Additional Mathematics Formula Sheet

A formula sheet can save memory. It cannot make the mathematical decision for you.

That distinction is one of the quiet dividing lines in Additional Mathematics. A student can know exactly where the quadratic formula is printed and still not recognise that a discriminant question is really asking about the number of intersections. Another can see the Binomial Theorem on the page and still choose the wrong value of r. A third can read a trigonometric identity correctly but use it in a direction that makes the expression harder.

The formula sheet is therefore not a substitute for mathematical knowledge. It is an interface. It places a small set of high-value relationships in front of the candidate so that memory is not carrying every symbol at once. The candidate still has to interpret the question, choose a relationship, respect its conditions, substitute correctly, preserve exactness where needed and show enough working for the argument to be followed.

This guide is written for the Singapore Additional Mathematics route. For the 2026 GCE O-Level examination, the subject is Additional Mathematics 4049. From 2027, the corresponding SEC G3 subject is K341. The published 2026 and 2027 syllabuses use the same assessment architecture and both state that relevant mathematical formulae are provided to candidates.

Official references: 2026 O-Level Additional Mathematics 4049 syllabus · 2027 SEC G3 Additional Mathematics K341 syllabus.

Wait, What? The Sheet Gives a Formula, Not a Route

Suppose the question says that the line y = 2x + k is tangent to the curve y = x² − 4x + 9, and asks for possible values of k. The formula page contains the quadratic formula. That does not tell you the first move.

The first move comes from meaning. A tangent gives one repeated intersection. Equating the line and curve creates a quadratic equation in x. A repeated root means the discriminant is zero. Only after that structural decision do the printed formulas become useful.

This is the recurring pattern:

Question → mathematical object → condition → route → formula → substitution → interpretation → check.

Students who jump directly from “I see a formula” to “I should use it” often reverse the order. They let the sheet choose the topic instead of letting the question choose the mathematics.

What the Published Formula Page Actually Gives

The 2026 O-Level 4049 and 2027 SEC G3 K341 syllabuses each contain a mathematical-formulae page. It includes several relationships that matter repeatedly across the subject.

Quadratic equation: the standard formula for solving ax² + bx + c = 0.
Binomial expansion: the positive-integer expansion of (a+b)ⁿ, including combination notation and the general term.
Trigonometric identities: Pythagorean identities, angle-sum and angle-difference identities, and double-angle identities.
Triangle formulae: sine rule, cosine rule and the area formula ½bc sin A.

That is a powerful set. But it is deliberately not the whole syllabus. The subject still expects students to understand algebra, functions, coordinate geometry, logarithms, calculus, proof, modelling and the conditions under which different methods apply.

The most useful way to read the sheet is not “What can I avoid learning?” It is “Which relationships are supplied, and what thinking still has to happen around them?”

Formula Sheet Literacy Has Four Levels

Level 1: Recognition

You know what the printed formula is called. You can find the quadratic formula, the Binomial Theorem and the trigonometric identities quickly.

Level 2: Meaning

You know what the symbols represent. In the quadratic formula, a, b and c are coefficients of a quadratic equation written in standard form. In the sine rule, the side a is opposite angle A. In a binomial term, r counts selections and must be an allowed integer.

Level 3: Selection

You know when the formula is useful. A sine rule is not chosen because a triangle appears. It is chosen when the known and required quantities fit an opposite side-angle relationship. A cosine rule becomes useful when the geometry provides the appropriate side-angle structure.

Level 4: Integration

You can use the formula as one step inside a longer route. A quadratic formula result may feed a coordinate-geometry interpretation. A trigonometric identity may simplify an integral or equation. A binomial coefficient may be only one term inside a product.

Examination-ready formula literacy reaches Level 4. The page is not merely a lookup surface; it becomes a reliable component inside multi-step reasoning.

The Quadratic Formula: What Is Given and What Is Not

The sheet gives the symbolic solution formula. It does not identify the coefficients for you. It does not decide whether factorisation would be faster. It does not tell you what the roots mean in the original problem.

Consider:

3x² + 7 = 10x.

Before using the formula, rewrite it as 3x² − 10x + 7 = 0. Then a = 3, b = −10 and c = 7. The sign on b is part of the coefficient. Substituting b = 10 because “10 is visible” corrupts the formula before the calculator is touched.

The formula also does not replace discriminant reasoning. The expression b² − 4ac appears inside the formula, but the student must know what its sign implies for real roots and how that maps to intersections or tangency.

