For $\theta$ up to about $0.2$ the three graphs are almost indistinguishable; by $\theta = 0.5$ they have clearly separated.
| $\theta$ (rad) | $\sin\theta$ | $\tan\theta$ | $\cos\theta$ | $1 - \tfrac12\theta^2$ |
|---|---|---|---|---|
| $0.05$ | $0.04998$ | $0.05004$ | $0.998750$ | $0.99875$ |
| $0.1$ | $0.09983$ | $0.10033$ | $0.995004$ | $0.995$ |
| $0.2$ | $0.19867$ | $0.20271$ | $0.980067$ | $0.98$ |
| $0.5$ | $0.47943$ | $0.54630$ | $0.877583$ | $0.875$ |
Take a sector of a unit circle with a small angle $\theta$. Compare three areas: the triangle inside the sector, the sector, and the right-angled triangle formed by the tangent at one end.
As $\theta \to 0$, $\cos\theta \to 1$, so $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ and $\sin\theta$ get closer together โ and $\theta$ is trapped between them. Hence $\sin\theta \approx \theta \approx \tan\theta$.
For cosine, use the double angle formula $\cos\theta = 1 - 2\sin^2\dfrac{\theta}{2}$ (see Compound and Double Angles):
- Replace every $\sin(k\theta)$ by $k\theta$, every $\tan(k\theta)$ by $k\theta$, and every $\cos(k\theta)$ by $1 - \tfrac12(k\theta)^2$.
- Expand and simplify, ignoring powers higher than the question needs.
- If $\theta$ appears in both numerator and denominator, cancel it.
Find an approximation for $\cos 3\theta$ when $\theta$ is small.
Show that, for small $\theta$, $\dfrac{1 - \cos 4\theta}{\theta\sin 2\theta} \approx 4$.
Find the value that $\dfrac{\sin\theta\tan\theta}{1 - \cos\theta}$ approaches as $\theta \to 0$.
Given that $\theta$ is small, use the small angle approximations to find an approximate solution of $2\cos\theta + 3\sin\theta = 2.1$.
Use a small angle approximation to estimate $\cos 5^\circ$ to $4$ d.p.
Sine
$\sin\theta \approx \theta$
Tangent
$\tan\theta \approx \theta$
Cosine
$\cos\theta \approx 1 - \tfrac12\theta^2$
Units
Radians only; convert degrees first.
Multiples
$\sin k\theta \approx k\theta$, $\cos k\theta \approx 1 - \tfrac12k^2\theta^2$
$1 - \cos\theta$
$\approx \tfrac12\theta^2$ โ keep the squared term.
Equations
Keep only the root that is actually small.
Accuracy
Excellent for $|\theta| \lesssim 0.2$; poor by $0.5$.
Use the small angle approximations to estimate $\sin 0.03$, $\tan 0.03$ and $\cos 0.03$.
โถ Show solution
$\sin 0.03 \approx 0.03$, $\tan 0.03 \approx 0.03$.
$\cos 0.03 \approx 1 - \tfrac12(0.0009) = 0.99955$ (calculator: $0.9995500\ldots$).
Find a small angle approximation for $\cos 2\theta$.
โถ Show solution
$\cos 2\theta \approx 1 - \tfrac12(2\theta)^2 = 1 - 2\theta^2$.
For small $\theta$, find an approximation for $\sin 2\theta\tan 3\theta$.
โถ Show solution
$\sin 2\theta\tan 3\theta \approx 2\theta \times 3\theta = 6\theta^2$.
Show that, for small $\theta$, $\dfrac{1 - \cos 2\theta}{\theta^2} \approx 2$.
โถ Show solution
$1 - \cos 2\theta \approx 1 - (1 - 2\theta^2) = 2\theta^2$, so the expression $\approx \dfrac{2\theta^2}{\theta^2} = 2$.
Show that, for small $\theta$, $\dfrac{1 - \cos 4\theta}{2\theta\sin 3\theta} \approx \dfrac43$.
