Overview
What you’ll learn
- Expand and use sin(A ± B), cos(A ± B) and tan(A ± B).
- Derive exact values such as sin 75°, cos 105° and tan 15° from compound angles.
- State and apply the double-angle formulae for sin 2A, cos 2A and tan 2A.
- Express a cos θ + b sin θ in the forms R cos(θ − α) and R sin(θ + α) with R > 0.
- Find maximum and minimum values of expressions like a cos θ + b sin θ + c.
- Solve trigonometric equations involving 2x, compound angles and the R-formula in a given interval.
- Prove identities using compound-angle, double-angle and Pythagorean formulae.
In 30 seconds
- 1Compound angles: sin(A + B) = sin A cos B + cos A sin B and sin(A − B) flips the middle sign.
- 2cos(A + B) = cos A cos B − sin A sin B and cos(A − B) flips the middle sign back to plus.
- 3tan(A + B) = (tan A + tan B)/(1 − tan A tan B); the denominator sign flips for tan(A − B).
- 4Double angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A/(1 − tan²A).
- 5R-formula: a cos θ + b sin θ = R cos(θ − α) with R = √(a² + b²), tan α = b/a; and a sin θ + b cos θ = R sin(θ + α) with tan α = b/a.
- 6Maximum of a cos θ + b sin θ + c is R + c; the minimum is −R + c.
- 7An equation in 2x needs the interval doubled first: 0° ≤ x ≤ 360° means 0° ≤ 2x ≤ 720°, then halve every solution.
- 8Choose the cos 2A form that matches the rest of the expression: sines present → 1 − 2sin²A; cosines present → 2cos²A − 1.
Notes · 1 of 5
Compound-angle expansions
Six formulae that turn sin(A ± B), cos(A ± B) and tan(A ± B) into functions of A and B.
| Formula | Sum form | Difference form |
|---|---|---|
| Sine | sin(A + B) = sin A cos B + cos A sin B | sin(A − B) = sin A cos B − cos A sin B |
| Cosine | cos(A + B) = cos A cos B − sin A sin B | cos(A − B) = cos A cos B + sin A sin B |
| Tangent | 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 sign pattern rewards care: in the sine rows the outer sign matches, in the cosine rows the middle sign is the opposite of the connecting sign, and in the tangent rows the numerator sign matches while the denominator sign flips. Note sin(A + B) is never sin A + sin B — test A = B = 30°: sin 60° = √3/2 but sin 30° + sin 30° = 1.
Compound angles also simplify expressions of known angles: sin(90° − B) = sin 90° cos B − cos 90° sin B = cos B, recovering the complementary-angle facts from one formula.
Notes · 2 of 5
Double-angle formulae
Three forms of cos 2A, and the discipline to pick the right one.
Setting B = A in the compound-angle formulae gives the double-angle formulae: sin 2A = 2 sin A cos A; tan 2A = 2 tan A/(1 − tan²A); and cos 2A in three equivalent forms: cos²A − sin²A, 2cos²A − 1, or 1 − 2sin²A.
The three cos 2A forms are the working toolbox. If an expression contains sines, use 1 − 2sin²A so everything is one name; if it contains cosines, use 2cos²A − 1. Solving cos 2θ = sin θ, for example, becomes 1 − 2sin²θ = sin θ — a quadratic in sin θ. The same substitution powers the identity sin²A = ½(1 − cos 2A), used later in integration.
Notes · 3 of 5
The R-formula: one wave from two
a cos θ + b sin θ collapses to R cos(θ − α); maxima and minima fall out immediately.
Expanding R cos(θ − α) with the compound-angle formula gives R cos θ cos α + R sin θ sin α. Matching coefficients with a cos θ + b sin θ forces R = √(a² + b²) and tan α = b/a. So a cos θ + b sin θ = R cos(θ − α), and the partner form a sin θ + b cos θ = R sin(θ + α) follows the same pattern (tan α = b/a again).
Because cos(θ − α) ranges over [−1, 1], the expression a cos θ + b sin θ has maximum R and minimum −R; adding a constant c shifts both: max R + c, min −R + c. This is the standard route to max/min questions that look nothing like quadratics.
Notes · 4 of 5
Solving harder trigonometric equations
Double angles, shifts inside the argument, and the R-form — same routine, one extra step.
Every equation still follows the Sec 3 routine — rearrange, principal value, reflect for all solutions — with one addition: when the argument is 2x (or x + 45°, or θ − α), the interval transforms with it. For sin 2x = k with 0° ≤ x ≤ 360°, solve for 2x in 0° ≤ 2x ≤ 720° and halve the four answers. For sin(x + 45°) = k, solve for x + 45° in 45° ≤ x + 45° ≤ 405° and subtract 45°.
Mixed-name equations become quadratics through a double-angle substitution. Solve cos 2θ = sin θ for 0° ≤ θ ≤ 360°: cos 2θ = 1 − 2sin²θ turns the equation into 1 − 2sin²θ = sin θ, so 2sin²θ + sin θ − 1 = 0 and (2 sin θ − 1)(sin θ + 1) = 0. Then sin θ = 1/2 gives θ = 30°, 150°, and sin θ = −1 gives θ = 270°.
Notes · 5 of 5
Proving identities with the new formulae
Same one-side discipline; the formula list is now much bigger.
The proof method is unchanged: rewrite one side towards the other, converting double angles and compound angles with the formula list as you go. Two habits pay off: replace every cos 2A with the form that matches the surrounding functions, and turn tangents into sines and cosines the moment the proof stalls.
