac{\pi}{4} + \cos \frac{\pi}{4}}{\cos \frac{\pi}{4} - \sin \frac{\pi}{4}}=\frac{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}}=\frac{\frac{2}{\sqrt{2}}}{0}=\infty\]

\[\text { Let }\theta \text { be the angle made by the tangent with thex-axis.}\]

\[ \therefore \tan\theta=\infty\]

\[ \Rightarrow \theta = \frac{\pi}{2}\]