Higher Maths · Relationships and Calculus
Compound Angle Formulae | Higher Maths
Compound and double angle formulae let you expand expressions like sin(A + B) and cos 2x into expressions you can simplify or solve. They are tested every year in Higher Maths, often combined with the wave function or identity proofs.
Compound angle formulae (on the formula sheet)
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
Double angle formulae
sin 2A = 2 sin A cos A
cos 2A = cos2A − sin2A = 2cos2A − 1 = 1 − 2sin2A
Worked example
Worked example — Solving with double angle
Problem: Solve 2 cos 2x + 1 = 0 for 0 ≤ x ≤ 2π.
- Rearrange.cos 2x = −1/2
- Replace 2x with θ and find θ in [0, 4π].θ = 2π/3, 4π/3, 2π + 2π/3 = 8π/3, 2π + 4π/3 = 10π/3
- Divide by 2 to get x.x = π/3, 2π/3, 4π/3, 5π/3
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Expand sin(x + π/3).
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sin x cos π/3 + cos x sin π/3 = (1/2)sin x + (√3/2)cos x - Show that sin 2x = 2 sin x cos x using the compound angle formula.
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sin(x + x) = sin x cos x + cos x sin x = 2 sin x cos x. - Solve cos 2x = sin x for 0 ≤ x ≤ 2π.
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1 − 2sin2x = sin x ⇒ 2sin2x + sin x − 1 = 0 ⇒ (2 sin x − 1)(sin x + 1) = 0. x = π/6, 5π/6 or 3π/2 - If sin A = 3/5 and A is acute, find the exact value of sin 2A.
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cos A = 4/5. sin 2A = 2(3/5)(4/5) = 24/25 - Expand cos(x − π/4) and simplify.
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= cos x cos π/4 + sin x sin π/4 = (1/√2)(cos x + sin x)
Common mistakes
Common mistakes & how to avoid them
- Mixing up the sign in cos(A ± B): the sign on the right is opposite to the left.
- Forgetting that cos 2A has three equivalent forms — pick the one that matches the rest of the equation.
- Solving for θ over the wrong interval. If the equation has cos 2x, you must solve for 2x in [0, 4π] before dividing.
Frequently asked questions
Are these formulae on the SQA formula sheet?
The compound angle formulae for sin and cos are on the Higher formula sheet. The double angle versions are derived from them — they are not separately listed but follow immediately.
When do I use which form of cos 2A?
Use cos 2A = 1 − 2sin2A if the equation already involves sin x. Use 2cos2A − 1 if it involves cos x. The choice that eliminates the squared term you don't want is the right one.
Is there a compound angle formula for tan?
Yes: tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B). It is rarely needed at Higher level — check whether it is required by your specific paper.
Related Higher Maths topics
These topics often appear together in SQA exam questions.
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