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Edexcel Pure Paper 2 2024 Question 5
Edexcel Pure Paper 2 2024 Question 5 – Small Angle Approximations
❓ The Question
🧠 Before you start
This is one of those questions where, if you recognise what’s going on early, it becomes quite quick.
If not, it can feel a bit awkward — mainly because the expression looks more complicated than it really is.
The key idea here is that the angle is small. That changes everything.
Once you see that, you’re not really dealing with trig functions anymore in the usual sense — you’re replacing them with simpler expressions.
✏️ Working
We’re told that \theta is small and in radians.
That immediately tells us to use the standard approximations:
We’re told that \theta is small and in radians.
That tells us to use the standard small angle approximations:
\sin \theta \approx \theta
\tan \theta \approx \theta
\cos \theta \approx 1 – \frac{\theta^2}{2}
These are the only tools we need here.
Now look at the expression in the question.
Rather than trying to simplify everything first, it’s usually easier to apply the approximations straight away.
We are given:
\frac{\theta \tan(2\theta)}{1 – \cos(3\theta)}
with \theta small (in radians).
Step 1: Apply correct approximations
Use:
\tan(2\theta) \approx 2\theta
\cos(3\theta) \approx 1 – \frac{(3\theta)^2}{2}
Step 2: Substitute
Numerator:
\theta \tan(2\theta) \approx \theta \cdot 2\theta = 2\theta^2
Denominator:
1 – \cos(3\theta) \approx 1 – \left(1 – \frac{(3\theta)^2}{2}\right)
= \frac{(3\theta)^2}{2} = \frac{9\theta^2}{2}
Step 3: Form the fraction
\frac{2\theta^2}{\frac{9\theta^2}{2}}
Step 4: Simplify
= 2\theta^2 \cdot \frac{2}{9\theta^2}
= \frac{4}{9}
✅ Final Answer:
\frac{4}{9}
🎯 Where the Marks Are
This question is quite direct in how it’s marked.
- recognising small angle approximations
- applying them correctly
- simplifying without errors
There aren’t many steps, but each one needs to be right.
⚠️ What Went Wrong
Most mistakes here came from either not using the approximations at all, or using them incorrectly.
A common issue was keeping trig functions in the working instead of replacing them straight away.
Another mistake was mixing up the approximations — for example, using \cos \theta \approx \theta, which isn’t correct.
There were also a few algebra slips once the substitutions had been made. Even though the trig part is simple, errors can still creep in during simplification.
💡 One Small Tip
If a question mentions “small” and “radians”, stop there.
That’s your signal. You don’t need to think about identities or graphs — just apply the standard approximations and move on.
🚀 If This Felt Difficult
If this didn’t feel immediate, it’s usually because the trigger isn’t automatic yet.
With enough practice, you start to recognise these questions almost instantly.
Working through more examples like this, especially with structured A level maths support, helps build that recognition so you don’t hesitate in the exam.
And once that part becomes more natural, it’s much easier to improve your maths grades, since these are typically quick marks when done confidently.
🔗 Next Steps
👨🏫Author Bio
S Mahandru is an A Level Maths teacher focused on helping students stay accurate under exam pressure. The aim is to keep methods clear and build confidence through consistent exam-style practice.
❓ Frequently Asked Questions
📌Why must the angle be in radians?
Because the approximations only work properly in radians, not degrees.
📌Do I always replace trig functions straight away?
In this type of question, yes — it’s usually the quickest approach.
📌Are these questions common?
Yes, they appear regularly and are often worth a few marks.
📌What’s the main risk?
Using the wrong approximation or overcomplicating the algebra.