A-Level Maths trigonometry revision: identities, equations, R-form
A-Level trigonometry covers radians and circular measure, all six trig functions and their graphs, the core identities (sin²θ + cos²θ = 1 and the double-angle and addition formulae), solving equations over a given range, expressing a cos θ + b sin θ as R cos(x − α), and small-angle approximations. Almost every exam question reduces to: pick the right identity, keep every solution in the required range, and keep your calculator in radians.
Trigonometry is the topic students most often describe as 'memorisation', which is a misdiagnosis — the exam rarely asks you to recall anything directly. It asks you to transform an expression until it becomes solvable, and the identities are just the legal moves. Learn them as tools with a purpose and the topic shrinks dramatically.
The specification, in plain English
| Area | What's in it | Where it bites |
|---|---|---|
| Radians | Arc length rθ, sector area ½r²θ, all trig in radians | Degree mode on the calculator ruins everything |
| All six functions | sin, cos, tan, sec, cosec, cot — graphs, domains, ranges | sec/cosec/cot identities in Year 2 questions |
| Identities | sin²+cos²=1, 1+tan²=sec², addition and double-angle formulae | Choosing which identity opens the question |
| Equations | sin θ = k over a range, quadratic-in-cos, transformed equations | Missing solutions outside the principal value |
| R-form | a cos θ + b sin θ = R cos(θ − α) | The phase angle's sign; using it to solve equations |
| Small angles | sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ | Valid in radians only; 'show that' proofs |
The identities, organised by what they're for
- sin²θ + cos²θ = 1 — turns squares into each other; the first thing to try when an equation mixes sin and cos.
- tan θ = sin θ/cos θ — converts tan into something you can solve; the reason equations in tan reduce nicely.
- sin 2θ = 2 sin θ cos θ — opens equations with a product of sin and cos.
- cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ — the version you pick depends on which function you want to keep.
- The addition formulae sin(A ± B) and cos(A ± B) — the engine inside R-form and most 'show that' proofs.
Worked questions
Solve 2 sin²θ = 1 + cos θ for 0 ≤ θ ≤ 2π.
- Convert the sin² using the identity: 2(1 − cos²θ) = 1 + cos θ.
- Rearrange to a quadratic in cos θ: 2cos²θ + cos θ − 1 = 0.
- Factorise: (2cos θ − 1)(cos θ + 1) = 0, so cos θ = ½ or cos θ = −1.
- cos θ = ½ gives θ = π/3 and 5π/3; cos θ = −1 gives θ = π.
- Check the range — all three solutions lie in [0, 2π]. Examiners deduct for missing solutions and for extras outside the range.
θ = π/3, π, 5π/3. The standard move: one identity converts a mixed equation into a quadratic you already know how to solve.
Express 3 sin θ + 4 cos θ in the form R sin(θ + α), and hence find the maximum value and the smallest positive θ at which it occurs.
- R = √(3² + 4²) = 5. Write 3 sin θ + 4 cos θ = 5 sin(θ + α).
- Expand: 5 sin(θ + α) = 5 sin θ cos α + 5 cos θ sin α. Match coefficients: 5 cos α = 3, 5 sin α = 4.
- tan α = 4/3, so α ≈ 0.927 rad.
- Maximum of sin is 1, so the maximum value is 5.
- It occurs when θ + α = π/2, so θ = π/2 − 0.927 ≈ 0.644 rad.
5 sin(θ + 0.927); maximum 5 at θ ≈ 0.644 rad. 'Hence' means you must use the R-form — solving numerically earns nothing.
Show that for small θ, (1 − cos θ)/θ² ≈ ½.
- Use the small-angle approximation cos θ ≈ 1 − θ²/2.
- Substitute: 1 − cos θ ≈ 1 − (1 − θ²/2) = θ²/2.
- Divide: (θ²/2)/θ² = ½.
- Note the approximation is only valid because θ is in radians — worth stating in a 'show that'.
(1 − cos θ)/θ² ≈ ½. This is the approximation that makes the derivative of sin θ work — exams love it.
Where marks get dropped
- Calculator in degrees — every A-Level trig question is in radians, and degrees produce confidently wrong answers.
- Only the principal solution — cos θ = ½ has two solutions in [0, 2π], and the mark scheme wants both.
- In R-form, getting α from tan α = b/a without checking the quadrant or the requested form (sin vs cos).
- Using the wrong cos 2θ version — pick the one that leaves only the function already in the equation.
- Forgetting solutions created by dividing through by sin θ or cos θ — dividing loses the solutions where that factor is zero.
- Small-angle approximations applied with θ in degrees — meaningless, and examiners test that you know it.
Your trigonometry checklist
- My calculator lives in radian mode and I check it before every paper.
- I can sketch all six trig graphs with their key values in radians.
- I know which cos 2θ version to pick for a given equation.
- I find every solution in the given range, using symmetry, not just the calculator's first answer.
- I can derive and use R sin(θ + α) and R cos(θ − α) for maxima and equations.
- I can use small-angle approximations in 'show that' questions and state the radians condition.
- I never divide through by a trig function without checking for lost solutions.
Trig is the topic where one missing identity makes a question look impossible — and one hint makes it fall apart. If you keep seeing mark-scheme moves you would never have tried, that is the specific thing Lumi is good at: it shows the move, explains why that identity was the door, and sets another door for you to find.
What to take from this
- All trig at A-Level is in radians — check your calculator mode before anything else.
- Three identities do most of the work: sin²θ + cos²θ = 1, tan θ = sin θ/cos θ, and the double-angle formulae.
- Equation questions always give a range — find every solution inside it, not just the principal value.
- R-form converts a cos θ + b sin θ into one function, which makes maxima, minima and equations trivial.
- Small-angle approximations (sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ) are only valid in radians.
Questions people also ask
Because calculus on trig functions only works in radians — the derivative of sin θ is cos θ only when θ is measured in radians. Degrees are a human convenience; radians are the natural unit, and the whole of A-Level trig assumes them.
The formula booklet contains the addition and double-angle formulae, but sin²θ + cos²θ = 1, tan θ = sin θ/cos θ and the small-angle approximations should be instant. The exam's difficulty is in knowing which identity to reach for, which no booklet tells you.
Get the principal value from your calculator, then use the graph's symmetry — for sin, the second is π − θ; for cos, it is 2π − θ; for tan, add π repeatedly. Then check whether a shifted argument (like 2θ or θ − π/6) expands the range before you apply it.
It merges a cos θ + b sin θ into a single trig function, which makes the maximum, minimum and equation-solving immediate — you cannot easily see the max of 3 sin θ + 4 cos θ, but 5 sin(θ + 0.927) obviously peaks at 5.