HSC Mathematics Reference Sheet — Explained (Advanced, Ext 1 & Ext 2)
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The NESA Mathematics reference sheet is the same for Advanced, Extension 1 and Extension 2 — but having a formula in front of you is not the same as knowing when to reach for it. Below, every formula is explained: what each symbol means, when to use it, a worked example, and practice to try yourself.
In short: the NESA Mathematics reference sheet is identical for Advanced, Extension 1 and Extension 2 — but having a formula in front of you isn't the same as knowing when to use it. This guide explains every formula on the sheet, grouped by topic (measurement, financial maths, series, calculus, trigonometry, probability), so you can reach for the right one fast in the exam.
Measurement
7What each symbol means
$r$ — radius of the circle; $\theta$ — central angle in radians.
When to use it
Use when finding the area of a pie-slice region of a circle and $\theta$ is given in radians.
Units:
Square length units (cm², m², etc.).
Worked sample
Find the area of a sector of radius 6 cm and angle $\tfrac{\pi}{3}$ rad.
$A = \tfrac{1}{2}(6)^2 \cdot \tfrac{\pi}{3} = \tfrac{1}{2} \cdot 36 \cdot \tfrac{\pi}{3} = 6\pi \approx 18.85$ cm².
Your turn:
A sector has $r = 5$ and $\theta = 1.2$ rad. Find its area.
$A = \tfrac{1}{2}(25)(1.2) = 15$ square units.
Your turn:
A sector has area 20 cm² and $r = 4$ cm. Find $\theta$.
$\theta = \dfrac{2 \times 20}{16} = 2.5$ rad.
The formula only works in radians. If the angle is in degrees, convert first: $\theta_{\text{rad}} = \theta_{\deg} \times \tfrac{\pi}{180}$.
What each symbol means
$r$ — radius; $\theta$ — central angle in radians.
When to use it
Finding the curved boundary length of a sector or arc.
Units:
Length units.
Worked sample
Find the arc length for $r = 8$ and $\theta = \tfrac{\pi}{4}$.
$l = 8 \times \tfrac{\pi}{4} = 2\pi \approx 6.28$.
Your turn:
A wheel of radius 30 cm turns through 2.5 radians. How far does a point on the rim travel?
$l = 30 \times 2.5 = 75$ cm.
Remember: $l = r\theta$ gives the arc only. The full perimeter of a sector is $l + 2r$.
What each symbol means
$r$ — base radius; $l$ — slant height.
When to use it
Finding the curved surface area of a cone (excludes the circular base).
Units:
Square length units.
Worked sample
A cone has $r = 3$ and $l = 5$. Find the curved surface area.
$S = \pi(3)(5) = 15\pi \approx 47.1$ square units.
Your turn:
A cone has $r = 4$ and perpendicular height $h = 3$. Find the total surface area.
Slant height $l = \sqrt{r^2+h^2} = 5$. Curved SA $= 20\pi$. Base $= 16\pi$. Total $= 36\pi \approx 113.1$ square units.
Slant height $l \ne$ perpendicular height $h$. Use $l = \sqrt{r^2+h^2}$ when only $h$ is given.
What each symbol means
$r$ — radius.
When to use it
Finding the total surface area of a sphere or hemisphere (use half and add a great-circle base).
Units:
Square length units.
Worked sample
Find the surface area of a sphere of radius 7 cm.
$S = 4\pi(49) = 196\pi \approx 615.8$ cm².
Your turn:
A hemisphere has radius 6 cm. Find its total surface area.
Curved: $\tfrac{1}{2}(4\pi \cdot 36) = 72\pi$. Base circle: $36\pi$. Total: $108\pi \approx 339.3$ cm².
For a hemisphere, the total surface area is $3\pi r^2$ (half of $4\pi r^2$ plus the base $\pi r^2$).
What each symbol means
$r$ — base radius; $h$ — perpendicular height.
When to use it
Finding the volume of any right circular cone.
Units:
Cubic length units.
Worked sample
A cone has $r = 6$ and $h = 10$. Find its volume.
$V = \tfrac{1}{3}\pi(36)(10) = 120\pi \approx 376.99$ cubic units.
Your turn:
A cone has volume $96\pi$ and $r = 6$. Find $h$.
$96\pi = \tfrac{1}{3}\pi(36)h \Rightarrow h = 8$.
The cone volume is exactly $\tfrac{1}{3}$ of the enclosing cylinder ($\pi r^2 h$). Useful cross-check.
What each symbol means
$r$ — radius.
When to use it
Finding the volume of a sphere or hemisphere.
Units:
Cubic length units.
Worked sample
Find the volume of a sphere of radius 3 cm.
$V = \tfrac{4}{3}\pi(27) = 36\pi \approx 113.1$ cm³.
Your turn:
A sphere has volume 288$\pi$. Find the radius.
$\tfrac{4}{3}\pi r^3 = 288\pi \Rightarrow r^3 = 216 \Rightarrow r = 6$.
For a hemisphere, divide by 2: $V = \tfrac{2}{3}\pi r^3$.
What each symbol means
$A$ — area of the base; $h$ — perpendicular height.
When to use it
Finding the volume of any pyramid (square, rectangular, or triangular base).
Units:
Cubic length units.
Worked sample
A square pyramid has base side 4 and height 6. Find its volume.
$A = 16$, so $V = \tfrac{1}{3}(16)(6) = 32$ cubic units.
Your turn:
A rectangular pyramid has base $5 \times 3$ and height 8. Find its volume.
$V = \tfrac{1}{3}(15)(8) = 40$ cubic units.
Like the cone, a pyramid is always $\tfrac{1}{3} \times$ the prism with the same base and height.
Financial Mathematics
4What each symbol means
$FV$ — future value; $PV$ — present value (principal); $r$ — interest rate per period (as a decimal); $n$ — number of periods.
When to use it
Use for a lump-sum investment or loan compounding at a constant rate — no regular contributions.
Units:
Dollars (or any currency).
Worked sample
$2{,}000 is invested at 6% p.a. compound interest for 5 years. Find the future value.
$FV = 2000(1.06)^5 = 2000 \times 1.3382 \approx \$2{,}676.45$.
Your turn:
$5{,}000 is borrowed at 4% p.a. compounded quarterly for 3 years. Find the amount owing.
$r = 0.01, n = 12$: $FV = 5000(1.01)^{12} \approx \$5{,}634.13$.
Your turn:
An investment doubles in value. If $r = 0.08$ p.a., how many years?
$2 = (1.08)^n \Rightarrow n = \ln 2 / \ln 1.08 \approx 9.01$ years.
When interest is compounded more frequently than annually, divide the annual rate by the number of periods per year and multiply $n$ by that same number.
What each symbol means
$PV$ — present value; $FV$ — future value; $r$ — rate per period; $n$ — number of periods.
When to use it
Finding today's equivalent of a future lump sum — the 'discount' formula.
Units:
Currency.
Worked sample
What lump sum today is equivalent to $\$5{,}000$ in 4 years at 5% p.a.?
$PV = 5000(1.05)^{-4} = 5000/1.2155 \approx \$4{,}113.51$.
Your turn:
You need $\$10{,}000$ in 3 years. Rate is 6% p.a. How much must you invest today?
$PV = 10000(1.06)^{-3} \approx \$8{,}396.19$.
$PV = FV \times (1+r)^{-n}$ is identical to $PV = \dfrac{FV}{(1+r)^n}$. Use whichever your calculator handles better.
What each symbol means
$C$ — regular contribution per period; $r$ — interest rate per period; $n$ — number of periods.
When to use it
Superannuation, savings plans — equal deposits made at the end of each period (ordinary annuity).
Units:
Currency.
Worked sample
$200 is deposited monthly at 6% p.a. (0.5% per month) for 12 months.
$FV = 200 \times \dfrac{(1.005)^{12}-1}{0.005} = 200 \times 12.336 \approx \$2{,}467.16$.
Your turn:
$1{,}000 is contributed annually at 4% for 10 years. Find the final balance.
$FV = 1000 \times \dfrac{(1.04)^{10}-1}{0.04} \approx 1000 \times 12.006 \approx \$12{,}006.11$.
The denominator $r$ must match the period of $C$ and $n$ — all monthly, or all annual.
What each symbol means
$C$ — regular payment; $r$ — rate per period; $n$ — total number of payments.
When to use it
Loan repayments — finding the loan size that a series of equal payments will repay.
