A-level · Trigonometry

Small angle approximationsExam-style questions

The radian approximations sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − ½θ², why they hold, and using them to approximate expressions and evaluate limiting ratios for small θ.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
When $\theta$ is small, show that $\dfrac{1 - \cos 2\theta}{\theta\sin\theta} \approx 2$ .
Level 4 · 2 marksTry it
2
When $\theta$ is small, show that $\dfrac{5\theta - \sin 2\theta}{\tan 3\theta} \approx 1$ .
Level 4 · 2 marksTry it
3
(a) Given that $\theta$ is small and measured in radians, use small-angle approximations to show that

$\cos 6\theta + 3\sin 2\theta - \tan 5\theta$

is approximately equal to $A + B\theta + C\theta^2$ , and state the values of the constants $A$ , $B$ and $C$

(3 marks)
(b) Use your answer to part (a) to find an estimate for

$\cos 0.24 + 3\sin 0.08 - \tan 0.2$

Give your answer to three decimal places.

(2 marks)
Level 4 · 5 marksTry it
4
(a) Find the first two terms, in ascending powers of $x$ , of the binomial expansion of

$(1 - x)^{-\frac{1}{2}}$

(2 marks)
(b) Hence, for small values of $x$ , show that

$\tan 2x + \frac{3}{\sqrt{\cos 4x}} \approx A + Bx + Cx^2$

where $A$ , $B$ and $C$ are constants to be found.

(4 marks)
Level 4 · 6 marksTry it
5
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A curve $C$ has equation $y = \text{f}(x)$ , where $x$ is measured in radians and

$\text{f}'(x) = 8x + 1 - 3\cos x$

The curve $C$ has a minimum point $M$ , and the $x$ coordinate of $M$ is small and positive.
(a) By replacing $\cos x$ with its small angle approximation, find an estimate of the $x$ coordinate of $M$ . Give your answer to 3 decimal places.

(3 marks)

$C$ meets the $y$ -axis at the point $A(0, 6)$
(b) Find an equation of the tangent to $C$ at $A$ . Give your answer in the form $y = mx + c$ , where $m$ and $c$ are integers.

(2 marks)
Level 4 · 5 marksTry it