Formula supplied: symbolic solution. Student responsibility: standard form, coefficient identification, condition recognition, exactness, interpretation and checking.

The Binomial Formula: What Is Given and What Is Not

The formula page gives the finite positive-integer expansion and the general term. It does not decide which term contains the requested power.

For example, to find the coefficient of x⁴ in (2x − 1)⁶, a student still has to reason that the power of x in a general term is 6 − r. Setting 6 − r = 4 gives r = 2. Only then does the printed combination structure become directly useful.

A student who looks at x⁴ and writes r = 4 has not forgotten the formula. The failure happened earlier, at interpretation.

Use the full worked route in Binomial Expansion in Additional Mathematics when coefficient selection, constant terms or mixed products need repair.

Trigonometric Identities: A Library Is Not a Search Strategy

The formula page lists several identities. The difficulty is rarely “Is there an identity somewhere?” The difficulty is choosing a direction that simplifies the structure.

Suppose a proof contains 1 − cos²x. The identity sin²x + cos²x = 1 makes 1 − cos²x = sin²x immediately available. But if the target contains tangent and secant, a different identity may be strategically better.

A printed identity has two jobs: it confirms the relationship and reduces memory load. It does not inspect the expression for you.

Before substituting an identity, ask:

Which side is structurally more complicated?
Which functions dominate the target?
Can one identity reduce the number of function families?
Will the substitution create a useful factor or denominator?

This is why identity work is closer to algebraic route selection than to formula recall.

Sine Rule and Cosine Rule: Read the Triangle Before the Formula

Students often learn “two sides and an angle” or “two angles and a side” as surface cues. A more reliable method is to mark the opposite pairs.

If side a is opposite angle A, then the sine-rule ratio links them. If the data do not create a usable opposite pair, the sine rule may not yet be the right route.

The cosine rule is especially useful when the structure relates three sides and an included angle, or when three sides are known and an angle is required. But again, the formula does not label the diagram. The candidate must identify which side lies opposite the chosen angle.

Formula errors are often diagram-reading errors wearing algebraic clothing.

What Is Not Solved by the Formula Page

The published formula page is not a complete memory dump of Additional Mathematics. Several major capabilities still depend on learned structure and judgement.

Completing the square and interpreting turning points.
Choosing and applying factor and remainder theorem routes.
Manipulating surds and partial fractions.
Using logarithm laws and understanding exponential-logarithmic inverse structure.
Transforming relationships into linear form.
Recognising graph transformations and function structure.
Differentiation and integration techniques.
Using derivatives for tangents, normals, stationary points, optimisation and connected rates.
Using integrals for accumulation, area and kinematics.
Building geometry and trigonometry proofs.
Interpreting models, restrictions and answers in context.

This does not mean “memorise everything else as isolated formulas”. Many of these are procedures and relationships that should be understood well enough to reconstruct or recognise.

The Formula Sheet Should Change Your Revision Strategy

If a relationship is provided, revision time should move away from copying it repeatedly and toward using it accurately in changed contexts.

For each supplied formula, practise four kinds of questions:

Direct: substitute cleanly.
Hidden: recognise when the formula applies without the topic named.
Reverse: use the relationship to recover an unknown parameter or condition.
Integrated: use the formula inside a longer cross-topic route.

For example, the quadratic formula can appear in a routine solve, a line-curve intersection, a parameter problem or a model. The symbol pattern may be familiar while the mathematical job changes.

Build a Formula-Sheet Map Before the Examination

Do not wait until the paper to discover the layout of the supplied page. During revision, keep the official formula page beside selected mixed questions and build a small mental map.

Printed area	What it supplies	What you must decide
Quadratic equation	root formula	standard form, coefficients, whether roots are the right target
Binomial expansion	expansion and general term	which term/power is required, all contributing terms
Trig identities	relationships among functions	which identity and which direction simplify the task
Triangle formulae	sine/cosine/area relationships	diagram labels, opposite pairs, valid geometric route

The objective is location without search cost. Looking up a formula should take seconds, not interrupt the mathematical thread.

Under Time: The 20-Second Formula Decision

When a formula may be useful, run a fast check:

Name the object. Quadratic? Triangle? Binomial? Identity?
Name the target. Root? Coefficient? Angle? Proof step?
Name the condition. Tangency? Exact term? Opposite pair? Required interval?
Choose the formula.
Write the substitution before calculating.

This prevents a calculator-first response. The first written line should often make the structure visible enough that a later numerical error can be diagnosed.