Units:
Currency.
Worked sample
Monthly repayments of $\$600$ at 0.5% per month for 36 months — what loan amount does this repay?
$PV = 600 \times \dfrac{1-(1.005)^{-36}}{0.005} = 600 \times 32.871 \approx \$19{,}723$.
Your turn:
A loan of $\$20{,}000$ at 1% per month is repaid over 24 months. What is the monthly repayment?
$PV = C \times \dfrac{1-(1.01)^{-24}}{0.01} \Rightarrow 20000 = C \times 21.243 \Rightarrow C \approx \$941.47$.
Rearrange to find $C$ (the repayment): $C = \dfrac{PV \cdot r}{1-(1+r)^{-n}}$.
Sequences and Series
5What each symbol means
$a$ — first term; $d$ — common difference; $n$ — term number.
When to use it
Finding any particular term of an arithmetic progression (AP) without listing them all.
Worked sample
The AP $3, 7, 11, \ldots$ — find the 20th term.
$a = 3, d = 4$: $T_{20} = 3 + 19 \times 4 = 79$.
Your turn:
Find the 15th term of the AP $-2, 1, 4, \ldots$
$T_{15} = -2 + 14(3) = 40$.
Your turn:
Which term of the AP $5, 9, 13, \ldots$ equals 101?
$5 + (n-1) \times 4 = 101 \Rightarrow n = 25$.
Don't forget $n-1$, not $n$. The first term uses $n = 1$, giving $(1-1)d = 0$, so $T_1 = a$. ✓
What each symbol means
$a$ — first term; $l$ — last term; $d$ — common difference; $n$ — number of terms.
When to use it
Summing an AP. Use $\tfrac{n}{2}(a+l)$ when the last term is known; use $\tfrac{n}{2}[2a+(n-1)d]$ when only $a$, $d$, $n$ are given.
Worked sample
Find the sum of the first 10 terms of $2 + 5 + 8 + \cdots$
$S_{10} = \tfrac{10}{2}[2(2) + 9(3)] = 5 \times 31 = 155$.
Your turn:
Find the sum $1 + 2 + 3 + \cdots + 100$.
$S_{100} = \tfrac{100}{2}(1 + 100) = 50 \times 101 = 5050$.
The two forms are equivalent — pick whichever you have the data for. Easy check: $S_1$ should equal $a$.
What each symbol means
$a$ — first term; $r$ — common ratio; $n$ — term number.
When to use it
Finding any term of a geometric progression (GP).
Worked sample
The GP $2, 6, 18, \ldots$ — find the 7th term.
$a = 2, r = 3$: $T_7 = 2 \times 3^6 = 2 \times 729 = 1458$.
Your turn:
The GP has $a = 5, r = 0.4$. Find $T_4$.
$T_4 = 5 \times (0.4)^3 = 5 \times 0.064 = 0.32$.
A GP can decrease if $0 < r < 1$ or alternate signs if $r < 0$. Always check what $r$ actually is.
What each symbol means
$a$ — first term; $r$ — common ratio; $n$ — number of terms.
When to use it
Summing a finite GP. Use the $1-r$ form when $|r| < 1$ (avoids negatives); use the $r-1$ form when $r > 1$.
Worked sample
Sum the first 6 terms of $3 + 6 + 12 + \cdots$
$S_6 = \dfrac{3(2^6 - 1)}{2-1} = 3 \times 63 = 189$.
Your turn:
Sum the first 8 terms of $1, \tfrac{1}{2}, \tfrac{1}{4}, \ldots$
$S_8 = \dfrac{1(1-(0.5)^8)}{0.5} = \dfrac{255/256}{0.5} = \dfrac{255}{128} \approx 1.992$.
Both forms give the same answer — but one form always has a positive numerator and denominator, making calculator input cleaner.
What each symbol means
$a$ — first term; $r$ — common ratio with $|r| < 1$.
When to use it
Finding the sum of an infinite GP — only valid when $|r| < 1$ (the terms shrink to zero).
Worked sample
Find the limiting sum of $8 + 4 + 2 + \cdots$
$a = 8, r = \tfrac{1}{2}$: $S = \dfrac{8}{1 - 0.5} = 16$.
Your turn:
Find the limiting sum of $1 - \tfrac{1}{3} + \tfrac{1}{9} - \cdots$
$a = 1, r = -\tfrac{1}{3}$: $S = \dfrac{1}{1+\tfrac{1}{3}} = \dfrac{3}{4}$.
Always verify $|r| < 1$ before applying the formula. If the question asks whether a limiting sum exists, you are being asked to check this condition.
Logarithmic, Exponential and Indices
9What each symbol means
$x$ — the number; $a$ — the new base (usually 10 or $e$); $b$ — the original base.
When to use it
Evaluating a logarithm to a non-standard base on a calculator, or proving log identities.
Worked sample
Find $\log_2 50$ using a calculator.
$\log_2 50 = \dfrac{\ln 50}{\ln 2} = \dfrac{3.912}{0.693} \approx 5.644$.
Your turn:
Evaluate $\log_3 20$ to 2 decimal places.
$\dfrac{\ln 20}{\ln 3} \approx \dfrac{2.996}{1.099} \approx 2.73$.
Any base $a$ works — most students use $\ln$ (base $e$) since it's always available.
What each symbol means
$a$ — base; $x, y > 0$.
When to use it
Expanding or simplifying a log of a product.
Worked sample
Simplify $\log_2 8 + \log_2 4$.
$= \log_2 32 = 5$.
Your turn:
Write $\ln 6$ in terms of $\ln 2$ and $\ln 3$.
$\ln 6 = \ln 2 + \ln 3$.
This identity runs in both directions — use it to split a product into a sum, or combine a sum into one log.
What each symbol means
$a$ — base; $x, y > 0$.
When to use it
Expanding or simplifying a log of a quotient.
Worked sample
Simplify $\log_{10} 200 - \log_{10} 2$.
$= \log_{10} 100 = 2$.
Your turn:
Simplify $\ln 15 - \ln 5$.
$\ln 3$.
Order matters: $\log x - \log y = \log(x/y)$, not $\log(y/x)$.
What each symbol means
$n$ — any real exponent; $x > 0$; $a > 0, a \ne 1$.
When to use it
Bringing an exponent down to a coefficient — essential for solving exponential equations.
Worked sample
Solve $2^x = 10$.
$x \log 2 = \log 10 = 1 \Rightarrow x = \dfrac{1}{\log 2} \approx 3.32$.
Your turn:
Solve $3^{2x} = 50$.
$2x \ln 3 = \ln 50 \Rightarrow x = \dfrac{\ln 50}{2\ln 3} \approx 1.81$.
This is the key step in every exponential equation — take the log of both sides, then bring the exponent down.
What each symbol means
$a$ — base ($a > 0, a \ne 1$); $x > 0$.
When to use it
Cancelling a log and its inverse in simplifications and solving equations.
Worked sample
Simplify $\ln(e^{5})$.
$= 5$.
Your turn:
Simplify $e^{\ln 7}$.
$= 7$.
$\ln$ is simply $\log_e$. These two identities reflect that $\log$ and exponential are inverses.
What each symbol means
$a$ — base; $m, n$ — real exponents.
When to use it
Multiplying expressions with the same base.
Worked sample
Simplify $x^3 \cdot x^5$.
$x^8$.
Your turn:
Simplify $2^4 \cdot 2^{-1}$.
$2^3 = 8$.
Bases must match. $x^3 \cdot y^3 \ne (xy)^3$ unless you use the separate product law $(ab)^n = a^n b^n$.
What each symbol means
$a \ne 0$; $m, n$ — real exponents.
When to use it
Dividing expressions with the same base.
Worked sample
Simplify $\dfrac{x^7}{x^3}$.
$x^4$.
Your turn:
Simplify $\dfrac{3^5}{3^8}$.
$3^{-3} = \dfrac{1}{27}$.
Subtracting a larger exponent gives a negative power, which is equivalent to $\dfrac{1}{a^n}$.
What each symbol means
$a$ — base; $m, n$ — real exponents.
When to use it
Raising a power to another power — multiply the indices.
Worked sample
Simplify $(x^3)^4$.
$x^{12}$.
Your turn:
Simplify $(2^3)^{-2}$.
$2^{-6} = \dfrac{1}{64}$.
Contrast: $(a^m)^n = a^{mn}$ (multiply), but $a^{m^n}$ (tower) means $a^{(m^n)}$ — a different operation.
