Simplify expression to polynomial form

A question is this type if and only if it asks to show that a trigonometric expression involving small angle θ can be approximated by a polynomial of the form a + bθ + cθ², where a, b, c are constants to be found.

6 questions · Standard +0.3

1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x

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OCR H240/01 2020 November Q1

5 marks Moderate -0.3

For a small angle $\theta$, where $\theta$ is in radians, show that $2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } \approx 3 - 2 \theta$.
Hence determine an approximate solution to $2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } = 28 \sin \theta$.

Edexcel Paper 2 2021 October Q4

3 marks Standard +0.3

Given that $\theta$ is small and measured in radians, use the small angle approximations to show that

$$4 \sin \frac { \theta } { 2 } + 3 \cos ^ { 2 } \theta \approx a + b \theta + c \theta ^ { 2 }$$ where $a , b$ and $c$ are integers to be found.

OCR H240/03 Q4

4 marks Standard +0.3

4 For a small angle $\theta$, where $\theta$ is in radians, show that $1 + \cos \theta - 3 \cos ^ { 2 } \theta \approx - 1 + \frac { 5 } { 2 } \theta ^ { 2 }$.

AQA Paper 1 2024 June Q9

5 marks Standard +0.8

Show that, for small values of $\theta$ measured in radians $$\cos 4\theta + 2 \sin 3\theta - \tan 2\theta \approx A + B\theta + C\theta^2$$ where $A$, $B$ and $C$ are constants to be found. [3 marks]
Use your answer to part (a) to find an approximation for $$\cos 0.28 + 2 \sin 0.21 - \tan 0.14$$ Give your answer to three decimal places. [2 marks]

AQA Paper 1 Specimen Q3

3 marks Standard +0.3

When $\theta$ is small, find an approximation for $\cos 3\theta + \theta \sin 2\theta$, giving your answer in the form $a + b\theta^2$ [3 marks]

OCR H240/03 2017 Specimen Q4

4 marks Standard +0.3

For a small angle $\theta$, where $\theta$ is in radians, show that $1 + \cos \theta - 3\cos^2 \theta \approx -1 + \frac{3}{2}\theta^2$. [4]