Calculator Use Does Not Remove Formula Responsibility

The 2026 and 2027 G3 syllabuses state that an approved calculator may be used in both papers. That changes execution speed, not the need for mathematical judgement.

A calculator can evaluate a discriminant, a trigonometric ratio or a binomial coefficient. It cannot tell whether the wrong angle was selected, whether the equation was entered in the wrong mode, whether a negative sign belonged inside a square, or whether an extraneous solution violates a restriction.

Use the calculator as an executor and checker. Keep the mathematical route visible outside it.

Exactness and Rounding: The Formula Sheet Does Not Decide When to Approximate

The official examination notes state that non-exact numerical answers should generally be given to 3 significant figures, or 1 decimal place for angles in degrees, unless a different accuracy is specified. If a question explicitly asks for an answer to be shown correct to a particular accuracy, the answer must first be shown to a higher degree of accuracy.

That creates a practical rule: keep exact forms and calculator precision alive until the problem is ready to finish.

If a surd, fraction or exact trigonometric value will feed a later step, premature rounding can change the final result. A provided formula may be exact even when the final answer is not.

Original Worked Example: Formula Present, Decision Hidden

Question: The curve y = x² − 6x + 5 and the line y = mx − 4 meet at exactly one point. Find the possible value of m.

Equate the expressions:

x² − 6x + 5 = mx − 4

x² − (m+6)x + 9 = 0.

Exactly one intersection means a repeated root, so the discriminant is zero:

(m+6)² − 4(1)(9) = 0.

Hence (m+6)² = 36, so m+6 = ±6. Therefore m = 0 or m = −12.

The quadratic formula was available, but using it directly was not the most economical route. The decisive idea was the repeated-root condition. Formula-sheet literacy includes knowing when not to use the most visible formula.

Original Worked Example: Formula Is Only the Middle

Question: In triangle ABC, a = 7, b = 10 and C = 42°. Find side c, then explain why the result should be less than 10.

The cosine rule gives:

c² = 7² + 10² − 2(7)(10)cos42°.

After calculation, c is approximately 6.71. The formula produced the value; the geometry checks it. Since the included angle is acute and the two sides are 7 and 10, a side a little below 7 is plausible here. More generally, the triangle inequality also requires 3 < c < 17.

A complete learner uses formula, interpretation and reasonableness together.

Formula-Sheet Errors to Diagnose

Error	What happened	Repair
Lookup dependence	student knows location but not meaning	explain every symbol and condition
Formula matching	surface words trigger a formula prematurely	name object and target first
Coefficient error	equation not standardised before substitution	rewrite and label coefficients
Diagram mismatch	sides/angles paired incorrectly	mark opposite relationships first
Identity sprawl	many substitutions without simplification direction	choose target function family
Calculator opacity	only final number is visible	write substitution and intermediate structure
Premature approximation	rounded value reused downstream	retain exact/calculator precision

A Seven-Day Formula-Sheet Training Loop

Day 1: annotate the official formula page. Beside each relationship, write one sentence saying when it is useful.

Day 2: complete direct substitution questions without looking at notes other than the official sheet.

Day 3: complete hidden questions where the topic heading is removed.

Day 4: complete reverse questions: recover parameters, conditions or missing information.

Day 5: complete mixed questions where a supplied formula is only one step.

Day 6: complete a timed set with the official formula page beside you. Record every moment you looked at it and why.

Day 7: retest the errors without the previous solutions. The aim is not to stop using the sheet; it is to stop depending on it for decisions it cannot make.

What Parents and Tutors Should Notice

If a student says, “I know the formulas but I still cannot do the paper,” that can be a highly informative statement. The difficulty may sit in AO2-style interpretation and connection rather than AO1 recall. More formula drilling may add little.

Bring a marked paper or attempted question. Ask where the route first became uncertain. Was the formula unknown, or was the student unable to recognise when it applied? Did the student identify the correct formula but misread the condition? Did calculator execution hide an algebraic sign error?

The diagnosis should occur before assigning another hundred formula exercises.

Evidence Boundary

This guide is original eduKate teaching material. The official syllabuses establish the formula page, assessment notes and subject scope. The practice routines, worked examples, diagnostic labels and timing suggestions here are teaching scaffolds, not an official marking scheme or prediction of future questions.

Students should always use the formula page and instructions issued for their actual examination year and subject level.

Continue the Examination Interface Layer

The Quiet Return

The sheet supplies relationships. The question supplies the job. Your mathematics must connect the two.

Additional Mathematics Formula Sheet | What Is Given, What Is Not, and How to Use It Under Time