What each symbol means
$a > 0$, $a \ne 1$ — constant base; $\ln a$ — natural log of the base.
When to use it
Differentiating an exponential whose base is a constant other than $e$, e.g. $2^x$, $10^x$.
Worked sample
Differentiate $y = 3^x$.
$y' = 3^x \ln 3$.
Your turn:
Differentiate $y = 5^{2x}$.
By chain rule: $y' = 5^{2x} \cdot \ln 5 \cdot 2 = 2\ln 5 \cdot 5^{2x}$.
When $a = e$, $\ln e = 1$ and the formula reduces to $\dfrac{d}{dx}(e^x) = e^x$, confirming the special case.
Trigonometry
13What each symbol means
$a, b, c$ — side lengths; $A, B, C$ — the angles opposite those sides.
When to use it
Use when you know two angles and any side (AAS/ASA), or two sides and a non-included angle (SSA — watch for the ambiguous case).
Worked sample
In $\triangle ABC$, $a = 7$, $A = 50^\circ$, $B = 65^\circ$. Find $b$.
$b = \dfrac{7\sin 65^\circ}{\sin 50^\circ} = \dfrac{7 \times 0.906}{0.766} \approx 8.28$.
Your turn:
In $\triangle ABC$, $A = 40^\circ$, $B = 70^\circ$, $a = 9$. Find $b$.
$b = \dfrac{9\sin 70^\circ}{\sin 40^\circ} \approx 13.15$.
Your turn:
Explain why an SSA triangle might have two solutions.
When $a < b$ and $a > b\sin A$, two different triangles satisfy the conditions — the ambiguous case.
The ambiguous case (SSA): when you are finding an angle with the sine rule, consider whether $\sin^{-1}$ also has a second-quadrant solution.
What each symbol means
$a, b, c$ — the three side lengths; $C$ — the angle opposite side $c$.
When to use it
Use it when you know two sides and the angle between them (to find the third side), or all three sides (to find an angle). The plain sine rule will not work in those two cases.
Worked sample
In $\triangle ABC$, $a = 7$, $b = 9$ and $C = 40^\circ$. Find $c$.
$c^2 = 7^2 + 9^2 - 2(7)(9)\cos 40^\circ = 130 - 96.5 \approx 33.5$, so $c \approx 5.79$.
Your turn:
In $\triangle ABC$, $a = 5$, $b = 8$ and $C = 60^\circ$. Find $c$ (1 dp).
$c^2 = 25 + 64 - 2(5)(8)\cos 60^\circ = 89 - 40 = 49$, so $c = 7.0$.
Your turn:
In $\triangle ABC$, $a = 3$, $b = 5$, $c = 7$. Find $\cos C$.
$\cos C = \dfrac{9+25-49}{30} = \dfrac{-15}{30} = -0.5$, so $C = 120^\circ$.
To find an angle, rearrange to $\cos C = \dfrac{a^2 + b^2 - c^2}{2ab}$.
What each symbol means
$a, b$ — two side lengths; $C$ — the angle between them.
When to use it
Use it when you know two sides and the included angle. If you only know the base and perpendicular height, use $A = \tfrac12 bh$ instead.
Units:
Area is in square units (e.g. cm²).
Worked sample
Find the area of $\triangle ABC$ with $a = 6$, $b = 10$, $C = 30^\circ$.
$A = \tfrac12 (6)(10)\sin 30^\circ = 30 \times 0.5 = 15$ square units.
Your turn:
Find the area with $a = 4$, $b = 9$, $C = 90^\circ$.
$A = \tfrac12 (4)(9)\sin 90^\circ = 18$ square units.
$\sin 90^\circ = 1$, so for a right angle this is just $\tfrac12 \times$ the two shorter sides.
What each symbol means
Standard angles derived from equilateral and isosceles right triangles.
When to use it
Any time an exact value is required — leave surds in your answer rather than rounding.
Worked sample
Find $\sin 60^\circ$ exactly.
$\sin 60^\circ = \tfrac{\sqrt{3}}{2}$.
Your turn:
Evaluate $\tan 45^\circ$ exactly.
$\tan 45^\circ = 1$.
Your turn:
Evaluate $2\cos^2 30^\circ - 1$ exactly.
$2 \times \tfrac{3}{4} - 1 = \tfrac{3}{2} - 1 = \tfrac{1}{2}$. (This equals $\cos 60^\circ$.)
Draw the two triangles (half-equilateral for 30°/60°, isosceles right for 45°) — they are faster than memorising a table.
What each symbol means
$\theta$ — any angle.
When to use it
Simplifying trig expressions, proving identities, or converting between sin and cos.
Worked sample
If $\sin\theta = \tfrac{3}{5}$ and $\theta$ is acute, find $\cos\theta$.
$\cos^2\theta = 1 - \tfrac{9}{25} = \tfrac{16}{25}$, so $\cos\theta = \tfrac{4}{5}$.
Your turn:
Simplify $\dfrac{1 - \cos^2\theta}{\sin\theta}$.
$= \dfrac{\sin^2\theta}{\sin\theta} = \sin\theta$.
Variants: $\sin^2\theta = 1 - \cos^2\theta$ and $\cos^2\theta = 1 - \sin^2\theta$ — write these on your working paper.
What each symbol means
$A, B$ — any angles.
When to use it
Expanding $\sin$ of a sum or difference of angles, deriving double-angle formulas, simplifying products.
Worked sample
Expand $\sin(45^\circ + 30^\circ)$ using the sum formula, and verify it equals $\sin 75^\circ$.
$\sin 45^\circ\cos 30^\circ + \cos 45^\circ\sin 30^\circ = \tfrac{\sqrt{2}}{2}\cdot\tfrac{\sqrt{3}}{2} + \tfrac{\sqrt{2}}{2}\cdot\tfrac{1}{2} = \tfrac{\sqrt{6}+\sqrt{2}}{4}$.
Your turn:
Find $\sin 15^\circ$ exactly using $\sin(45^\circ - 30^\circ)$.
$\sin 15^\circ = \tfrac{\sqrt{6}-\sqrt{2}}{4}$.
The $\pm$ on the right-hand side matches the sign in $A \pm B$. For cosine, the sign flips.
What each symbol means
$A, B$ — any angles.
When to use it
Expanding $\cos$ of a sum or difference, and deriving double-angle cosine forms.
Worked sample
Find $\cos 75^\circ$ exactly.
$\cos(45^\circ+30^\circ) = \cos 45^\circ\cos 30^\circ - \sin 45^\circ\sin 30^\circ = \tfrac{\sqrt{6}-\sqrt{2}}{4}$.
Your turn:
Find $\cos 15^\circ$ exactly.
$\cos(45^\circ-30^\circ) = \tfrac{\sqrt{6}+\sqrt{2}}{4}$.
Note the opposite sign: $\cos(A+B)$ has $-\sin A \sin B$, while $\cos(A-B)$ has $+\sin A\sin B$.
What each symbol means
$A, B$ — angles where $\tan$ is defined.
When to use it
Finding exact tan values or solving tan equations by expanding the angle.
Worked sample
Find $\tan 75^\circ$ exactly.
$\tan(45^\circ+30^\circ) = \dfrac{1 + \tfrac{1}{\sqrt{3}}}{1 - \tfrac{1}{\sqrt{3}}} = \dfrac{\sqrt{3}+1}{\sqrt{3}-1} = 2+\sqrt{3}$.
Your turn:
Find $\tan 15^\circ$ exactly.
$\tan(45^\circ-30^\circ) = \dfrac{1 - 1/\sqrt{3}}{1+1/\sqrt{3}} = 2-\sqrt{3}$.
Denominator sign is opposite: $(A+B)$ gives $1 - \tan A \tan B$; $(A-B)$ gives $1 + \tan A\tan B$.
What each symbol means
$A$ — any angle.
When to use it
Differentiating or integrating $\sin^2$ and $\cos^2$ (after converting), or solving equations of the form $\sin 2A = k$.
Worked sample
Solve $\sin 2x = \sin x$ for $0 \le x \le 2\pi$.
$2\sin x\cos x - \sin x = 0 \Rightarrow \sin x(2\cos x-1) = 0$. So $x = 0, \pi, 2\pi$ or $x = \pi/3, 5\pi/3$.
Your turn:
Differentiate $y = \cos^2 x$.
Using chain rule: $y' = -2\cos x\sin x = -\sin 2x$.
