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(C) Given the curve $y = 2 \sin x + \sin 2x$.First,find the derivative $\frac{dy}{dx}$ to determine the slope of the tangent:$\frac{dy}{dx} = 2 \cos x

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$2y = 3\sqrt{3}$

Vedclass · Answer

(C) Given the curve $y = 2 \sin x + \sin 2x$.First,find the derivative $\frac{dy}{dx}$ to determine the slope of the tangent:$\frac{dy}{dx} = 2 \cos x + 2 \cos 2x$.At $x = \pi / 3$,the slope $m$ is:$m = \left( \frac{dy}{dx} \right)_{x = \pi / 3} = 2 \cos(\pi / 3) + 2 \cos(2\pi / 3) = 2(1/2) + 2(-1/2) = 1 - 1 = 0$.Next,find the $y$-coordinate at $x = \pi / 3$:$y = 2 \sin(\pi / 3) + \sin(2\pi / 3) = 2(\sqrt{3}/2) + (\sqrt{3}/2) = \sqrt{3} + \sqrt{3}/2 = 3\sqrt{3}/2$.The equation of the tangent line at point $(x_1, y_1)$ with slope $m$ is $(y - y_1) = m(x - x_1)$.Since $m = 0$,the equation becomes $y - 3\sqrt{3}/2 = 0$,which simplifies to $y = 3\sqrt{3}/2$ or $2y = 3\sqrt{3}$.

What is the equation of the tangent to the curve $y = 2 \sin x + \sin 2x$ at the point $x = \pi / 3$?

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