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 Mechanics - Question 4 - 2020 - Paper 2
Question 4
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.
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 Mechanics - Question 4 - 2020 - Paper 2
Step 1
Using small angle approximation for $\tan 5x$
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Answer
For small angles, we use the approximation:
tanθ≈θ
Thus, for tan5x, we have:
tan5x≈5x.
Consequently, we can rewrite the expression as:
$$\frac{x \tan 5x}{\cos 4x - 1} \approx \frac{x(5x)}{\cos 4x - 1} = \frac{5x^2}{\cos 4x - 1}.$
Step 2
Using small angle approximation for $\cos 4x$
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Answer
Using the small angle approximation for cosine:
cosθ≈1−2θ2
we get:
cos4x≈1−2(4x)2=1−8x2.
Thus, we can express the denominator as follows:
cos4x−1≈(1−8x2)−1=−8x2.
Step 3
Constructing the final expression
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Answer
Substituting the results from the previous steps into the expression, we have:
cos4x−15x2≈−8x25x2=−85.