The double-angle formula is the compound-angle formula with $B = A$. You can derive it on the fly.
What each symbol means
$A$ — any angle.
When to use it
Integration of $\sin^2$ and $\cos^2$ (use a rearranged form: $\cos^2 A = \tfrac{1+\cos 2A}{2}$, $\sin^2 A = \tfrac{1-\cos 2A}{2}$).
Worked sample
Integrate $\displaystyle\int \sin^2 x\,dx$.
$= \displaystyle\int \tfrac{1-\cos 2x}{2}\,dx = \tfrac{x}{2} - \tfrac{\sin 2x}{4} + C$.
Your turn:
Integrate $\displaystyle\int \cos^2 x\,dx$.
$= \dfrac{x}{2} + \dfrac{\sin 2x}{4} + C$.
Three equivalent forms — pick the one that eliminates the trig function you don't want.
What each symbol means
$A$ — angle where $\tan A \ne \pm 1$.
When to use it
Solving equations involving $\tan 2A$, or converting between $\tan A$ and $\tan 2A$.
Worked sample
If $\tan A = 2$, find $\tan 2A$.
$\tan 2A = \dfrac{2(2)}{1-4} = \dfrac{4}{-3} = -\tfrac{4}{3}$.
Your turn:
Solve $\tan 2x = 1$ for $0 \le x < \pi$.
$2x = \pi/4, 5\pi/4 \Rightarrow x = \pi/8, 5\pi/8$.
This is just the compound-angle formula for $\tan$ with $B = A$.
What each symbol means
$A, B$ — any angles.
When to use it
Converting a product of trig functions into a sum — needed for some integration and limit problems.
Worked sample
Write $2\sin 3x\cos x$ as a sum.
$= \sin(3x+x) + \sin(3x-x) = \sin 4x + \sin 2x$.
Your turn:
Integrate $\sin 3x \cos x\,dx$.
$= \tfrac{1}{2}\int(\sin 4x + \sin 2x)\,dx = -\tfrac{\cos 4x}{8} - \tfrac{\cos 2x}{4} + C$.
The other product formulae ($2\cos A\cos B$, $2\sin A\sin B$) are derived by adding/subtracting the compound-angle cosine identities.
What each symbol means
$t = \tan(\theta/2)$.
When to use it
Solving equations of the form $a\sin\theta + b\cos\theta = c$, or integrating rational trig functions.
Worked sample
Solve $\sin\theta + \cos\theta = 1$ for $0 \le \theta \le 2\pi$ using the $t$-formulae.
$\dfrac{2t}{1+t^2} + \dfrac{1-t^2}{1+t^2} = 1 \Rightarrow 2t + 1 - t^2 = 1 + t^2 \Rightarrow 2t(1-t)=0 \Rightarrow t=0$ or $t=1$. So $\theta = 0, 2\pi$ or $\theta = \pi/2$. ($\theta=\pi$ must be checked separately since $t=\tan(\pi/2)$ is undefined; checking: $\sin\pi+\cos\pi = 0+(-1)=-1\ne 1$, so $\theta=\pi$ is not a solution.)
Your turn:
Express $\sin\theta + \cos\theta$ in terms of $t$ and simplify.
$\dfrac{2t + 1 - t^2}{1+t^2}$.
This substitution is on the NESA reference sheet for Ext 1. Check $\theta = \pi$ separately (the substitution is undefined there since $\tan(\pi/2)$ is undefined).
Calculus — Differentiation
9What each symbol means
$n$ — any real number.
When to use it
Differentiating any power of $x$ — including negative and fractional exponents.
Worked sample
Differentiate $y = 3x^4 - 2x^{-1} + 5$.
$y' = 12x^3 + 2x^{-2}$.
Your turn:
Differentiate $y = \sqrt{x} = x^{1/2}$.
$y' = \dfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}$.
Write roots and fractions as powers before differentiating: $\sqrt[3]{x^2} = x^{2/3}$.
What each symbol means
$u$ and $v$ — two functions of $x$; $u', v'$ — their derivatives.
When to use it
Use it to differentiate a product of two functions, e.g. $x^2 e^x$ or $x\sin x$.
Worked sample
Differentiate $y = x^2 e^x$.
$u = x^2,\ v = e^x$, so $y' = 2x\,e^x + x^2 e^x = x e^x(2 + x)$.
Your turn:
Differentiate $y = x\sin x$.
$y' = \sin x + x\cos x$.
Pick $u$ as the part that gets simpler when differentiated — it keeps the algebra clean.
What each symbol means
$u, v$ — functions of $x$ with $v \ne 0$.
When to use it
Differentiating a function written as one expression divided by another.
Worked sample
Differentiate $y = \dfrac{x^2}{x+1}$.
$y' = \dfrac{2x(x+1) - x^2(1)}{(x+1)^2} = \dfrac{x^2 + 2x}{(x+1)^2}$.
Your turn:
Differentiate $y = \dfrac{\sin x}{x}$.
$y' = \dfrac{x\cos x - \sin x}{x^2}$.
Numerator is $u'v - uv'$ — the minus goes in the middle. Denominator is always $v^2$.
What each symbol means
$u$ — an intermediate function of $x$; $y$ — a function of $u$.
When to use it
Differentiating a composite (function of a function), e.g. $\sin(x^2)$, $(3x+1)^5$.
Worked sample
Differentiate $y = (3x+1)^5$.
Let $u = 3x+1$, $y = u^5$. $\dfrac{dy}{du} = 5u^4$, $\dfrac{du}{dx} = 3$. So $\dfrac{dy}{dx} = 15(3x+1)^4$.
Your turn:
Differentiate $y = e^{\sin x}$.
$y' = e^{\sin x}\cos x$.
Informally: 'derivative of the outside, leave the inside alone, multiply by derivative of inside'.
What each symbol means
$x$ in radians.
When to use it
Differentiating $\sin$ (or any expression involving $\sin$, via the chain rule).
Worked sample
Differentiate $y = \sin(3x)$.
$y' = 3\cos(3x)$ (chain rule with inner derivative 3).
Your turn:
Differentiate $y = \sin^2 x$.
$y' = 2\sin x\cos x = \sin 2x$.
Always work in radians in calculus. Degrees give a wrong derivative.
What each symbol means
$x$ in radians.
When to use it
Differentiating $\cos$ directly or as part of a composite function.
Worked sample
Differentiate $y = \cos(x^2)$.
$y' = -\sin(x^2) \cdot 2x = -2x\sin(x^2)$.
Your turn:
Differentiate $y = 4\cos(2x - \pi/3)$.
$y' = -8\sin(2x - \pi/3)$.
The negative sign distinguishes $\dfrac{d}{dx}\cos x$ from $\dfrac{d}{dx}\sin x$. Memorise: cos goes negative.
What each symbol means
$x$ in radians, $x \ne \pi/2 + n\pi$.
When to use it
Differentiating $\tan x$ or expressions like $\tan(ax+b)$.
Worked sample
Differentiate $y = \tan(2x)$.
$y' = 2\sec^2(2x)$.
Your turn:
Differentiate $y = x\tan x$.
$y' = \tan x + x\sec^2 x$ (product rule).
$\sec^2 x = 1 + \tan^2 x$ — this alternative form is sometimes more useful for algebraic simplification.
What each symbol means
$e \approx 2.71828$ — Euler's number.
When to use it
Differentiating the exponential function $e^x$ or $e^{f(x)}$ (via chain rule).
Worked sample
Differentiate $y = e^{3x^2}$.
$y' = 6x\,e^{3x^2}$.
Your turn:
Differentiate $y = e^{-x}$.
$y' = -e^{-x}$.
The uniqueness of $e^x$: it is its own derivative. This makes $e^x$ the natural choice for modelling continuous growth.
What each symbol means
$\ln x = \log_e x$.
When to use it
Differentiating logarithms. For $\ln|x|$ the derivative is also $\tfrac{1}{x}$ (valid for $x \ne 0$).
Worked sample
Differentiate $y = \ln(x^2 + 1)$.
$y' = \dfrac{2x}{x^2+1}$.
Your turn:
Differentiate $y = x\ln x$.
$y' = \ln x + x \cdot \dfrac{1}{x} = \ln x + 1$.
For $\ln[f(x)]$, the derivative is $\dfrac{f'(x)}{f(x)}$ — 'derivative of inside over inside'.
