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SACE Stage 2 Specialist Mathematics formula sheet: identities, matrices and calculus

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Formula sheet

The card below opens today's paper with the sheet in a drawer. The contents follow in our own layout. The SACE Board prints the sheet on one page and says it has been provided since November 2018, so check any line against the Board's Specialist Mathematics external assessment page and its past papers.

Formula sheet

Maths Extension 2 & Specialist Maths

Open formula sheet

Circular functions

\sin^2A+\cos^2A=1Pythagorean identity
\tan^2A+1=\sec^2Atangent and secant
1+\cot^2A=\text{cosec}^2Acotangent and cosecant
\sin(A\pm B)=\sin A\cos B\pm\cos A\sin Bsine of a sum or difference
\cos(A\pm B)=\cos A\cos B\mp\sin A\sin Bcosine of a sum or difference
\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}tangent of a sum or difference
\sin2A=2\sin A\cos Adouble angle, sine
\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2Adouble angle, cosine
\tan2A=\frac{2\tan A}{1-\tan^2A}double angle, tangent
2\sin A\cos B=\sin(A+B)+\sin(A-B)product to sum, sine and cosine
2\cos A\cos B=\cos(A+B)+\cos(A-B)product to sum, cosines
2\sin A\sin B=\cos(A-B)-\cos(A+B)product to sum, sines
\sin A\pm\sin B=2\sin\frac{1}{2}(A\pm B)\cos\frac{1}{2}(A\mp B)sum to product, sines
\cos A+\cos B=2\cos\frac{1}{2}(A+B)\cos\frac{1}{2}(A-B)sum to product, cosines
\cos A-\cos B=-2\sin\frac{1}{2}(A+B)\sin\frac{1}{2}(A-B)difference of cosines

Matrices and determinants

\det A=|A|=ad-bcdeterminant, where A has first row a, b and second row c, d
A^{-1}=\frac{1}{|A|}\times\text{the matrix with rows }(d,\ -b)\text{ and }(-c,\ a)inverse of that matrix

Measurement

A=\frac{1}{2}r^2\thetaarea of a sector, θ in radians
l=r\thetaarc length, θ in radians
\text{area}=\frac{1}{2}ab\sin Carea of a triangle ABC
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}sine rule
a^2=b^2+c^2-2bc\cos Acosine rule

Quadratic equations

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}solution of ax² + bx + c = 0

Distance from a point to a plane

\frac{\left|Ax_1+By_1+Cz_1+D\right|}{\sqrt{A^2+B^2+C^2}}distance from (x₁, y₁, z₁) to Ax + By + Cz + D = 0

Derivatives

\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}inverse sine
\frac{d}{dx}\arccos x=\frac{-1}{\sqrt{1-x^2}}inverse cosine
\frac{d}{dx}\arctan x=\frac{1}{1+x^2}inverse tangent

Properties of derivatives

\frac{d}{dx}\left(f(x)g(x)\right)=f'(x)g(x)+f(x)g'(x)product rule
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f'(x)g(x)-f(x)g'(x)}{\left(g(x)\right)^2}quotient rule
\frac{d}{dx}f(g(x))=f'(g(x))g'(x)chain rule

Arc length along a parametric curve

l=\int_a^b\sqrt{\mathbf{v}\cdot\mathbf{v}}\ dtlength of the curve for a ≤ t ≤ b

Integration by parts

\int f'(x)g(x)dx=f(x)g(x)-\int f(x)g'(x)dxformula

Volumes of revolution

V=\int_a^b\pi y^2dxabout the x-axis, where y is a function of x
V=\int_c^d\pi x^2dyabout the y-axis, where y is a one-to-one function of x

How the exam works with the sheet

The Board's 2025 Specialist Mathematics paper lists its materials as the question booklets, the formula sheet and your registration label:

ItemDetail
Subject code2MSC20, on Tuesday 10 November in the 2026 timetable
Total time130 minutes
Total marks100
Formula sheetProvided, one page
NotesTwo unfolded A4 sheets (four sides) of your own handwritten notes
CalculatorsApproved calculators may be used

Because the sheet is so short, your notes carry a lot of weight. Compare the QCE Specialist Maths sheet guide, which gives eight pages, and the WACE Specialist sheet guide, which gives seven.

What is not on the SACE Specialist formula sheet

  • Complex numbers. No polar form, no de Moivre's theorem and no root formula.
  • Vectors. No dot product, cross product, line or plane equations, apart from the point-to-plane distance.
  • Basic derivatives and integrals. Only arcsin, arccos and arctan are given, not x^n, e^x, ln x or the trigonometric functions.
  • Differential equations and growth models. Nothing on separable equations or logistic growth.
  • Statistics and Methods formulas. There is no probability, normal distribution or confidence interval line.
  • Sequences and series, induction and proof. None are printed.
  • Numerical methods. No trapezoidal or Simpson's rule and no Euler's method.

How to use the sheet in the exam

  1. Find the identity you need by its shape: sum, difference, double angle, product or sum to product.
  2. Write the formula, substitute, and show each step of the logic.
  3. For parametric arc length, remember that v · v is the speed squared, so the integrand is the speed.
  4. For volumes of revolution, check the axis: the two lines are written with y² dx and x² dy.

Example 1: a derivative using the table and the chain rule

Find the derivative of arcsin(2x). The sheet gives d/dx arcsin x = 1 ÷ √(1 − x²) and the chain rule, with g(x) = 2x:

\frac{d}{dx}\arcsin(2x)=\frac{1}{\sqrt{1-(2x)^2}}\times2=\frac{2}{\sqrt{1-4x^2}}

The chain rule line is why the arcsin line on its own is enough.

Example 2: distance from a point to a plane

Find the distance from (1, 2, 3) to the plane 2x − y + 2z + 4 = 0. The point-to-plane line gives:

\frac{|2\times1-1\times2+2\times3+4|}{\sqrt{2^2+(-1)^2+2^2}}=\frac{10}{3}=3.33

Make sure the plane is in the form Ax + By + Cz + D = 0 with D on the left-hand side first.

Example 3: integration by parts

Find ∫ x eˣ dx. The sheet writes the rule as ∫ f'(x)g(x) dx, so take f'(x) = eˣ and g(x) = x:

\int xe^xdx=xe^x-\int e^x\cdot1\ dx=xe^x-e^x+c=(x-1)e^x+c

Choosing g as the factor that simplifies when you differentiate it is what makes the method work. Practise related skills in the Calculus course.

Practise with the sheet open

Our Maths Extension 2 and Specialist Maths daily paper is a short timed paper of 8 questions worth 25 to 35 marks. It is not a SACE paper. Practise with a page of your own notes next to the sheet. The Specialist Maths mechanics paper covers the applications.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. They show which line of the sheet each question wanted. We do not publish official answers, past paper answers or leaked papers.

Common questions

Do you get a formula sheet in SACE Specialist Maths?
Yes. The SACE Board provides a one-page formula sheet with the Stage 2 examination.
Can I bring notes into the SACE Specialist exam?
Yes. The Board's exam rules page allows two unfolded A4 sheets (four sides) of your own handwritten notes.
Is the quadratic formula on the sheet?
Yes, with the point-to-plane distance formula beside it.
Are complex number formulas on the sheet?
No. De Moivre's theorem and polar form are not printed.
How long is the exam?
130 minutes, 100 marks.

Revision cards

Say the answer first, then flip the card to check it.

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