โถ Show solution
Numerator: $1 - \left(1 - \tfrac12(16\theta^2)\right) = 8\theta^2$.
Denominator: $2\theta \times 3\theta = 6\theta^2$.
Ratio $\approx \dfrac{8\theta^2}{6\theta^2} = \dfrac43$. $\blacksquare$
Find a quadratic approximation for $3\sin\theta + \cos 2\theta$ and use it to estimate the value when $\theta = 0.1$. Compare with the true value.
โถ Show solution
$3\theta + 1 - 2\theta^2$.
At $\theta = 0.1$: $0.3 + 1 - 0.02 = 1.28$.
True value: $3\sin 0.1 + \cos 0.2 = 0.29950 + 0.98007 = 1.27957$. The error is about $0.0004$.
A student uses $\sin\theta \approx \theta$ to estimate $\sin 30^\circ$. Find the percentage error and comment.
โถ Show solution
$30^\circ = \dfrac{\pi}{6} = 0.5236$ rad, so the estimate is $0.5236$ against a true value of $0.5$.
Percentage error $= \dfrac{0.5236 - 0.5}{0.5} \times 100 = 4.7\%$.
$0.52$ rad is not small, so the approximation is poor. It should only be used for angles close to zero.
Given that $\theta$ is small, find an approximate solution of $5\sin\theta + \cos\theta = 1.2$, giving your answer to $3$ significant figures.
โถ Show solution
$5\theta + 1 - \tfrac12\theta^2 = 1.2 \Rightarrow \tfrac12\theta^2 - 5\theta + 0.2 = 0 \Rightarrow \theta^2 - 10\theta + 0.4 = 0$.
$\theta = \dfrac{10 \pm \sqrt{100 - 1.6}}{2} = \dfrac{10 \pm \sqrt{98.4}}{2}$.
The small root is $\theta = \dfrac{10 - \sqrt{98.4}}{2} = 0.0402$ (3 s.f.). The other root, about $9.96$, is not small and is rejected.
Find the limit of $\dfrac{\tan 5\theta}{\sin 2\theta}$ as $\theta \to 0$.
โถ Show solution
$\dfrac{\tan 5\theta}{\sin 2\theta} \approx \dfrac{5\theta}{2\theta} = \dfrac52$.
The limit is $\dfrac52$.
This question shows that $\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin x) = \cos x$, with $x$ in radians.
(a) Use $\sin(A + B) = \sin A\cos B + \cos A\sin B$ to show that $\dfrac{\sin(x + h) - \sin x}{h} = \sin x\left(\dfrac{\cos h - 1}{h}\right) + \cos x\left(\dfrac{\sin h}{h}\right)$.
(b) Use small angle approximations to find the limits of $\dfrac{\cos h - 1}{h}$ and $\dfrac{\sin h}{h}$ as $h \to 0$.
(c) Hence find the derivative of $\sin x$ from first principles, and explain why the result needs $x$ in radians.
โถ Show solution
(a) $\sin(x + h) - \sin x = \sin x\cos h + \cos x\sin h - \sin x = \sin x(\cos h - 1) + \cos x\sin h$. Divide by $h$ โ
(b) $\dfrac{\cos h - 1}{h} \approx \dfrac{-\frac12h^2}{h} = -\tfrac12h \to 0$. $\dfrac{\sin h}{h} \approx \dfrac{h}{h} = 1$.
(c) $\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin x) = \displaystyle\lim_{h \to 0}\left[\sin x \cdot \dfrac{\cos h - 1}{h} + \cos x \cdot \dfrac{\sin h}{h}\right] = \sin x \times 0 + \cos x \times 1 = \cos x$.
The step $\sin h \approx h$ is only true in radians. In degrees, $\dfrac{\sin h^\circ}{h} \to \dfrac{\pi}{180}$, and the derivative would be $\dfrac{\pi}{180}\cos x$.