A-Level AQA Maths: Pure 5.1 Basic Trigonometry: Using small angle approximations

Using small angle approximations, show that for small, non-zero values of $x$: $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined. - AQA - A-Level Maths: Pure - Question 4 - 2020 - Paper 2

Question 4

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Using small angle approximations, show that for small, non-zero values of $x$: $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined... show full transcript

Worked Solution & Example Answer:Using small angle approximations, show that for small, non-zero values of $x$: $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined. - AQA - A-Level Maths: Pure - Question 4 - 2020 - Paper 2

Step 1

Using small angle approximation for $ \tan 5x $

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Answer

For small angles, we can use the approximation:

$\tan \theta \approx \theta$

Therefore, for (\tan 5x):

$\tan 5x \approx 5x$

Step 2

Using small angle approximation for $ \cos 4x $

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Answer

The cosine function can be approximated as:

$\cos \theta \approx 1 - \frac{\theta^2}{2}$

Thus for (\cos 4x):

$\cos 4x \approx 1 - \frac{(4x)^2}{2} = 1 - 8x^2$

Step 3

Substituting expressions into the original formula

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Answer

Substituting the approximations into the expression, we have:

$\frac{x \tan 5x}{\cos 4x - 1} \approx \frac{x (5x)}{(1 - 8x^2) - 1}$

This simplifies to:

$\frac{5x^2}{-8x^2} = -\frac{5}{8}$

Step 4

Determining $ A $

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Answer

From the previous step, we find:

$A = -\frac{5}{8}$

Thus, we have shown that:

$\frac{x \tan 5x}{\cos 4x - 1} \approx -\frac{5}{8}$

Using small angle approximations, show that for small, non-zero values of $x$: $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined. - AQA - A-Level Maths: Pure - Question 4 - 2020 - Paper 2

Worked Solution & Example Answer:Using small angle approximations, show that for small, non-zero values of $x$: $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined. - AQA - A-Level Maths: Pure - Question 4 - 2020 - Paper 2

Using small angle approximation for \( \tan 5x \)

Using small angle approximation for \( \cos 4x \)

Substituting expressions into the original formula

Determining \( A \)

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