Calculus — Standard Integrals
10What each symbol means
$n$ — any real number except $-1$; $C$ — constant of integration.
When to use it
Integrating any power of $x$ (except $x^{-1}$).
Worked sample
Find $\displaystyle\int 4x^3\,dx$.
$= x^4 + C$.
Your turn:
Find $\displaystyle\int \sqrt{x}\,dx$.
$= \displaystyle\int x^{1/2}\,dx = \dfrac{2}{3}x^{3/2} + C$.
Add 1 to the exponent, divide by the new exponent. Common error: forgetting to divide. Check by differentiating your answer.
What each symbol means
$x \ne 0$; absolute value handles negative $x$.
When to use it
Integrating $x^{-1}$ — the excluded case from the power rule.
Worked sample
Find $\displaystyle\int \frac{3}{x}\,dx$.
$= 3\ln|x| + C$.
Your turn:
Find $\displaystyle\int \frac{1}{2x}\,dx$.
$= \dfrac{1}{2}\ln|x| + C$.
Do not forget the absolute value bars — they matter when $x$ can be negative.
What each symbol means
$a \ne 0$ — a constant.
When to use it
Integrating exponential functions with a linear exponent.
Worked sample
Find $\displaystyle\int 3e^{2x}\,dx$.
$= \dfrac{3}{2}e^{2x} + C$.
Your turn:
Find $\displaystyle\int e^{-3x}\,dx$.
$= -\dfrac{1}{3}e^{-3x} + C$.
The $\tfrac{1}{a}$ factor compensates for the chain-rule factor $a$ that would appear on differentiation.
What each symbol means
$a \ne 0$.
When to use it
Integrating cosine with a linear argument.
Worked sample
Find $\displaystyle\int \cos(3x)\,dx$.
$= \dfrac{1}{3}\sin(3x) + C$.
Your turn:
Evaluate $\displaystyle\int_0^{\pi/2} \cos(2x)\,dx$.
$= \Big[\tfrac{1}{2}\sin(2x)\Big]_0^{\pi/2} = \tfrac{1}{2}\sin\pi - 0 = 0$.
Inverse of $\dfrac{d}{dx}\sin(ax) = a\cos(ax)$ — always divide by $a$.
What each symbol means
$a \ne 0$.
When to use it
Integrating sine with a linear argument.
Worked sample
Find $\displaystyle\int \sin(4x)\,dx$.
$= -\dfrac{1}{4}\cos(4x) + C$.
Your turn:
Evaluate $\displaystyle\int_0^{\pi} \sin x\,dx$.
$= [-\cos x]_0^{\pi} = -\cos\pi + \cos 0 = 1 + 1 = 2$.
Note the minus sign — $\int \sin(ax)\,dx$ is negative cosine.
What each symbol means
$a \ne 0$.
When to use it
Integrating $\sec^2$, which arises after trig identities or when antidifferentiating $\tan$-family expressions.
Worked sample
Find $\displaystyle\int \sec^2(2x)\,dx$.
$= \dfrac{1}{2}\tan(2x) + C$.
Your turn:
Find $\displaystyle\int \sec^2 x\,dx$.
$= \tan x + C$.
This is just the reverse of $\dfrac{d}{dx}(\tan x) = \sec^2 x$.
What each symbol means
$a > 0$; $\sin^{-1}$ denotes the arcsine function (range $[-\pi/2, \pi/2]$).
When to use it
Any time the integrand is a reciprocal square root of the form $\dfrac{1}{\sqrt{a^2-x^2}}$.
Worked sample
Find $\displaystyle\int \frac{1}{\sqrt{9-x^2}}\,dx$.
$= \sin^{-1}\!\left(\dfrac{x}{3}\right) + C$.
Your turn:
Find $\displaystyle\int \frac{1}{\sqrt{16-x^2}}\,dx$.
$= \sin^{-1}\!\left(\dfrac{x}{4}\right) + C$.
Identify $a$ from the constant under the root: $\sqrt{a^2 - x^2}$, so $a = \sqrt{\text{constant}}$.
What each symbol means
$a > 0$; $\tan^{-1}$ denotes the arctangent function.
When to use it
Integrating a rational function whose denominator is a sum of a constant squared and $x^2$.
Worked sample
Find $\displaystyle\int \frac{1}{4+x^2}\,dx$.
$= \dfrac{1}{2}\tan^{-1}\!\left(\dfrac{x}{2}\right) + C$.
Your turn:
Evaluate $\displaystyle\int_0^{3} \frac{1}{9+x^2}\,dx$.
$= \Big[\dfrac{1}{3}\tan^{-1}\!\dfrac{x}{3}\Big]_0^3 = \dfrac{1}{3} \times \dfrac{\pi}{4} = \dfrac{\pi}{12}$.
Identify $a^2$ as the standalone constant: $4 + x^2$ means $a = 2$, giving the prefactor $\tfrac{1}{2}$.
What each symbol means
$a > 0$, $a \ne 1$ — constant base; $\ln a$ — natural log of the base.
When to use it
Integrating a general exponential $a^x$ where the base is not $e$.
Worked sample
Find $\displaystyle\int 2^x\,dx$.
$= \dfrac{2^x}{\ln 2} + C$.
Your turn:
Find $\displaystyle\int 10^x\,dx$.
$= \dfrac{10^x}{\ln 10} + C$.
Verify by differentiating: $\dfrac{d}{dx}\!\left(\dfrac{a^x}{\ln a}\right) = \dfrac{a^x \ln a}{\ln a} = a^x$. ✓
What each symbol means
$f(x)$ — a differentiable function; $f'(x)$ — its derivative.
When to use it
When the numerator is (or can be made to be) the derivative of the denominator — the standard reverse-chain-rule for logarithms.
Worked sample
Find $\displaystyle\int \frac{2x}{x^2+1}\,dx$.
Numerator $2x$ is the derivative of $x^2+1$, so the integral $= \ln|x^2+1| + C = \ln(x^2+1)+C$ (since $x^2+1 > 0$).
Your turn:
Find $\displaystyle\int \frac{3x^2}{x^3-5}\,dx$.
$= \ln|x^3-5| + C$.
If the numerator is a multiple of $f'(x)$, factor out the constant first: $\int \dfrac{6x}{x^2+1}\,dx = 3\ln(x^2+1)+C$.
Probability and Statistics
7What each symbol means
$P(A)$ — probability of event $A$; $P(\bar{A})$ — probability of the complement (A does not occur).
When to use it
When it is easier to count the cases where $A$ does not happen.
Worked sample
The probability of at least one head in three coin tosses.
$P(\text{at least one H}) = 1 - P(\text{no H}) = 1 - (1/2)^3 = 7/8$.
Your turn:
A bag contains 4 red and 6 blue balls. Three are drawn with replacement. Find $P(\text{at least one red})$.
$1 - (0.6)^3 = 1 - 0.216 = 0.784$.
'At least one' almost always signals the complement rule.
What each symbol means
$A \cup B$ — A or B (or both); $A \cap B$ — both A and B.
When to use it
Finding the probability that at least one of two events occurs, accounting for overlap.
Worked sample
$P(A) = 0.5$, $P(B) = 0.4$, $P(A \cap B) = 0.2$. Find $P(A \cup B)$.
$P(A \cup B) = 0.5 + 0.4 - 0.2 = 0.7$.
Your turn:
In a class, 60% passed Maths, 50% passed English, 30% passed both. What % passed at least one?
$60 + 50 - 30 = 80\%$.
For mutually exclusive events $P(A \cap B) = 0$, so the formula reduces to $P(A) + P(B)$.
What each symbol means
$P(A \mid B)$ — probability of A given B has occurred.
When to use it
When the sample space has been restricted by knowing event B occurred.
Worked sample
A die is rolled. Given the result is even, what is $P(\text{greater than 4})$?
Even results: {2,4,6}. Greater than 4 and even: {6}. $P = 1/3$.
Your turn:
$P(A \cap B) = 0.12$, $P(B) = 0.4$. Find $P(A \mid B)$.
$P(A \mid B) = 0.12/0.4 = 0.3$.
Can also be written $P(A \cap B) = P(B) \cdot P(A \mid B)$ — the multiplication rule.
What each symbol means
$x_i$ — each possible value; $P(X = x_i)$ — its probability.
When to use it
Finding the long-run average of a discrete probability distribution.
Units:
Same units as $x$.
Worked sample
$X$ takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find $E(X)$.
$E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1.0 + 0.9 = 2.1$.
