Compound & Double Angle Formulae

Revise Compound & Double Angle Formulae for Pure Mathematics 3 WMA13 (A2 Level) — revision notes and instant AI marking.

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Compound & Double Angle Formulae

Master the art of splitting and recombining angles

The Big Idea: You can break any angle into the sum or difference of two known angles, then use special formulae to simplify expressions, solve equations, and model real-world oscillations.

What You'll Learn

This chapter teaches you six compound angle formulae (for sin, cos, and tan), how to derive and use double angle formulae, how to convert trig expressions into harmonic form, and how to apply these to model and solve real-world problems.

Compound angles: Break sin(A + B), cos(A + B), tan(A + B) into simpler pieces
Double angles: Use these to simplify expressions with 2θ
Harmonic form: Rewrite asinx + bcosx as Rsin(x + α) to find maxima, minima, and solve equations
Real-world modelling: Use harmonic form and period formulas to solve problems about wheels, tides, and oscillations

Compound Angle Formulae

What Are Compound Angle Formulae?

Compound angle formulae (also called addition formulae) let you express trigonometric functions of combined angles—like sin(A + B)—in terms of the individual angles A and B. They're incredibly powerful because they transform difficult angles into sums or differences of angles you know.

Think of them as a breakdown rule: just as you might break £47 into "£40 + £7" to understand what you have, you can break sin(70°) into sin(45° + 25°) and use the formula to work with it more easily.

The Six Formulae

There are three pairs—one for sin, one for cos, one for tan:

Sine Compound Angles

sin(A + B) ≡ sin A cos B + cos A sin B
sin(A − B) ≡ sin A cos B − cos A sin B
✓ The ± sign on the left matches the ± on the right

Why does the sign match? Think about what's happening: sin(A + B) has a "+" in the angle itself, so when you expand it, you also get a "+". It's because both parts of the angle contribute positively to the final result.

Cosine Compound Angles

cos(A + B) ≡ cos A cos B − sin A sin B
cos(A − B) ≡ cos A cos B + sin A sin B
✗ The ± sign on the left is OPPOSITE to the ± in the middle

Why opposite? Cosine behaves differently—the sign "flips" because of how the unit circle works. This is one of the trickiest parts, so commit this to memory!

Tangent Compound Angles

tan(A + B) ≡ (tan A + tan B) / (1 − tan A tan B)
tan(A − B) ≡ (tan A − tan B) / (1 + tan A tan B)
✓ The ± in the numerator matches the left; ✗ the denominator is opposite

Exam tip: All these formulae are in your formula booklet. You don't have to memorise them for the exam itself, but you absolutely must understand how they work, because you'll need to recognise when and how to apply them.

Worked Example 1: Using the tan Formula

Example

Express tan(225° − 30°) in terms of tan 225° and tan 30°, then show that tan 195° = 2 − √3.

Step 1: Use the tan(A − B) formula:
tan(A − B) ≡ (tan A − tan B) / (1 + tan A tan B)
So tan(225° − 30°) = (tan 225° − tan 30°) / (1 + tan 225° tan 30°)

Step 2:

Step 3: Substitute into the formula:
tan 195° = (1 − √3/3) / (1 + 1 · √3/3) = (1 − √3/3) / (1 + √3/3)

Step 4:

Step 5: 2 − √3

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Also in the full note

Harmonic Form
Modelling with Trigonometric Functions
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
You've Got This
Practice Question 1
What Are Double Angle Formulae?

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Compound & Double Angle Formulae