The tangent to the curve $y=2\sin x$ at $A$, where the $x$-coordinate of $A$ is \(\dfrac{\pi

Question

The tangent to the curve $y=2\sin x$ at $A$, where the $x$-coordinate of $A$ is $\dfrac{\pi}{3}$, meets the $y$-axis at $B$. Find the equation of the tangent at $A$.[3] Find the coordinates of $B$.[1] Hence find the exact area enclosed by the curve, the tangent and the $y$-axis.[5]

Accepted Answer

(i) $y=x-\dfrac{\pi}{3}+\sqrt3$ (ii) $B=\left(0,\sqrt3-\dfrac{\pi}{3}\right)$ (iii) $\dfrac{\pi\sqrt3}{3}-\dfrac{\pi^2}{18}-1\text{ units}^2$

Sine curve and its tangent beside the y-axis

Sine curve and its tangent beside the y-axis

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