Your turn:
Find the expected gain in a game where you win $\$5$ with probability 0.3 and lose $\$2$ with probability 0.7.
$E = 5(0.3) + (-2)(0.7) = 1.5 - 1.4 = \$0.10$.
Probabilities must sum to 1 — check this before computing $E(X)$.
What each symbol means
$E(X^2) = \sum x_i^2 P(X=x_i)$; $E(X) = \mu$.
When to use it
Measuring the spread of a discrete distribution.
Units:
Variance is in squared units; $\sigma = \sqrt{\text{Var}(X)}$ is in the same units as $X$.
Worked sample
With the distribution above ($\mu = 2.1$), find $\text{Var}(X)$.
$E(X^2) = 1(0.2)+4(0.5)+9(0.3) = 0.2+2+2.7 = 4.9$. $\text{Var} = 4.9 - 2.1^2 = 4.9 - 4.41 = 0.49$.
Your turn:
Find the standard deviation from the result above.
$\sigma = \sqrt{0.49} = 0.7$.
The shortcut formula $E(X^2) - \mu^2$ avoids expanding $(x-\mu)^2$ for each value.
What each symbol means
$n$ — number of trials; $p$ — probability of success; $r$ — number of successes; $\binom{n}{r} = \dfrac{n!}{r!(n-r)!}$.
When to use it
Fixed number of independent trials, each with the same probability of success/failure.
Worked sample
A fair coin is tossed 5 times. Find $P(X = 3)$.
$P(X=3) = \binom{5}{3}(0.5)^3(0.5)^2 = 10 \times 0.03125 = 0.3125$.
Your turn:
$X \sim B(10, 0.2)$. Find $P(X = 2)$.
$\binom{10}{2}(0.2)^2(0.8)^8 = 45 \times 0.04 \times 0.1678 \approx 0.302$.
Mean of binomial: $E(X) = np$. Variance: $\text{Var}(X) = np(1-p)$.
What each symbol means
$x$ — observed value; $\mu$ — population mean; $\sigma$ — standard deviation.
When to use it
Standardising a normally distributed variable to use the standard normal tables (or calculator).
Units:
$z$ is dimensionless.
Worked sample
Heights are normally distributed with $\mu = 170$ cm, $\sigma = 8$ cm. Find the z-score of 186 cm.
$z = \dfrac{186-170}{8} = 2$.
Your turn:
A score of 55 has $z = -1$ with $\sigma = 10$. Find the mean.
$-1 = (55-\mu)/10 \Rightarrow \mu = 65$.
A positive $z$-score means above the mean; negative means below. $|z| > 2$ is roughly the top/bottom 2.5%.
Combinatorics Ext 1
3What each symbol means
$n$ — total number of objects; $r$ — number selected; order matters.
When to use it
Counting arrangements where order matters: passwords, race finishing positions, seating in a row.
Worked sample
How many ways can 3 people be chosen from 8 to be president, vice-president and secretary?
$^8P_3 = \dfrac{8!}{5!} = 8 \times 7 \times 6 = 336$.
Your turn:
How many 4-digit PINs use distinct digits from {1,...,9}?
$^9P_4 = 9 \times 8 \times 7 \times 6 = 3024$.
If repetition is allowed, it's simply $n^r$, not $^nP_r$.
What each symbol means
$n$ — total objects; $r$ — number chosen; order does not matter.
When to use it
Selecting a committee, lottery numbers, dealing a hand of cards — whenever order is irrelevant.
Worked sample
How many ways can a committee of 4 be chosen from 9 people?
$\binom{9}{4} = \dfrac{9!}{4!\,5!} = 126$.
Your turn:
How many ways can 2 books be chosen from a shelf of 10?
$\binom{10}{2} = 45$.
$\binom{n}{r} = \binom{n}{n-r}$ — choosing $r$ is equivalent to choosing who is left out.
What each symbol means
$n$ — a positive integer; $\binom{n}{r}$ — binomial coefficients from Pascal's triangle.
When to use it
Expanding a power of a binomial, finding a specific term, or proving identities.
Worked sample
Find the term in $x^3$ in the expansion of $(2+x)^5$.
Term: $\binom{5}{3}(2)^2(x)^3 = 10 \times 4 \times x^3 = 40x^3$.
Your turn:
Find the constant term in $\left(x - \dfrac{2}{x}\right)^6$.
General term: $\binom{6}{r}x^{6-r}\left(-\dfrac{2}{x}\right)^r = \binom{6}{r}(-2)^r x^{6-2r}$. For constant: $6-2r=0 \Rightarrow r=3$. Term $= \binom{6}{3}(-8) = -160$.
The general term is $T_{r+1} = \binom{n}{r}a^{n-r}b^r$. To find a specific term, set the power of $x$ equal to the target and solve for $r$.
Ext 1 — Further Calculus and Functions Ext 1
4What each symbol means
$a > 0$; $|x| < a$.
When to use it
Differentiating the inverse sine function — arises in arc-length, geometry and inverse trig proofs.
Worked sample
Differentiate $y = \sin^{-1}\!\left(\dfrac{x}{3}\right)$.
$y' = \dfrac{1}{\sqrt{9-x^2}}$.
Your turn:
Differentiate $y = \sin^{-1}(2x)$.
$y' = \dfrac{2}{\sqrt{1-4x^2}}$.
This derivative is the reverse of the standard integral $\int \dfrac{1}{\sqrt{a^2-x^2}}\,dx$.
What each symbol means
$a > 0$; $|x| < a$.
When to use it
Differentiating inverse cosine — identical magnitude to $\sin^{-1}$ derivative but negative.
Worked sample
Differentiate $y = \cos^{-1}(x)$.
$y' = -\dfrac{1}{\sqrt{1-x^2}}$.
Your turn:
Differentiate $y = \cos^{-1}(3x)$.
$y' = -\dfrac{3}{\sqrt{1-9x^2}}$.
The only difference from $\sin^{-1}$: a negative sign. $\sin^{-1} x + \cos^{-1} x = \dfrac{\pi}{2}$ for all $|x| \le 1$.
What each symbol means
$a > 0$.
When to use it
Differentiating the inverse tangent function. Also confirms the standard integral $\int \dfrac{1}{a^2+x^2}\,dx$.
Worked sample
Differentiate $y = \tan^{-1}\!\left(\dfrac{x}{2}\right)$.
$y' = \dfrac{2}{4+x^2}$.
Your turn:
Differentiate $y = \tan^{-1}(5x)$.
$y' = \dfrac{5}{1+25x^2}$.
The formula on the sheet uses $\dfrac{a}{a^2+x^2}$; when $a = 1$ this simplifies to $\dfrac{1}{1+x^2}$.
What each symbol means
$g(x)$ — inner function (substitution); $F$ — antiderivative of $f$.
When to use it
Reversing the chain rule — use when the integrand contains a composite function and its derivative.
Worked sample
Find $\displaystyle\int 2x(x^2+1)^4\,dx$.
Let $u = x^2+1$, $du = 2x\,dx$. $\displaystyle\int u^4\,du = \dfrac{u^5}{5} + C = \dfrac{(x^2+1)^5}{5} + C$.
Your turn:
Find $\displaystyle\int \frac{x}{\sqrt{x^2+4}}\,dx$.
Let $u = x^2+4$: $\displaystyle\int \dfrac{1}{2\sqrt{u}}\,du = \sqrt{u} + C = \sqrt{x^2+4} + C$.
Change the limits when evaluating a definite integral by substitution — it avoids back-substituting.
Ext 1 — Vectors (2D) Ext 1
3What each symbol means
$\mathbf{u} = x\,\mathbf{i} + y\,\mathbf{j}$ — a 2D vector with components $x$ and $y$.
When to use it
Finding the length (size) of a vector — i.e. the distance formula applied to a displacement.
Units:
Same units as the components.
Worked sample
Find $|\mathbf{u}|$ if $\mathbf{u} = 3\mathbf{i} - 4\mathbf{j}$.
$|\mathbf{u}| = \sqrt{9+16} = 5$.
Your turn:
Find $|\mathbf{v}|$ if $\mathbf{v} = (1, -2)$ and $|\mathbf{w}| = ?$ for $\mathbf{w} = 2\mathbf{v}$.
$|\mathbf{v}| = \sqrt{5}$; $|\mathbf{w}| = 2\sqrt{5}$.
