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NSW Y12 Maths - Advanced Trig Functions Applications of Differentiation of Trig Functions

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Applications of Differentiation of Trig Functions Theory

Applications of the differentiation of trigonometric functions involves determining equations of tangents and normals and stationary points on these functions.\\ \begin{multicols}{2} \textbf{Example 1}\\%31067 Consider the function \(y = \sin x + \cos 2x\). Find the equation of the tangent to the graph of the function at \(x = \dfrac{\pi}{6}\).\\ \textbf{Example 1 solution}\\ $\begin{aligned} y &= \sin x + \cos 2x\\\frac{dy}{dx} &= \cos x -2\sin 2x\\\text{at } x = \frac{\pi}{6} \qquad \frac{dy}{dx} &= \cos\frac{\pi}{6} - 2\sin\frac{\pi}{3}\\&= \frac{\sqrt{3}}{2} - 2\times\frac{\sqrt{3}}{2}\\&= -\frac{\sqrt{3}}{2}\\\text{and } y &= \sin \frac{\pi}{6} +\cos\frac{\pi}{3}\\&= \frac{1}{2} + \frac{1}{2}\\&= 1 \end{aligned}$\\ $\begin{aligned}\therefore \text{ the equation} &\text{ of the tangent is }:\\ y - 1 &= -\frac{\sqrt{3}}{2}\left(x - \frac{\pi}{6}\right)\\y -1 &= -\frac{\sqrt{3}}{2}x + \frac{\sqrt{3}\pi}{12}\\y &= -\frac{\sqrt{3}}{2}x + \frac{\sqrt{3}}{12}\pi + 1\end{aligned}$\\ \columnbreak \textbf{Example 2}\\%24815 Find the equation of the normal to the curve \(y = 2\sin 2x\) at \(x = \dfrac{\pi }{2}\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} y &=2 \sin 2 x \\ y^{\prime} &=2 \cos 2 x \times 2 \\ &=4 \cos 2 x \\ \text{At}\ x &=\pi / 2, y=2 \sin \pi=0 \\ y^{\prime} &=4 \cos \pi=-4 \\ y-0 &=\frac{1}{4}\left(x-\frac{\pi}{2}\right) \\ 4 y &=x-\frac{\pi}{2} \\ x &-4 y-\frac{\pi}{2}=0 \end{aligned}$\\ \end{multicols}

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  • Applications of Differentiation of Trig Functions - Video - Derivative Rules with TRIG functions

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