A Level · Cambridge (CIE) · 9709

Expand compound angles and evaluate exact trigonometric values

Choose the correct signs in addition and subtraction formulae, simplify angle-shifted expressions and keep exact surd values.

Section 3.3 · Trigonometry

Pure Mathematics 2 & 3 · Trigonometry · 2.3 / 3.3

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027. Paper 1 knowledge is assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Shared sections 2.3 and 3.3 cover all six trigonometric functions, reciprocal squared identities, compound and double angles, and amplitude-phase forms used in simplification, exact evaluation and equations. State angle units, keep original domains and find every solution in the supplied interval. Written practice is self-checked.

Before you begin

Use exact 30°, 45° and 60° values and simplify surds.
Keep denominator restrictions while manipulating trigonometric expressions.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which expansion is correct for cos(A + B)?

What you will learn

Use expansions for sin(A ± B), cos(A ± B) and tan(A ± B) with valid domains.
Select suitable special-angle combinations for exact values and simplify shifted expressions.

Addition and subtraction have different sign patterns

The sine formulae are sin(A + B) = sin A cos B + cos A sin B and sin(A − B) = sin A cos B − cos A sin B. For cosine, cos(A + B) = cos A cos B − sin A sin B and cos(A − B) = cos A cos B + sin A sin B. The sign in a cosine expansion is opposite to the angle's written sign. These sine and cosine identities are valid for all real angles, with both inputs measured in the same unit.

Where the terms are defined, tan(A + B) = (tan A + tan B)/(1 − tan A tan B) and tan(A − B) = (tan A − tan B)/(1 + tan A tan B). For these expressions require cos A ≠ 0, cos B ≠ 0 and the relevant denominator non-zero; the original tangent of the combined angle must also be defined. The zero denominator for A = B = 45° correctly signals that tan 90° is undefined. If an individual tangent is undefined but the combined angle's tangent is valid, use sine and cosine expansions instead of this quotient formula.

Expand before collecting coefficients

To evaluate sin 75° exactly, use 75° = 45° + 30°. The addition formula gives (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4. For tan 15°, use 45° − 30° to obtain (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1) = 2 − √3. Using tangent addition instead gives tan 75° = (1 + 1/√3)/(1 − 1/√3) = (√3 + 1)/(√3 − 1) = 2 + √3, with non-zero denominator. Rationalisation preserves an exact value; an early decimal approximation would obscure the surd result.

For cos(x − 30°) − 3 sin(x − 60°), expand both shifted functions in degrees. The first is (√3/2)cos x + (1/2)sin x. The second, including its −3 multiplier, is −(3/2)sin x + (3√3/2)cos x. Collecting gives 2√3 cos x − sin x. Distribute an outside negative multiplier across both terms before combining, and keep x in degree measure throughout this identity.

Pause and explain

For A = B = 45°, the denominator of the tangent addition formula is zero. What does this mean?

Put the idea to work

Worked example

Simplify cos(x − 30°) − 3 sin(x − 60°) into a combination of sin x and cos x, with x measured in degrees.

Show the worked solution

Expand cos(x − 30°) as (√3/2)cos x + (1/2)sin x.
Expand −3 sin(x − 60°) as −(3/2)sin x + (3√3/2)cos x.
Collect the cosine and sine coefficients separately to obtain 2√3 cos x − sin x.

Answer 2√3 cos x − sin x, for degree input x.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Find sin 75° exactly using 45° + 30°.

Give me a hint

Use the sine addition formula and keep the special-angle values as surds.

Compare my reasoning

Write sin 75° = sin 45° cos 30° + cos 45° sin 30°.
Substitute to get (√2/2)(√3/2) + (√2/2)(1/2).
Combine over denominator 4 to obtain (√6 + √2)/4.

sin 75° = (√6 + √2)/4.

Look for these in your work

Choose the correct addition formula and signs.
Retain an exact surd result rather than round its terms.

2 · Independent

Find tan 15° exactly using 45° − 30°.

Give me a hint

The tangent subtraction denominator has a plus sign; rationalise only after forming the full quotient.

Compare my reasoning

Use (tan 45° − tan 30°)/(1 + tan 45° tan 30°) = (1 − 1/√3)/(1 + 1/√3).
Multiply numerator and denominator by √3 to get (√3 − 1)/(√3 + 1).
Rationalise with √3 − 1 to obtain (4 − 2√3)/2 = 2 − √3; the original denominator is non-zero.

tan 15° = 2 − √3.

Look for these in your work

Use the subtraction formula's positive denominator sign.
Check the domain and simplify the exact quotient correctly.

3 · Transfer

A unit vector initially points at 30° from the positive x-axis and is rotated anticlockwise by 45°. Find its new exact coordinates and verify its length.

Give me a hint

The new direction is 75°; use the compound cosine and sine formulae for the two coordinates.

Compare my reasoning

The new angle is 30° + 45° = 75°, so its coordinates are (cos 75°, sin 75°).
The exact pair is ((√6 − √2)/4, (√6 + √2)/4).
The sum of their squares is [(8 − 4√3) + (8 + 4√3)]/16 = 1, retaining unit length.

((√6 − √2)/4, (√6 + √2)/4); squared length 1.

Look for these in your work

Connect rotation to a compound angle and the correct coordinate order.
Check the exact squared length without rounding the surds.

Common mistakes

The cosine addition formula has a minus sign, while its subtraction formula has a plus sign.
Use one consistent angle unit in each compound expression.
A tangent quotient formula needs both individual tangents and its denominator to be defined.

Recall without your notes

What can you explain now?

Find cos 75° exactly by expanding 45° + 30°.

Compare with the explanation

(√6 − √2)/4.

Cosine addition subtracts the product of the two sines from the product of the two cosines.

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Make it stick

Write all six compound-angle formulae with their signs. Give one exact-value calculation and explain a case where the tangent quotient representation cannot be used.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.