Measurement
Standard
Advanced
Ext 1
Ext 2
Arc Length
\[ l = \frac{\theta}{360} \times 2\pi r \]
Length of an arc when \(\theta\) is in degrees. For radians, use \(l = r\theta\).
Common mistake: Mixing up degrees and radians. If \(\theta\) is in radians, the formula simplifies to \(l = r\theta\).
Sector Area
\[ A = \frac{\theta}{360} \times \pi r^2 \]
Area of a sector when \(\theta\) is in degrees. For radians: \(A = \frac{1}{2}r^2\theta\).
Area of a Trapezium
\[ A = \frac{h}{2}(a + b) \]
Where \(a\) and \(b\) are the parallel sides and \(h\) is the perpendicular height between them.
Surface Area of a Cylinder
\[ A = 2\pi r^2 + 2\pi r h \]
The \(2\pi r^2\) covers both circular ends; the \(2\pi r h\) is the curved surface unrolled into a rectangle.
Surface Area of a Sphere
\[ A = 4\pi r^2 \]
Exactly four times the area of a great circle of the sphere.
Volume of a Pyramid/Cone
\[ V = \frac{1}{3}Ah \]
Where \(A\) is the area of the base and \(h\) is the perpendicular height. Works for cones (\(A = \pi r^2\)) and any pyramid.
Common mistake: Using the slant height instead of the perpendicular height.
Volume of a Sphere
\[ V = \frac{4}{3}\pi r^3 \]
Remember: surface area is \(4\pi r^2\) (derivative of volume with respect to \(r\)).
Area of a Triangle (Sine Formula)
\[ A = \frac{1}{2}ab \sin C \]
Use when you know two sides and the included angle. Much faster than Heron's formula.
Example: If \(a=5\), \(b=8\), \(C=30°\): \(A = \frac{1}{2}(5)(8)\sin 30° = 10\).
Financial Mathematics
Standard
Advanced
Ext 1
Compound Interest
\[ A = P(1+r)^n \]
\(A\) = final amount, \(P\) = principal, \(r\) = rate per period, \(n\) = number of periods. Make sure \(r\) and \(n\) use the same period (e.g. monthly).
Common mistake: Using the annual rate with monthly periods without dividing \(r\) by 12.
Simple Interest
\[ I = Prn \]
\(I\) = interest earned, \(P\) = principal, \(r\) = rate per period (as a decimal), \(n\) = number of periods. Total amount = \(P + I\).
Straight-Line Depreciation
\[ S = V_0 - Dn \]
\(S\) = salvage value, \(V_0\) = initial value, \(D\) = depreciation per period, \(n\) = periods elapsed.
Arithmetic Sequence — nth Term
\[ T_n = a + (n-1)d \]
\(a\) = first term, \(d\) = common difference. Each term adds \(d\) to the previous.
Arithmetic Series — Sum of n Terms
\[ S_n = \frac{n}{2}\big[2a + (n-1)d\big] = \frac{n}{2}(a + l) \]
Two forms: using first term \(a\) and common difference \(d\), or using first term \(a\) and last term \(l\).
Geometric Sequence — nth Term
\[ T_n = a\,r^{n-1} \]
\(a\) = first term, \(r\) = common ratio. Each term multiplies by \(r\).
Geometric Series — Sum of n Terms
\[ S_n = \frac{a(1-r^n)}{1-r} = \frac{a(r^n-1)}{r-1}, \quad r \ne 1 \]
Use the first form when \(|r| < 1\), the second when \(|r| > 1\). They are algebraically identical.
Sum to Infinity (Geometric)
\[ S = \frac{a}{1-r}, \quad |r| < 1 \]
Only converges when the common ratio \(|r| < 1\). If \(|r| \ge 1\), the series diverges (no finite sum).
Common mistake: Forgetting to check \(|r| < 1\) before using this formula.
Algebra & Functions
Advanced
Ext 1
Ext 2
Quadratic Formula
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Solves \(ax^2 + bx + c = 0\). The discriminant \(\Delta = b^2 - 4ac\) tells you the nature of the roots.