A unit vector has magnitude 1: $\hat{\mathbf{u}} = \dfrac{\mathbf{u}}{|\mathbf{u}|}$.
What each symbol means
$\mathbf{u} = (x_1, y_1)$, $\mathbf{v} = (x_2, y_2)$; $\theta$ — angle between them.
When to use it
Finding the angle between vectors, testing perpendicularity ($\mathbf{u}\cdot\mathbf{v}=0$), or scalar projection.
Units:
Square of the component units (but $\theta$ is dimensionless).
Worked sample
Find the angle between $\mathbf{u} = (3, 4)$ and $\mathbf{v} = (1, 0)$.
$\mathbf{u}\cdot\mathbf{v} = 3$, $|\mathbf{u}| = 5$, $|\mathbf{v}| = 1$. $\cos\theta = 3/5$, $\theta \approx 53.1^\circ$.
Your turn:
Show that $(2, -3)$ and $(3, 2)$ are perpendicular.
$(2)(3)+(-3)(2) = 6-6 = 0$. Dot product is zero, so they are perpendicular.
Two vectors are perpendicular iff their dot product is 0.
What each symbol means
Scalar projection — the signed length of $\mathbf{u}$ in the direction of $\mathbf{v}$.
When to use it
Finding the component of one vector along the direction of another (e.g. work done by a force).
Units:
Same units as $|\mathbf{u}|$.
Worked sample
Find the scalar projection of $\mathbf{u} = (4, 3)$ onto $\mathbf{v} = (1, 0)$.
$\dfrac{(4)(1)+(3)(0)}{1} = 4$.
Your turn:
Find the vector projection of $\mathbf{u} = (4, 3)$ onto $\hat{\mathbf{v}} = \mathbf{v}/|\mathbf{v}|$ where $\mathbf{v} = (3, 4)$.
Scalar proj $= \dfrac{12+12}{5} = 24/5$. Vector proj $= \dfrac{24}{5} \times \dfrac{(3,4)}{5} = \left(\dfrac{72}{25}, \dfrac{96}{25}\right)$.
The vector projection is $\dfrac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{v}|^2}\mathbf{v}$. Scalar projection is the magnitude of this.
Ext 1 — Projectile Motion Ext 1
3What each symbol means
$V$ — initial speed; $\alpha$ — angle of projection; $t$ — time.
When to use it
Finding horizontal position of a projectile (no air resistance, constant gravity).
Units:
Metres (or consistent length units).
Worked sample
A ball is projected at $20$ m/s at $30^\circ$. Find its horizontal displacement after 2 s.
$x = 20\cos 30^\circ \times 2 = 20 \times \tfrac{\sqrt{3}}{2} \times 2 = 20\sqrt{3} \approx 34.6$ m.
Your turn:
At what speed must a ball be projected at $45^\circ$ to travel 40 m horizontally in 2 s?
$40 = V\cos 45^\circ \times 2 = V \times \tfrac{\sqrt{2}}{2} \times 2 \Rightarrow V = 20\sqrt{2} \approx 28.3$ m/s.
Horizontal motion is uniform (no acceleration), so $x = $ constant $\times t$.
What each symbol means
$g = 9.8$ m/s² (or $10$ m/s²); upward positive.
When to use it
Finding the height of a projectile at time $t$.
Units:
Metres.
Worked sample
Using the same ball as above ($V=20$, $\alpha = 30^\circ$), find the height at $t = 2$ s (use $g = 10$).
$y = 20\sin 30^\circ \times 2 - \tfrac{1}{2}(10)(4) = 20 - 20 = 0$ m. (It has just landed.)
Your turn:
Find the maximum height of a projectile with $V = 30$ m/s, $\alpha = 60^\circ$ (use $g = 10$).
At max height $\dot{y}=0$: $t = \dfrac{V\sin\alpha}{g} = \dfrac{30 \times \sqrt{3}/2}{10} = \dfrac{3\sqrt{3}}{2}$ s. $y_{\max} = 30 \times \tfrac{\sqrt{3}}{2} \times \dfrac{3\sqrt{3}}{2} - \tfrac{1}{2}(10)\left(\dfrac{3\sqrt{3}}{2}\right)^2 = \dfrac{135}{4} - \dfrac{135}{8} = \dfrac{135}{8} = 33.75$ m.
At maximum height, vertical velocity is zero: set $\dot{y} = V\sin\alpha - gt = 0$.
What each symbol means
$V$ — initial speed; $\alpha$ — launch angle; $g$ — acceleration due to gravity.
When to use it
Maximum horizontal range for a projectile landing at the same height it was launched.
Units:
Metres.
Worked sample
Find the range of a projectile with $V = 30$ m/s at $\alpha = 45^\circ$ ($g = 10$).
$R = \dfrac{900 \times \sin 90^\circ}{10} = 90$ m.
Your turn:
Two launch angles give the same range. If one angle is $20^\circ$, what is the other?
$\sin 2\alpha$ is the same for $\alpha$ and $90^\circ - \alpha$, so the other angle is $70^\circ$.
Maximum range occurs at $\alpha = 45^\circ$ (since $\sin 2\alpha$ is maximised at $2\alpha = 90^\circ$).
Ext 2 — Complex Numbers Ext 2
4What each symbol means
$r = |z|$ — modulus; $\theta = \arg(z)$ — argument (in radians, usually $-\pi < \theta \le \pi$).
When to use it
Multiplying or dividing complex numbers, raising to powers, finding roots.
Worked sample
Write $z = 1 + i$ in polar form.
$r = \sqrt{2}$, $\theta = \pi/4$. So $z = \sqrt{2}\,\text{cis}(\pi/4)$.
Your turn:
Write $z = -2$ in polar form.
$r = 2$, $\theta = \pi$: $z = 2\,\text{cis}(\pi)$.
$\text{cis}\,\theta$ is standard NESA notation for $\cos\theta + i\sin\theta$. It is shorthand only — not a separate function.
What each symbol means
$r_1, r_2$ — moduli; $\theta_1, \theta_2$ — arguments.
When to use it
Multiplying complex numbers when they are in polar form — multiply moduli, add arguments.
Worked sample
Find $2\,\text{cis}(\pi/6) \times 3\,\text{cis}(\pi/3)$.
$= 6\,\text{cis}(\pi/6+\pi/3) = 6\,\text{cis}(\pi/2) = 6i$.
Your turn:
Find $\sqrt{2}\,\text{cis}(3\pi/4) \times \sqrt{2}\,\text{cis}(\pi/4)$.
$= 2\,\text{cis}(\pi) = -2$.
Division: $\dfrac{r_1\,\text{cis}\,\theta_1}{r_2\,\text{cis}\,\theta_2} = \dfrac{r_1}{r_2}\,\text{cis}(\theta_1-\theta_2)$ — divide moduli, subtract arguments.
What each symbol means
$n$ — any integer (and by extension, any rational).
When to use it
Raising a complex number to a power, proving trig identities (e.g. $\cos 3\theta$ in terms of $\cos\theta$), finding $n$th roots.
Worked sample
Find $(\cos(\pi/6) + i\sin(\pi/6))^6$.
$= \cos(\pi) + i\sin(\pi) = -1$.
Your turn:
Use De Moivre to find $(1+i)^8$.
$(1+i) = \sqrt{2}\,\text{cis}(\pi/4)$. $(1+i)^8 = (\sqrt{2})^8\,\text{cis}(2\pi) = 16$.
For the $n$ roots of unity: $z^n = 1 \Rightarrow z = \text{cis}(2\pi k/n)$ for $k = 0,1,\ldots,n-1$.
What each symbol means
$e$ — Euler's number; $\theta$ — angle in radians.
When to use it
Proving De Moivre's theorem, complex exponential representations, advanced integration.
Worked sample
Write $e^{i\pi} + 1 = 0$ (Euler's identity) using the formula.
$e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0 = -1$. So $e^{i\pi} + 1 = 0$.
Your turn:
Express $\cos\theta$ in terms of $e^{i\theta}$ and $e^{-i\theta}$.
$\cos\theta = \dfrac{e^{i\theta}+e^{-i\theta}}{2}$.
Euler's formula is the foundation behind De Moivre's theorem: $(e^{i\theta})^n = e^{in\theta}$.
Ext 2 — Further Integration Ext 2
3What each symbol means
$u, v$ — functions of $x$; $du = u'\,dx$, $dv = v'\,dx$.
