Uni-Sine
Australian exams
Australian examsHSC

HSC maths formula sheet (reference sheet): every line and what it leaves out

Published 8 min read

Papers reset in

Formula sheet

The card below opens today's paper with the sheet in a drawer beside it, which is how you will work in the exam hall. The sheet's contents follow, grouped in our own layout. NESA prints them in two columns over three pages, so before you rely on any line, check it against the official reference sheet on the NESA site.

Formula sheet

Year 11 Maths Advanced & Methods

Open formula sheet

The PDF NESA publishes carries a 2020 HSC heading and a November 2019 file name. NESA lists it on its page for the current Stage 6 syllabuses, and it is the sheet the exam specifications refer to when they say a reference sheet is provided.

Trigonometric functions

\sin A=\frac{\text{opp}}{\text{hyp}},\ \cos A=\frac{\text{adj}}{\text{hyp}},\ \tan A=\frac{\text{opp}}{\text{adj}}sine, cosine and tangent in a right-angled triangle
A=\frac{1}{2}ab\sin Carea of a triangle
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}sine rule
c^2=a^2+b^2-2ab\cos Ccosine rule
\cos C=\frac{a^2+b^2-c^2}{2ab}cosine rule for an angle
l=r\thetaarc length
A=\frac{1}{2}r^2\thetaarea of a sector

Trigonometric identities

\sec A=\frac{1}{\cos A},\ \cos A\ne0secant
\text{cosec}\ A=\frac{1}{\sin A},\ \sin A\ne0cosecant
\cot A=\frac{\cos A}{\sin A},\ \sin A\ne0cotangent
\cos^2x+\sin^2x=1Pythagorean identity

Compound angles

\sin(A+B)=\sin A\cos B+\cos A\sin Bsine of a sum
\cos(A+B)=\cos A\cos B-\sin A\sin Bcosine of a sum
\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}tangent of a sum
\sin A=\frac{2t}{1+t^2}t formula for sine, with t = tan(A/2)
\cos A=\frac{1-t^2}{1+t^2}t formula for cosine
\tan A=\frac{2t}{1-t^2}t formula for tangent
\cos A\cos B=\frac{1}{2}\left[\cos(A-B)+\cos(A+B)\right]product of cosines
\sin A\sin B=\frac{1}{2}\left[\cos(A-B)-\cos(A+B)\right]product of sines
\sin A\cos B=\frac{1}{2}\left[\sin(A+B)+\sin(A-B)\right]sine times cosine
\cos A\sin B=\frac{1}{2}\left[\sin(A+B)-\sin(A-B)\right]cosine times sine
\sin^2nx=\frac{1}{2}(1-\cos2nx)sine squared of nx
\cos^2nx=\frac{1}{2}(1+\cos2nx)cosine squared of nx

Statistical analysis

z=\frac{x-\mu}{\sigma}z-score
x<Q_1-1.5\times IQRoutliers, below
x>Q_3+1.5\times IQRoutliers, above
\approx68\%normal distribution: scores with z-scores between -1 and 1
\approx95\%normal distribution: scores with z-scores between -2 and 2
\approx99.7\%normal distribution: scores with z-scores between -3 and 3
E(X)=\muexpected value
\text{Var}(X)=E\left[(X-\mu)^2\right]=E\left(X^2\right)-\mu^2variance

Probability

P(A\cap B)=P(A)P(B)independent events
P(A\cup B)=P(A)+P(B)-P(A\cap B)addition rule
P(A|B)=\frac{P(A\cap B)}{P(B)},\ P(B)\ne0conditional probability
P(X\le r)=\int_a^rf(x)\ dxcontinuous random variable, up to r
P(a<X<b)=\int_a^bf(x)\ dxcontinuous random variable, between a and b

Binomial distribution

P(X=r)={}^nC_rp^r(1-p)^{n-r}probability of r successes
P(X=x)=\binom{n}{x}p^x(1-p)^{n-x},\ x=0,1,\ldots,nthe same in binomial coefficient form
E(X)=npmean
\text{Var}(X)=np(1-p)variance

Differential calculus

Each line gives the function and its derivative.

y=\left[f(x)\right]^n:\quad\frac{dy}{dx}=nf'(x)\left[f(x)\right]^{n-1}power of a function
y=uv:\quad\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}product rule
y=g(u),\ u=f(x):\quad\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}chain rule
y=\frac{u}{v}:\quad\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}quotient rule
y=\sin f(x):\quad\frac{dy}{dx}=f'(x)\cos f(x)sine
y=\cos f(x):\quad\frac{dy}{dx}=-f'(x)\sin f(x)cosine
y=\tan f(x):\quad\frac{dy}{dx}=f'(x)\sec^2f(x)tangent
y=e^{f(x)}:\quad\frac{dy}{dx}=f'(x)e^{f(x)}exponential
y=\ln f(x):\quad\frac{dy}{dx}=\frac{f'(x)}{f(x)}natural logarithm
y=a^{f(x)}:\quad\frac{dy}{dx}=(\ln a)f'(x)a^{f(x)}general exponential
y=\log_af(x):\quad\frac{dy}{dx}=\frac{f'(x)}{(\ln a)f(x)}general logarithm
y=\sin^{-1}f(x):\quad\frac{dy}{dx}=\frac{f'(x)}{\sqrt{1-\left[f(x)\right]^2}}inverse sine
y=\cos^{-1}f(x):\quad\frac{dy}{dx}=-\frac{f'(x)}{\sqrt{1-\left[f(x)\right]^2}}inverse cosine
y=\tan^{-1}f(x):\quad\frac{dy}{dx}=\frac{f'(x)}{1+\left[f(x)\right]^2}inverse tangent

