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Small angle approximations

Small angle approximations

When working in radians, for small values of $\theta$ , the following approximations hold $$\sin(\theta) \approx \theta,\; \cos(\theta) \approx 1 - \frac{1}{2}\theta^2$$ These approximations are the first terms of the Maclaurins series for each function. The graphs show the goodness of fit of the approximations and how they diverge as the magnitude of $\theta$ increases.
sin x vs x
Graph of sin(x) and its small angle approximation, x
cos x vs small angle approx
Graph of $\cos(x)$ and its small angle approx $1 - \frac{1}{2}x^2$

Small angle approximations 1

By using small angle approximations, show that $$ \frac{\sin(\theta) + \sqrt{2}}{\cos(\theta)} \approx \frac{2}{\sqrt{2} - \theta} $$ for small values of $\theta$
solution - press button to display

Recall that $\sin(\theta) \approx \theta$ and $\cos(\theta) \approx 1 - \frac{1}{2}\theta^2$ for small values of $\theta$. Substitution of these approximations yields

$$ \begin{align} \frac{\sin(\theta) + \sqrt{2}}{\cos(\theta)} &\approx \frac{\theta + \sqrt{2}}{1 - \frac{1}{2}\theta^2} \\ \frac{\theta + \sqrt{2}}{1 - \frac{1}{2}\theta^2} &=\frac{\theta + \sqrt{2}}{(1+\frac{\theta}{\sqrt{2}})(1-\frac{\theta}{\sqrt{2}})} \\ &=\frac{\sqrt{2}}{1 - \frac{\theta}{\sqrt{2}}} \\ &= \frac{2}{\sqrt{2} - \theta} \end{align}$$

Hence the result: $$\frac{\sin(\theta) + \sqrt{2}}{\cos(\theta)} \approx \frac{2}{\sqrt{2} - \theta}$$

small angle approximations 2

Using small angle approximations for $\sin(\theta)$ and $\cos(\theta)$, show that $$\sin\left(\theta + \frac{\pi}{3}\right) \approx \frac{1}{2}\theta\left(1 + \sqrt{3}\theta\right) $$
solution - press button to display

This question is just an application of the compound angle formula and then a substitution of the small angle approximations, $\sin(\theta) \approx \theta, \; \cos(\theta) \approx 1 - \frac{1}{2}\theta^2$.

$$ \begin{align} \sin\left(\theta + \frac{\pi}{3}\right) &= \sin(\theta)\cos\left(\frac{\pi}{3}\right) + \cos(\theta)\sin\left(\frac{\pi}{3}\right)\\ \sin(\theta)\cos\left(\frac{\pi}{3}\right) + \cos(\theta)\sin\left(\frac{\pi}{3}\right) &\approx \frac{1}{2}\theta + \frac{\sqrt{3}}{2}\theta^2 \\ &= \frac{\theta}{2}\left(1 + \sqrt{3}\theta\right) \end{align} $$ Hence $$\sin\left(\theta + \frac{\pi}{3}\right) \approx \frac{1}{2}\theta\left(1 + \sqrt{3}\theta\right) $$

Small angle approximations 3

By using small angle approximations, determine the value of $$\tan\left(\frac{\pi}{3} + \frac{1}{10}\right)$$
solution - press button to display

We must first apply the compound angle formula for $\tan(A+B)$ $$ \tan(A+B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)} $$

Hence $$\tan\left(\frac{\pi}{3} + \frac{1}{10}\right) = \frac{\tan\left(\frac{\pi}{3}\right) + \tan\left(\frac{1}{10}\right)}{1 - \tan\left(\frac{\pi}{3}\right)\tan\left(\frac{1}{10}\right)} $$ As $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$ and $\tan\left(\frac{1}{10}\right) \approx \frac{1}{10}$ then $$\tan\left(\frac{\pi}{3} + \frac{1}{10}\right) \approx \frac{\sqrt{3} + \frac{1}{10}}{1 - \frac{1}{10}\sqrt{3}}$$

Simplification yields $$\tan\left(\frac{\pi}{3} + \frac{1}{10}\right) \approx \frac{10\sqrt{3} + 1}{10 - \sqrt{3}}$$