Discriminant: \(\Delta > 0\) → two real roots; \(\Delta = 0\) → one repeated root; \(\Delta < 0\) → two complex roots.
Roots of a Cubic (Vieta's Formulas)
\[ \begin{aligned} \alpha + \beta + \gamma &= -\frac{b}{a} \\ \alpha\beta + \alpha\gamma + \beta\gamma &= \frac{c}{a} \\ \alpha\beta\gamma &= -\frac{d}{a} \end{aligned} \]
For \(ax^3 + bx^2 + cx + d = 0\) with roots \(\alpha, \beta, \gamma\). Sum, sum of products in pairs, and product of roots.
Equation of a Circle
\[ (x-h)^2 + (y-k)^2 = r^2 \]
Centre \((h, k)\), radius \(r\). Complete the square to convert from general form \(x^2 + y^2 + Dx + Ey + F = 0\).
Logarithm Laws
\[ \log_a a^x = x = a^{\log_a x} \]
The fundamental identity linking logs and exponentials. Also: \(\log_a(xy) = \log_a x + \log_a y\), \(\log_a(x^n) = n\log_a x\).
Change of Base
\[ \log_a x = \frac{\log_b x}{\log_b a} \]
Convert any logarithm to a different base. Most useful with \(b = 10\) or \(b = e\).
Exponential Identity
\[ a^x = e^{x \ln a} \]
Converts any exponential to base \(e\). Essential for differentiating \(a^x\).
Trigonometric Functions
Advanced
Ext 1
Ext 2
Right-Angled Triangle Ratios
\[ \sin A = \frac{\text{opp}}{\text{hyp}}, \quad \cos A = \frac{\text{adj}}{\text{hyp}}, \quad \tan A = \frac{\text{opp}}{\text{adj}} \]
SOH-CAH-TOA. Only works for right-angled triangles. For non-right-angled triangles, use sine or cosine rule.
Sine Rule
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Use when you know: two angles and a side, or two sides and a non-included angle.
Common mistake: The ambiguous case — when given two sides and a non-included angle, there may be two valid triangles.
Cosine Rule
\[ c^2 = a^2 + b^2 - 2ab\cos C \]
Use when you know: three sides, or two sides and the included angle. Rearranges to find any angle: \(\cos C = \frac{a^2+b^2-c^2}{2ab}\).
Arc Length (Radians)
\[ l = r\theta \]
When \(\theta\) is in radians. Much simpler than the degree version.
Sector Area (Radians)
\[ A = \frac{1}{2}r^2\theta \]
When \(\theta\) is in radians. Cleaner than the degree version: \(\frac{\theta}{360}\pi r^2\).
Pythagorean Identity
\[ \cos^2 x + \sin^2 x = 1 \]
The most important trig identity. Divide by \(\cos^2 x\) to get \(1 + \tan^2 x = \sec^2 x\).
Reciprocal Identities
\[ \sec A = \frac{1}{\cos A}, \quad \cosec A = \frac{1}{\sin A}, \quad \cot A = \frac{\cos A}{\sin A} \]
Domain restrictions: \(\cos A \ne 0\) for sec, \(\sin A \ne 0\) for cosec and cot.
Compound & Double Angle Formulas
Ext 1
Ext 2
Sine of Sum/Difference
\[ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \]
The sign is the same on both sides for sine: plus stays plus, minus stays minus.
Cosine of Sum/Difference
\[ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \]
The sign flips: \(\cos(A+B)\) has a minus, \(\cos(A-B)\) has a plus.
Common mistake: Getting the sign wrong. Remember: cosine changes, sine keeps.
Tangent of Sum/Difference
\[ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \]
The sign in the denominator is opposite to the numerator. Same pattern as cosine.
Double Angle — Sine
\[ \sin 2A = 2\sin A \cos A \]
Derived from \(\sin(A+A)\). Used extensively in integration and proof.
Double Angle — Cosine (3 forms)
\[ \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A \]
Choose the form based on what you need. For integrating \(\cos^2 x\), use the middle form. For \(\sin^2 x\), use the last form.
