Standard +0.3 This is a straightforward application of standard small angle approximations (tan θ ≈ θ, cos θ ≈ 1 - θ²/2) with basic algebraic manipulation. The question is slightly easier than average as it's a routine technique tested regularly at A-level, requiring only substitution of known approximations and simplification to find A = -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.
[4 marks]
Question 4:
4 | Uses or states small angle
tan5x≈5x
approximation for | 1.1b | B1 | xtan5x x×5x
≈
cos4x−1 ( 4x )2
1− −1
2
5x2
≈
−8x2
5
≈−
8
Uses or states small angle
approximation for
( )2
4x
cos4x≈1−
2
Condone omission of bracket | 1.1b | B1
Substitutes their expressions
tan5x≈mx
Of the form and
nx2
cos4x≈1−
2
xtan5x
into
cos4x−1
Condone correct extra terms | 1.1b | M1
5
DeducesA=− from a reasoned
8
argument
CSO | 2.2a | R1
Total | 4
Q | Marking instructions | AO | Marks | Typical solution
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.
[4 marks]
\hfill \mbox{\textit{AQA Paper 2 2020 Q4 [4]}}