When to use it
Integrating a product where one factor becomes simpler when differentiated and the other can be integrated — e.g. $x e^x$, $x\ln x$, $x^2\sin x$.
Worked sample
Find $\displaystyle\int x e^x\,dx$.
Let $u = x$, $dv = e^x\,dx$. Then $du = dx$, $v = e^x$. $\int x e^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x-1)+C$.
Your turn:
Find $\displaystyle\int x\ln x\,dx$.
Let $u = \ln x$, $dv = x\,dx$. $du = \dfrac{1}{x}dx$, $v = \dfrac{x^2}{2}$. $= \dfrac{x^2 \ln x}{2} - \int \dfrac{x}{2}\,dx = \dfrac{x^2\ln x}{2} - \dfrac{x^2}{4} + C$.
LIATE guide for choosing $u$: Logarithm, Inverse trig, Algebraic, Trig, Exponential — pick whichever comes first in that list.
What each symbol means
$A, B$ — constants found by equating numerators; $a, b$ — distinct real roots of the denominator.
When to use it
Splitting a rational function for integration when the denominator factors into linear factors.
Worked sample
Integrate $\displaystyle\int \frac{1}{x^2-1}\,dx$.
$\dfrac{1}{(x-1)(x+1)} = \dfrac{A}{x-1}+\dfrac{B}{x+1}$. $A = \tfrac12$, $B = -\tfrac12$. $\int = \tfrac12\ln|x-1| - \tfrac12\ln|x+1| + C = \tfrac12\ln\left|\dfrac{x-1}{x+1}\right|+C$.
Your turn:
Decompose $\dfrac{3x+1}{(x+1)(x-2)}$ into partial fractions.
$\dfrac{3x+1}{(x+1)(x-2)} = \dfrac{2/3}{x+1} + \dfrac{7/3}{x-2}$. (Verify by recombining.)
For a repeated factor $(x-a)^2$, the decomposition uses $\dfrac{A}{x-a} + \dfrac{B}{(x-a)^2}$.
What each symbol means
$t = \tan(x/2)$; transforms a rational trig integrand into a rational function of $t$.
When to use it
Integrating expressions like $\dfrac{1}{a + b\cos x}$ or $\dfrac{1}{a + b\sin x}$ that resist other methods.
Worked sample
Find $\displaystyle\int \frac{dx}{1 + \cos x}$.
With $t=\tan(x/2)$: $1+\cos x = 1+\dfrac{1-t^2}{1+t^2} = \dfrac{2}{1+t^2}$ and $dx = \dfrac{2}{1+t^2}\,dt$. So the integrand becomes $\dfrac{1+t^2}{2}\cdot\dfrac{2}{1+t^2}\,dt = dt$, giving $\int dt = t + C = \tan(x/2)+C$.
Your turn:
Why must you check $x = \pi$ when using the $t$-substitution on a definite integral?
$t = \tan(x/2)$ is undefined at $x = \pi$. Adjust limits or treat $\pi$ as a boundary.
The $dx$ substitution $dx = \dfrac{2}{1+t^2}dt$ is easy to forget — it always appears and must be included.
Ext 2 — Vectors in 3D, Mechanics and SHM Ext 2
6What each symbol means
$\mathbf{u} = (x, y, z)$ — a vector in three-dimensional space.
When to use it
Finding the length of a 3D vector or distance between two points in 3D.
Units:
Length units.
Worked sample
Find $|\mathbf{u}|$ if $\mathbf{u} = (2, -1, 2)$.
$|\mathbf{u}| = \sqrt{4+1+4} = 3$.
Your turn:
Find the distance from $A(1,2,3)$ to $B(4,2,-1)$.
$|\overrightarrow{AB}| = \sqrt{9+0+16} = 5$.
Everything from 2D vectors extends naturally: dot product, projection, unit vectors — just include the $z$-component.
What each symbol means
$\mathbf{u} = (x_1,y_1,z_1)$, $\mathbf{v} = (x_2,y_2,z_2)$; $\theta$ — angle between them.
When to use it
Testing perpendicularity in 3D, finding angles between lines/planes.
Worked sample
Find the angle between $\mathbf{u} = (1,0,1)$ and $\mathbf{v} = (0,1,1)$.
$\mathbf{u}\cdot\mathbf{v} = 1$, $|\mathbf{u}| = |\mathbf{v}| = \sqrt{2}$. $\cos\theta = \dfrac{1}{2}$, $\theta = 60^\circ$.
Your turn:
Find a vector perpendicular to both $(1,1,0)$ and $(0,1,1)$ using inspection.
Dot with $(a,b,c)$: $a+b=0$ and $b+c=0 \Rightarrow c=a$, $b=-a$. A solution: $(1,-1,1)$.
In Ext 2, the cross product is sometimes introduced for finding normals, but the dot product is always sufficient for angle questions.
What each symbol means
$x$ — displacement from equilibrium; $n > 0$ — angular frequency; $\ddot{x}$ — acceleration.
When to use it
Identifying or setting up SHM: any time acceleration is proportional to displacement and directed back to centre.
Units:
$n$ in rad/s; period $T = \dfrac{2\pi}{n}$.
Worked sample
A particle moves so that $\ddot{x} = -4x$. Find the period.
$n^2 = 4$, $n = 2$. $T = \dfrac{2\pi}{2} = \pi$ seconds.
Your turn:
If $T = \pi/3$, find $n$.
$n = \dfrac{2\pi}{\pi/3} = 6$.
The negative sign is essential: it shows acceleration always points towards the equilibrium position.
What each symbol means
$a$ — amplitude; $n$ — angular frequency; $\alpha$ — phase shift.
When to use it
Writing the general solution to $\ddot{x} = -n^2 x$; initial conditions determine $a$ and $\alpha$.
Worked sample
A particle starts at $x = 3$ with zero velocity: $n = 2$. Give $x(t)$.
$x = 3\cos(2t)$ (zero phase shift, starts at max displacement).
Your turn:
For $x = 4\sin(3t + \pi/6)$, find the velocity when $t = 0$.
$\dot{x} = 12\cos(3t+\pi/6)$. At $t=0$: $\dot{x} = 12\cos(\pi/6) = 12 \times \tfrac{\sqrt{3}}{2} = 6\sqrt{3}$.
Velocity: $\dot{x} = -an\sin(nt+\alpha)$ or $an\cos(nt+\alpha)$ depending on the form. Max velocity $= an$ (at equilibrium).
What each symbol means
$v$ — velocity; $a$ — amplitude; $x$ — displacement from centre.
When to use it
Finding speed at any displacement without knowing time — very common in exam questions.
Units:
Speed in m/s if consistent SI units.
Worked sample
A particle undergoes SHM with $a = 5$, $n = 3$. Find the speed at $x = 4$.
$v^2 = 9(25 - 16) = 81$, $v = 9$.
Your turn:
Find the amplitude of SHM where $n = 2$ and the speed at $x = 3$ is 8.
$64 = 4(a^2-9) \Rightarrow a^2 = 25 \Rightarrow a = 5$.
Derived by energy methods or by differentiating $v = \dot{x}$ and using the chain rule: $\ddot{x} = v\dfrac{dv}{dx}$.
What each symbol means
$m$ — mass; $\ddot{x}$ — acceleration; $F_{\text{net}}$ — net force (gravity plus resistance).
When to use it
Setting up differential equations for objects moving with air resistance or drag proportional to speed.
Units:
Force in Newtons; consistent SI units throughout.
Worked sample
A particle of mass $m$ falls under gravity with drag $kv$. Write the equation of motion.
$m\ddot{x} = mg - kv$ (taking downward positive).
Your turn:
Find the terminal velocity for the particle above.
At terminal velocity $\ddot{x} = 0$: $mg = kv_T \Rightarrow v_T = mg/k$.
Always define your positive direction first, and be consistent with sign conventions for gravity and drag throughout the working.
Frequently asked questions
Yes. NESA provides the same reference sheet across Mathematics Advanced, Extension 1 and Extension 2, and you get it in every exam.
It covers measurement, financial mathematics, sequences and series, logarithms/exponentials/indices, trigonometry, differentiation, standard integrals, and probability and statistics — but not every formula you'll need, so some still have to be known.
Some. The sheet has many formulas, but not all — and it won't tell you which one a question needs or how to apply it. The marks are in choosing and using the right formula correctly.
Work through past papers with the sheet open so you learn where each formula sits and the question types it solves. That way you find it instantly in the exam instead of scanning under pressure.