Key: \(\cos^2 x = \frac{1+\cos 2x}{2}\) and \(\sin^2 x = \frac{1-\cos 2x}{2}\) — essential for integration.
t-Formula Substitution
\[ \text{If } t = \tan\frac{A}{2}: \quad \sin A = \frac{2t}{1+t^2}, \quad \cos A = \frac{1-t^2}{1+t^2}, \quad \tan A = \frac{2t}{1-t^2} \]
Converts all trig functions to rational expressions in \(t\). Useful for integration and solving trig equations.
Product-to-Sum Formulas
\[ \sin A \cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)] \]
Converts products of trig functions to sums. The reverse direction (sum-to-product) is useful for solving equations.
Power-Reduction Formulas
\[ \sin^2 nx = \frac{1}{2}(1 - \cos 2nx), \quad \cos^2 nx = \frac{1}{2}(1 + \cos 2nx) \]
Essential for integrating \(\sin^2 x\) and \(\cos^2 x\). Always your first step with even powers of sine or cosine.
Differential Calculus
Advanced
Ext 1
Ext 2
Derivative Rules Table
| Function | Derivative |
|---|---|
| \(y = f(x)^n\) | \(\frac{dy}{dx} = nf'(x)[f(x)]^{n-1}\) |
| \(y = uv\) | \(\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}\) |
| \(y = g(u),\; u=f(x)\) | \(\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\) |
| \(y = \frac{u}{v}\) | \(\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\) |
| \(y = \sin f(x)\) | \(\frac{dy}{dx} = f'(x)\cos f(x)\) |
| \(y = \cos f(x)\) | \(\frac{dy}{dx} = -f'(x)\sin f(x)\) |
| \(y = \tan f(x)\) | \(\frac{dy}{dx} = f'(x)\sec^2 f(x)\) |
| \(y = e^{f(x)}\) | \(\frac{dy}{dx} = f'(x)e^{f(x)}\) |
| \(y = \ln f(x)\) | \(\frac{dy}{dx} = \frac{f'(x)}{f(x)}\) |
| \(y = a^{f(x)}\) | \(\frac{dy}{dx} = (\ln a)\,f'(x)\,a^{f(x)}\) |
These cover virtually all HSC differentiation. The chain rule (3rd row) is the most-used — always identify the "inside function" \(f(x)\) and multiply by its derivative.
Inverse Trig Derivatives
| Function | Derivative |
|---|---|
| \(y = \sin^{-1} f(x)\) | \(\frac{dy}{dx} = \frac{f'(x)}{\sqrt{1-[f(x)]^2}}\) |
| \(y = \cos^{-1} f(x)\) | \(\frac{dy}{dx} = -\frac{f'(x)}{\sqrt{1-[f(x)]^2}}\) |
| \(y = \tan^{-1} f(x)\) | \(\frac{dy}{dx} = \frac{f'(x)}{1+[f(x)]^2}\) |
Note the minus sign on \(\cos^{-1}\). The domain restriction for \(\sin^{-1}\) and \(\cos^{-1}\) requires \(|f(x)| < 1\).
Common mistake: Forgetting the minus sign on \(\cos^{-1}\), or forgetting \(f'(x)\) in the numerator (chain rule).
Logarithmic Derivative
\[ y = \log_a f(x) \implies \frac{dy}{dx} = \frac{f'(x)}{(\ln a)\,f(x)} \]
The \(\ln a\) in the denominator is the key difference from \(\ln f(x)\). When \(a = e\), this simplifies to \(\frac{f'(x)}{f(x)}\).