Integral calculus

\int f'(x)\left[f(x)\right]^ndx=\frac{1}{n+1}\left[f(x)\right]^{n+1}+cpower of a function, n not equal to −1
\int f'(x)\sin f(x)dx=-\cos f(x)+csine
\int f'(x)\cos f(x)dx=\sin f(x)+ccosine
\int f'(x)\sec^2f(x)dx=\tan f(x)+csecant squared
\int f'(x)e^{f(x)}dx=e^{f(x)}+cexponential
\int\frac{f'(x)}{f(x)}dx=\ln\left|f(x)\right|+clogarithmic form
\int f'(x)a^{f(x)}dx=\frac{a^{f(x)}}{\ln a}+cgeneral exponential
\int\frac{f'(x)}{\sqrt{a^2-\left[f(x)\right]^2}}dx=\sin^{-1}\frac{f(x)}{a}+cinverse sine form
\int\frac{f'(x)}{a^2+\left[f(x)\right]^2}dx=\frac{1}{a}\tan^{-1}\frac{f(x)}{a}+cinverse tangent form
\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dxintegration by parts
\int_a^bf(x)dx\approx\frac{b-a}{2n}\left[f(a)+f(b)+2\left(f(x_1)+\ldots+f(x_{n-1})\right)\right]trapezoidal rule, with a = x0 and b = xn

Combinatorics

{}^nP_r=\frac{n!}{(n-r)!}permutations
\binom{n}{r}={}^nC_r=\frac{n!}{r!(n-r)!}combinations
(x+a)^n=x^n+\binom{n}{1}x^{n-1}a+\ldots+\binom{n}{r}x^{n-r}a^r+\ldots+a^nbinomial expansion

Vectors

\left|\mathbf{u}\right|=\left|x\mathbf{i}+y\mathbf{j}\right|=\sqrt{x^2+y^2}magnitude
\mathbf{u}\cdot\mathbf{v}=\left|\mathbf{u}\right|\left|\mathbf{v}\right|\cos\theta=x_1x_2+y_1y_2dot product
\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}vector equation of a line

Complex numbers

z=a+ib=r(\cos\theta+i\sin\theta)=re^{i\theta}forms of a complex number
\left[r(\cos\theta+i\sin\theta)\right]^n=r^n(\cos n\theta+i\sin n\theta)=r^ne^{in\theta}de Moivre's theorem

Mechanics

\frac{d^2x}{dt^2}=\frac{dv}{dt}=v\frac{dv}{dx}=\frac{d}{dx}\left(\frac{1}{2}v^2\right)acceleration in four forms
x=a\cos(nt+\alpha)+csimple harmonic motion, cosine form
x=a\sin(nt+\alpha)+csimple harmonic motion, sine form
\frac{d^2x}{dt^2}=-n^2(x-c)simple harmonic motion, acceleration

NESA also prints two special triangles at the top of the trigonometry page: a right-angled isosceles triangle with sides 1, 1 and √2, and a triangle with angles of 30° and 60° and sides 1, 2 and √3. They give the exact values of sin, cos and tan at 30°, 45° and 60°, so you never have to memorise that table.

Which papers give you the sheet

NESA's examination specifications say the reference sheet is provided with each of three papers, and that NESA-approved calculators may be used:

PaperTimeMarksStructure
Mathematics Advanced3 hours plus 10 minutes reading100Section I: 10 multiple choice. Section II: 90 marks, 37 to 42 items
Mathematics Extension 12 hours plus 10 minutes reading70Section I: 10 multiple choice. Section II: 60 marks, 23 to 28 items
Mathematics Extension 23 hours plus 10 minutes reading100Section I: 10 multiple choice. Section II: 90 marks, 37 to 42 items

The same four pages go with all three, so an Extension 2 student sees lines that an Advanced student never needs, such as de Moivre's theorem and the simple harmonic motion forms. Mathematics Standard 1 and Standard 2 are separate: NESA publishes a Standard reference sheet for them, so check which course your paper belongs to.

