Small angle approximationsEdexcel A-Level Maths: Revision notes

Q: How accurate are they?

The approximations improve as \theta gets smaller. Example: \cos0.2\approx1-\frac{0.04}{2}=0.98, while the calculator gives 0.98007\ldots The percentage error is \frac{|approx-exact|}{exact}\times100=0.0068\%.

Section 1

Why small angles are special

For a small angle $\theta$ measured in radians, the arc, the opposite side and the tangent are almost the same length, so $\sin\theta$ , $\tan\theta$ and $\theta$ are almost equal; $\cos\theta$ is close to $1$ but slightly smaller. The approximations below are only valid in radians. For example $\sin0.1=0.0998$ is close to $0.1$ , but $\sin10$ is not close to $10$ .

Key termssmall angle

Common mistake

Using the approximations with a degree value such as $\sin10^\circ\approx10$ . Convert to radians first.

Section 2

The three standard approximations

For small $\theta$ in radians: $\sin\theta\approx\theta,\qquad\cos\theta\approx1-\frac{\theta^2}{2},\qquad\tan\theta\approx\theta.$ These replace the trig function by a simple polynomial. They give $1-\cos\theta\approx\frac{\theta^2}{2}$ directly. Multiples of the angle follow by substitution: $\sin3\theta\approx3\theta$ , $\tan2\theta\approx2\theta$ and $\cos4x\approx1-\frac{(4x)^2}{2}=1-8x^2$ , so $1-\cos4x\approx8x^2$ .

Key termsapproximation

Common mistake

Forgetting to square the coefficient: $\cos3x\approx1-\frac{9x^2}{2}$ , not $1-\frac{3x^2}{2}$ .

Section 3

How accurate are they?

The approximations improve as $\theta$ gets smaller. Example: $\cos0.2\approx1-\frac{0.04}{2}=0.98$ , while the calculator gives $0.98007\ldots$ The percentage error is $\frac{|\text{approx}-\text{exact}|}{\text{exact}}\times100=0.0068\%$ . At $\theta=0.5$ , $\sin\theta\approx\theta$ is out by about $4.3\%$ , so the approximation is poorer. Compare an approximation with the calculator value by working in radian mode.

Key termspercentage error

Exam tip

The larger $\theta$ is, the larger the error. Say that the approximation is only reliable for small $\theta$ .

Section 4

Approximating expressions with several terms

Replace each trigonometric function and simplify, keeping terms up to the order the question needs. Example: $4\cos\theta+3\sin\theta\approx4\left(1-\frac{\theta^2}{2}\right)+3\theta=4+3\theta-2\theta^2$ . Setting this equal to $4.2$ gives $2\theta^2-3\theta+0.2=0$ with roots $0.0699$ and $1.43$ . Only the small root is valid, because $1.43$ is not small and the approximation does not apply there; check: $4\cos0.0699+3\sin0.0699=4.1998$ .

Exam tip

Reject roots that are not small: the approximation cannot be trusted there.

Section 5

Quotients and limits

For a fraction, approximate numerator and denominator separately and cancel powers of $x$ . Example (specification type): $\frac{\cos3x-1}{x\sin4x}\approx\frac{-\frac{9x^2}{2}}{x(4x)}=-\frac98$ . Similarly $\frac{1-\cos4x}{x\tan2x}\approx\frac{8x^2}{2x^2}=4$ and $\frac{\cos5x-1}{x\sin2x}\approx-\frac{25}{4}$ ; with $x=0.01$ the calculator gives $-6.249$ , close to $-6.25$ .

Key termsnumeratordenominator

Common mistake

Using only $\cos x\approx1$ for the numerator: then $\cos3x-1\approx0$ and the answer is lost. Keep the $-\frac{\theta^2}{2}$ term.

Section 6

Using the approximations in context

In applied problems with small angles, such as a pendulum of length $L$ with $h=L(1-\cos\theta)$ and $d=L\sin\theta$ , the approximations give $h\approx\frac{L\theta^2}{2}$ and $d\approx L\theta$ . For $L=1.5$ and $\theta=0.1$ : $h\approx0.0075$ m with exact value $0.0074937$ m (percentage error $0.083\%$ ), and $d\approx0.15$ m. State clearly that the model assumes $\theta$ is small.

Exam tip

Quote units and say the approximation is valid because the angle is small.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Small angle approximations

In this question $\theta$ is small and measured in radians, so the standard small angle approximations may be used.
Calculate the percentage error in the estimate $\cos0.2\approx0.98$ , using the calculator value of $\cos0.2$ . Give your answer to 2 significant figures.2 marks
For small $x$ in radians, consider the expression $E=\frac{1-\cos4x}{x\tan2x}$ .
Hence find the approximate value of $E$ when $x$ is small.2 marks
A pendulum of length $1.5$ m swings through a small angle $\theta$ radians from the vertical. The horizontal displacement of the bob from the vertical is $d=1.5\sin\theta$ m and its height above its lowest point is $h=1.5(1-\cos\theta)$ m.
Use a small angle approximation to show that, for small $\theta$ , $h\approx0.75\theta^2$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).