Integral Calculus
Advanced
Ext 1
Ext 2
Standard Integrals
| Integrand | Integral (+ C) |
|---|---|
| \(f'(x)[f(x)]^n, \; n \ne -1\) | \(\frac{[f(x)]^{n+1}}{n+1}\) |
| \(f'(x)\sin f(x)\) | \(-\cos f(x)\) |
| \(f'(x)\cos f(x)\) | \(\sin f(x)\) |
| \(f'(x)\sec^2 f(x)\) | \(\tan f(x)\) |
| \(f'(x)e^{f(x)}\) | \(e^{f(x)}\) |
| \(\frac{f'(x)}{f(x)}\) | \(\ln|f(x)|\) |
| \(f'(x)\,a^{f(x)}\) | \(\frac{a^{f(x)}}{\ln a}\) |
Each requires the chain rule in reverse — look for the \(f'(x)\) factor. If it's missing (up to a constant), you can factor it out.
Inverse Trig Integrals
| Integrand | Integral (+ C) |
|---|---|
| \(\frac{f'(x)}{\sqrt{a^2-[f(x)]^2}}\) | \(\sin^{-1}\frac{f(x)}{a}\) |
| \(\frac{f'(x)}{a^2+[f(x)]^2}\) | \(\frac{1}{a}\tan^{-1}\frac{f(x)}{a}\) |
Recognise the pattern: square root in denominator → arcsin; sum of squares in denominator → arctan.
Common mistake: Forgetting the \(\frac{1}{a}\) coefficient in the arctan integral.
Integration by Parts
\[ \int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx \]
The reverse of the product rule. Choose \(u\) as the function that simplifies when differentiated (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).
Example: \(\int x\,e^x\,dx\): let \(u=x\), \(dv=e^x dx\). Result: \(xe^x - e^x + C\).
Trapezoidal Rule
\[ \int_a^b f(x)\,dx \approx \frac{b-a}{2n}\Bigg\{f(a)+f(b)+2\big[f(x_1)+\cdots+f(x_{n-1})\big]\Bigg\} \]
Numerical approximation with \(n\) subintervals. The endpoints are counted once; interior points twice. More subintervals = more accurate.
Statistical Analysis
Standard
Advanced
Ext 1
Z-Score
\[ z = \frac{x - \mu}{\sigma} \]
How many standard deviations a value is from the mean. \(z > 0\) means above average; \(z < 0\) means below.
Example: If \(\mu=70\), \(\sigma=5\), \(x=85\): \(z = \frac{85-70}{5} = 3\). The score is 3 SDs above the mean.
Outlier Rule
\[ \text{Outlier if } x < Q_1 - 1.5 \times IQR \text{ or } x > Q_3 + 1.5 \times IQR \]
Where \(IQR = Q_3 - Q_1\). Any value outside these fences is flagged as an outlier.
Expectation & Variance
\[ E(X) = \mu, \quad \text{Var}(X) = E\big[(X-\mu)^2\big] = E(X^2) - \mu^2 \]
The shortcut \(\text{Var}(X) = E(X^2) - [E(X)]^2\) is universally useful for computing variance.
Empirical Rule (68-95-99.7)
\[ 68\% \text{ within } 1\sigma, \quad 95\% \text{ within } 2\sigma, \quad 99.7\% \text{ within } 3\sigma \]
Only applies to bell-shaped (normal) distributions. Lets you quickly estimate probabilities without a table.
Probability
Standard
Advanced
Ext 1
Independent Events
\[ P(A \cap B) = P(A) \times P(B) \]
Two events are independent if the occurrence of one doesn't affect the other.
Addition Rule
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
The \(- P(A \cap B)\) prevents double-counting the intersection. If mutually exclusive, \(P(A \cap B) = 0\).
Conditional Probability
\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) \ne 0 \]
Probability of A given that B has occurred. If A and B are independent, \(P(A|B) = P(A)\).
Common mistake: Confusing \(P(A|B)\) with \(P(B|A)\). They are generally not equal.
Continuous Random Variable — CDF
\[ P(X \le r) = \int_a^r f(x)\,dx, \quad P(a < X < b) = \int_a^b f(x)\,dx \]
The probability density function \(f(x)\) gives probability as area under the curve. Total area must equal 1.
Binomial Distribution
Ext 1
Binomial Probability
\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \]
\(n\) = number of trials, \(r\) = number of successes, \(p\) = probability of success. Requires: fixed \(n\), independent trials, constant \(p\).
