Who these notes are for
These free revision notes are for students revising Edexcel International A-Level Pure Mathematics 3 (P3, paper code WMA13). They focus on harmonic equations: combining sine and cosine terms into one trigonometric expression and using that form to answer questions.
The 8-page PDF moves from coefficient matching and angle signs to maximum, minimum and range questions, then solving equations in a shifted interval. It includes worked examples, short Try It exercises, common pitfalls, targeted practice and an answers and quick-check page.
8 pages · Worked examples · Targeted practice · Selected answers
What the notes cover
- Rewriting a sin x + b cos x as R sin(x + α) or R cos(x - α)
- Finding R = √(a² + b²) and comparing coefficients
- Checking the signs and quadrants of the angle α
- Maximum, minimum and range, including a constant vertical shift
- Solving harmonic equations using the shifted-angle interval
- Targeted practice, exam-style reasoning and selected answers
This is one P3 topic area, not a complete P3 revision pack. Answers are provided for Sections 1, 2, 3, 5 and 7; the Section 4 Try It exercises are left for self-checking by expansion.
Key ideas at a glance
Find R, then compare coefficients
For R sin(x + α), expansion gives R cos α = a and R sin α = b. With non-zero coefficients, R = √(a² + b²). For R cos(x - α), the coefficient equations change to R sin α = a and R cos α = b. Always expand the form requested in the question.
Check angle signs before using your calculator
The signs of the original coefficients determine the signs of sin α and cos α. For example, 3 sin x - 4 cos x = 5 sin(x - α), with α = tan⁻¹(4/3). Expand your final expression to check that both coefficient signs match.
Read maximum, minimum and range from harmonic form
For unrestricted real x, a sin x + b cos x + c has maximum c + R, minimum c - R and range [c - R, c + R]. The handout example y = 6 + 3 sin x - 4 cos x therefore has range [1, 11]. On a restricted interval, check whether the extrema occur within that interval.
Solve in the shifted interval, then return to x
For 3 sin x + 4 cos x = 2 with 0 ≤ x < 2π, put α = tan⁻¹(4/3) and θ = x + α. Solve sin θ = 0.4 in α ≤ θ < 2π + α. With β = sin⁻¹(0.4), use θ = π - β or 2π + β, then subtract α. The solutions are x ≈ 1.8028 and 5.7674 radians.
About the author
Dr Zhiying Xin is the founder of Study with Dr and a private Maths and Science tutor with nearly seven years of experience, teaching students from 11+ and GCSE through to A-Level and university. She completed her PhD at the University of Manchester.