What is not on the HSC maths formula sheet

Reading the sheet line by line shows how much it leaves out. These are the things to learn by heart:

  • The quadratic formula and the discriminant. Neither is printed.
  • Sequences and series. There are no arithmetic or geometric term or sum formulas, and no formula for the sum to infinity.
  • Logarithm and index laws. The sheet differentiates and integrates logarithms but never states the laws themselves.
  • Mensuration. No area or volume formulas for circles, cones, spheres or prisms, only arc length and sector area.
  • Double-angle identities. The sheet gives the compound-angle formulas, so you derive sin 2A and cos 2A by setting B equal to A.
  • First principles. The limit definition of the derivative is not printed.
  • Straight-line formulas. Gradient, midpoint, distance and point-to-line distance are absent.
  • Standard graphs and exact values. You draw these from knowledge, apart from the two special triangles.

How to use the reference sheet in the exam

  1. During reading time, glance down the sheet so you know which headings exist. Finding a formula in ten seconds beats deriving it in three minutes.
  2. Write the formula you are using at the start of each working line, then substitute. Markers award the method mark for the right formula even when the arithmetic slips.
  3. Match the letters before you substitute. The sheet writes the cosine rule with C opposite c, and the binomial lines with n, r and p.
  4. Use the sheet to check your memory rather than to learn the formula in the exam. A line you have never practised is slow to use.

Example 1: integrating a function of a function

Find\int 6x(3x^2+1)^4\ dx. The integrand has the shape f'(x)[f(x)]ⁿ, with f(x) = 3x² + 1 and f'(x) = 6x, so the first integral line applies:

\int 6x(3x^2+1)^4dx=\frac{1}{4+1}\left(3x^2+1\right)^{4+1}+c=\frac{(3x^2+1)^5}{5}+c

Differentiating back by the chain rule gives 5(3x² + 1)⁴ × 6x ÷ 5 = 6x(3x² + 1)⁴, which confirms it. Practise this pattern in the Calculus course.

Example 2: the binomial line

A fair spinner lands on green with probability 0.3. It is spun 8 times. What is the probability of exactly 2 greens? Here n = 8, p = 0.3 and r = 2, so the binomial probability line gives:

P(X=2)={}^8C_2(0.3)^2(0.7)^6=28\times0.09\times0.117649=0.2965

The sheet supplies the form, but you still choose n, p and r from the wording. The mean of this distribution is np = 2.4 and its variance is np(1 − p) = 1.68.

Example 3: the cosine rule for an angle

A triangle has sides 7, 8 and 9. To find the angle opposite the side of length 9, use the rearranged cosine rule, which the sheet prints in full:

\cos C=\frac{a^2+b^2-c^2}{2ab}=\frac{49+64-81}{2\times7\times8}=\frac{32}{112}=0.2857,\quad C=73.4^\circ

Set the calculator to degrees before you take the inverse cosine, since radians would give 1.28. Check your own triangles with the Trigonometry Calculator.

Practise with the sheet open

Our Year 12 Maths Advanced and Methods paper is a short timed paper of 8 questions worth 25 to 35 marks, a new one every day. It is not a full HSC paper and it is not written by NESA. Sit it with the sheet open, as you would in a trial. For the Extension courses use Maths Extension 2 and Specialist Maths, and for the wider picture read our HSC Maths guide, the Extension 1 guide and the HSC trials guide.

Exam answers

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

Common mistakes with the sheet

  • Using the sheet's line for the wrong course. Standard students have a different sheet, so a Standard formula may not match.
  • Forgetting the constant. The integral lines all end in + c, and a definite integral drops it.
  • Radians against degrees. The arc length and sector area lines l = rθ and A = ½r²θ need θ in radians.
  • Misreading the t formulas. They hold only when t = tan(A/2).

Common questions

Do you get a formula sheet in the HSC maths exam?
Yes. NESA provides a reference sheet with Mathematics Advanced, Extension 1 and Extension 2. Mathematics Standard 1 and 2 have their own reference sheet.
Is the quadratic formula on the HSC maths formula sheet?
No. The sheet has no quadratic formula, no sequences and series formulas and no logarithm laws, so you must learn them.
Is the sheet the same for Advanced, Extension 1 and Extension 2?
Yes. NESA publishes one four-page sheet that is provided with all three papers.
Can I use a calculator in HSC Maths?
NESA's specifications allow NESA-approved calculators in Advanced, Extension 1 and Extension 2.
Where do I find the official HSC maths reference sheet?
On the NESA website under HSC exam papers, titled "Mathematics Advanced, Extension 1 and 2 HSC reference sheet".

Revision cards

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

Where to go next

  • Solve it

    Trigonometry Calculator

    Interactive triangle solver with step-by-step

    Open the tool
  • Learn it

    Calculus

    Differentiation, integration and the mathematics that drives physics and engineering

    Start the course
  • Test it

    Year 12 Maths Advanced & Methods

    A new exam-style paper every day, with instant marking and a mark scheme

    Start today's paper
Read next
Australia · HSCHSC Maths: Courses, 2027 Exams and Practice QuestionsHSC maths explained: Standard, Advanced, Extension 1 and 2, the new syllabus first examined in 2027, exam formats, worked questions and free daily trials.