Example: 10 coin flips, \(p=0.5\). \(P(X=7) = \binom{10}{7}(0.5)^7(0.5)^3 = 120 \times \frac{1}{1024} \approx 0.117\).
Binomial — Mean & Variance
\[ E(X) = np, \quad \text{Var}(X) = np(1-p) \]
The expected number of successes is \(np\). The variance is always less than the mean (since \(1-p < 1\)).
Combinatorics
Ext 1
Ext 2
Permutations
\[ {}^nP_r = \frac{n!}{(n-r)!} \]
Number of ways to arrange \(r\) objects from \(n\) objects where order matters.
Combinations
\[ \binom{n}{r} = {}^nC_r = \frac{n!}{r!(n-r)!} \]
Number of ways to choose \(r\) objects from \(n\) objects where order doesn't matter.
Common mistake: Using permutations when the problem asks for selections/committees (order doesn't matter).
Binomial Theorem
\[ (x+a)^n = x^n + \binom{n}{1}x^{n-1}a + \cdots + \binom{n}{r}x^{n-r}a^r + \cdots + a^n \]
The general term is \(T_{r+1} = \binom{n}{r}x^{n-r}a^r\). Use this to find a specific term without expanding.
Example: The coefficient of \(x^3\) in \((x+2)^5\) is \(\binom{5}{2}(2)^2 = 10 \times 4 = 40\).
Vectors
Ext 1
Ext 2
Vector Magnitude (2D)
\[ |\vec{u}| = |x\,\hat{i} + y\,\hat{j}| = \sqrt{x^2 + y^2} \]
Pythagorean theorem for vectors. In 3D: \(|\vec{u}| = \sqrt{x^2+y^2+z^2}\).
Dot Product
\[ \vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos\theta = x_1 x_2 + y_1 y_2 \]
Two forms: geometric (with angle) and algebraic (component-wise). If \(\vec{u} \cdot \vec{v} = 0\), the vectors are perpendicular.
Example: \((1,2)\cdot(3,-1) = 3-2 = 1\). Angle: \(\cos\theta = \frac{1}{\sqrt{5}\sqrt{10}}\).
Vector Equation of a Line
\[ \vec{r} = \vec{a} + \lambda\vec{b} \]
\(\vec{a}\) is a point on the line, \(\vec{b}\) is the direction vector, \(\lambda\) is a scalar parameter.
Complex Numbers
Ext 2
Modulus-Argument Form
\[ z = a + ib = r(\cos\theta + i\sin\theta) = re^{i\theta} \]
Three equivalent forms: rectangular, polar, and exponential. \(r = |z| = \sqrt{a^2+b^2}\), \(\theta = \arg(z) = \tan^{-1}(b/a)\).
De Moivre's Theorem
\[ \big[r(\cos\theta + i\sin\theta)\big]^n = r^n(\cos n\theta + i\sin n\theta) = r^n e^{in\theta} \]
Raises a complex number to an integer power. Essential for finding \(n\)-th roots: set \(z^n = w\) and solve.
Example: Cube roots of \(re^{i\theta}\): \(z_k = r^{1/3}e^{i(\theta+2k\pi)/3}\) for \(k=0,1,2\).
Mechanics
Ext 2
Acceleration Forms
\[ \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\!\left(\frac{1}{2}v^2\right) \]
Four equivalent expressions for acceleration. Use \(v\frac{dv}{dx}\) when acceleration is a function of position; use \(\frac{dv}{dt}\) when it's a function of time.
Simple Harmonic Motion
\[ x = a\cos(nt + \alpha) + c \]
Centre of motion is \(c\), amplitude is \(a\), period is \(\frac{2\pi}{n}\). The equivalent sine form: \(x = a\sin(nt+\alpha) + c\).
SHM Differential Equation
\[ \ddot{x} = -n^2(x - c) \]
The defining ODE of SHM: acceleration is proportional to displacement from the centre, directed towards it. The negative sign means restoring force.