First slide

Tangents and normals

Question

The slope of the tangent to the curve $x = a \sin t, y = a {\cos t + \log (\tan \frac{t}{2})}$ at point $' t'$ is

Easy

A

$\tan t$
B

$\cot t$
C

$\tan \frac{t}{2}$
D

None of these

Solution

Slope of the tangent $= \frac{d y}{d x} = (\frac{d y}{d t}) / (\frac{d x}{d t}) = \frac{a (- \sin t + \cos e c t)}{a \cos t} = \cot t .$
(Note that $\frac{d}{d t} (\log (\tan (t / 2)) = \frac{1}{\tan t} \sec^{2} \frac{t}{2} . \frac{1}{2} = \cos e c t)$ .

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