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
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:
tanθ≈θ.
Thus, for tan5x, we have
tan5x≈5x.
Step 2
Using small angle approximation for $\cos 4x$
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Answer
Using the small angle approximation for cosine, we know that:
cosθ≈1−2θ2.
Therefore, for cos4x, we can write:
cos4x≈1−2(4x)2=1−8x2.
Step 3
Substituting into the expression
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Answer
We substitute these approximations into our expression:
cos4x−1xtan5x≈(1−8x2)−1x(5x)=−8x25x2=−85.
Thus, we can conclude that A=−85.
Step 4
Final deduction of constant $A$
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Answer
Finally, we have shown that for small values of x,
cos4x−1xtan5x≈A where
A=−85.
