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A wave on a rope is drawn with the displacement y = sin x, where x is an angle in radians and the…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A wave on a rope is drawn with the displacement $y = \sin x$ , where $x$ is an angle in radians and the two axes have the same scale. A marker is put on the wave at $x = \dfrac{\pi}{3}$ . Find the slope of the wave at the marker. Then find the angle that the tangent line at the marker makes with the $x$ -axis, in degrees, to two decimal places.

Need a reminder? The key ideaShow

With $x$ in radians: $(\sin x)' = \cos x$ , $(\cos x)' = -\sin x$ and $(\tan x)' = \sec^2 x$ . A tangent is horizontal where the derivative is $0$ . The angle $\theta$ of a tangent with the $x$ -axis satisfies $\tan\theta = f'(x)$ , when both axes have the same scale.

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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