Show that \(\frac{\mathrm{d}}{\mathrm{d}x}\left(\cot{\frac{x}{2}}\right)=-\frac{1}{2}\textbf{co

Question

Show that $\frac{\mathrm{d}}{\mathrm{d}x}\left(\cot{\frac{x}{2}}\right)=-\frac{1}{2}\textbf{cosec}^2\frac{x}{2}$ The diagram shows an isosceles triangle $ABC$ with $AC = BC$ and $AB = 8$ cm. The vertex $C$ is moving towards $AB$ in the direction perpendicular to $AB$ at a constant speed of $1.5$ cm/s. Let angle $ACB = \theta$. Find the rate of change of $\theta$ when triangle $ABC$ is equilateral.

Accepted Answer

(a) $\dfrac{\mathrm{d}}{\mathrm{d}x}\cot\dfrac{x}{2}=-\dfrac12\csc^2\dfrac{x}{2}$ (b) $\dfrac{\mathrm{d}\theta}{\mathrm{d}t}=\dfrac3{16}\text{ rad s}^{-1}$

Differentiation - Derivatives of Reciprocal Trigonometric Functions

Differentiation - Derivatives of Reciprocal Trigonometric Functions

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