Revise
Key terms
The words examiners expect you to use precisely.
- Compound angle
- An angle written as a sum or difference, such as A + B; the expansions convert its sine, cosine or tangent into functions of A and B separately.
- Double-angle formula
- A formula for sin 2A, cos 2A or tan 2A in terms of functions of A.
- R-formula
- The rewriting a cos θ + b sin θ = R cos(θ − α) with R = √(a² + b²) and tan α = b/a.
- Amplitude (R-form)
- The value R in the R-formula: the maximum distance of the combined wave from its midline.
- Exact value
- A surd or fraction form such as (√6 + √2)/4, required when the question forbids or does not need decimals.
- Identity
- An equation true for all allowed values; proved by rewriting one side with standard formulae.
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- Complementary angles
- Two angles summing to 90°; the compound-angle formula explains why cos 15° = sin 75°.
- Hidden quadratic
- A trigonometric equation such as cos 2θ + sin θ = 0 that becomes a quadratic equation once a double-angle form is substituted.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Writing sin(A + B) = sin A + sin B.
Sine is not additive over angles: sin 60° = √3/2 but sin 30° + sin 30° = 1.
Instead: Use the expansion sin(A + B) = sin A cos B + cos A sin B.
Mistake: Using sin 2A = 2 sin A.
Doubling the angle does not double the ratio: sin 60° ≈ 0.866 while 2 sin 30° = 1.
Instead: sin 2A = 2 sin A cos A.
Show 8 more mistakesShow fewer mistakes
Mistake: Mixing up tan α = b/a with a/b in the R-formula.
α is the angle whose tangent is the sine-coefficient over the cosine-coefficient: matching R cos(θ − α) = R cos θ cos α + R sin θ sin α to a cos θ + b sin θ gives tan α = b/a.
Instead: For a cos θ + b sin θ = R cos(θ − α): tan α = b/a; for the sine form a sin θ + b cos θ = R sin(θ + α): tan α = b/a again.
Mistake: Taking R = a + b.
R comes from matching coefficients after expanding, which forces R² = a² + b².
Instead: R = √(a² + b²): for 3 and 4, R = 5, not 7.
Mistake: Forgetting to double the interval for equations in 2x.
When x runs to 360°, the angle 2x runs to 720°, so solving over 0° to 360° finds only half the solutions.
Instead: Solve for 2x over 0° ≤ 2x ≤ 720°, collect every solution, then halve.
Mistake: Choosing the wrong cos 2A form and grinding to a dead end.
cos²A − sin²A keeps two names alive when the rest of the expression has one.
Instead: Match the form to the expression: sines around → 1 − 2sin²A; cosines around → 2cos²A − 1.
Mistake: Giving the minimum of a cos θ + b sin θ + c as −R.
The constant c shifts the whole wave, so the minimum is −R + c, not −R.
Instead: Maximum R + c, minimum −R + c: for 4 cos θ + 3 sin θ + 6, max 11, min 1.
Mistake: Dropping the cross term when expanding (sin A + cos A)².
A squared binomial has three terms; omitting 2 sin A cos A destroys the proof.
Instead: (sin A + cos A)² = 1 + 2 sin A cos A = 1 + sin 2A.
Mistake: Quoting only the principal solution of a compound-angle equation.
The shifted interval still contains two or more solutions; sin(x + 45°) = 0.7 has two in 0° ≤ x ≤ 360°.
Instead: Transform the interval with the argument, list every solution, then map back.
Mistake: Rounding α to 1 d.p. and then carrying the rounded angle through the rest of the calculation.
Premature rounding propagates into the solutions and can flip the final decimal place.
Instead: Keep α (and every intermediate value) to several figures; round only final answers to the accuracy demanded.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Expand compound angles
sin(A + B) expands to
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 4Foundation
- 4Standard
- 3Challenging
11 questions worth 58 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 1 Practice Paper
30 marks · 40 min · 4 pages
5 multiple-choice questions and structured questions with answer lines. Chapter 1 (G1 Trigonometric functions, identities and equations — Sec 4 portion).
Mark scheme
Chapter 1 Practice Paper — mark scheme
Answers · Marking points · 2 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
Questions
Frequently asked questions
What does Sec 4 A-Math Chapter 1 cover?
It completes syllabus sub-topic G1 of O-Level 4049 / SEC G3 K341: compound-angle expansions, the double-angle formulae, the R-formula for a cos θ + b sin θ, and the harder equations and identity proofs that use them. The Sec 3 half of G1 — functions, graphs, special angles and simple equations — is Chapter 7 of Sec 3.
How do I choose which cos 2A form to use?
Match the form to the rest of the expression: if sines dominate, use 1 − 2sin²A; if cosines dominate, use 2cos²A − 1; and if you need to cancel a constant such as 1 + cos 2A, the cosine form is usually the one that collapses.
How many solutions does an equation in 2x have between 0° and 360°?
Normally four. The angle 2x runs over 0° to 720°, two full cycles, so a sine or cosine equation usually has two solutions per cycle. Solve over the doubled interval, then halve every answer.
What is the point of the R-formula?
It compresses a cos θ + b sin θ into a single wave R cos(θ − α). The maximum and minimum are then R and −R immediately, and equations reduce to a one-name trigonometric equation in θ − α.
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Chapter 2: Proofs in Plane Geometry