This is the complete reference card for WACE Year 12 Mathematics Methods. Every formula, rule and shortcut you should have committed to memory by exam day.
Unit circle (memorise this)
| Angle | sin | cos | tan |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| pi/6 (30deg) | 1/2 | sqrt(3)/2 | sqrt(3)/3 |
| pi/4 (45deg) | sqrt(2)/2 | sqrt(2)/2 | 1 |
| pi/3 (60deg) | sqrt(3)/2 | 1/2 | sqrt(3) |
| pi/2 (90deg) | 1 | 0 | undefined |
Differentiation rules
| Function f(x) | Derivative f'(x) |
|---|---|
| x^n | n x^(n-1) |
| e^x | e^x |
| e^(ax) | a e^(ax) |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec^2(x) |
| sin(ax+b) | a cos(ax+b) |
| cos(ax+b) | -a sin(ax+b) |
Composite rules
- Chain rule: derivative of f(g(x)) = f'(g(x)) g'(x). "Derivative of outside, leave inside, times derivative of inside."
- Product rule: (uv)' = u'v + uv'. "First times derivative of second, plus second times derivative of first."
- Quotient rule: (u/v)' = (u'v - uv') / v^2. "Low d-high minus high d-low, all over low squared."
Integration rules
| Function | Antiderivative + C |
|---|---|
| x^n (n != -1) | x^(n+1) / (n+1) |
| 1/x | ln|x| |
| e^x | e^x |
| e^(ax) | e^(ax) / a |
| sin(ax) | -cos(ax) / a |
| cos(ax) | sin(ax) / a |
| (ax + b)^n | (ax + b)^(n+1) / (a(n+1)) |
Definite integral applications
- Area under curve: integral from a to b of f(x) dx (positive only when f(x) >= 0).
- Area between curves: integral from a to b of [f(x) - g(x)] dx where f(x) >= g(x).
- Average value of function: (1 / (b - a)) integral from a to b of f(x) dx.
- Total change: integral of rate-of-change function = total accumulated change.
Trigonometric identities
- sin^2(x) + cos^2(x) = 1
- tan(x) = sin(x) / cos(x)
- sin(-x) = -sin(x), cos(-x) = cos(x), tan(-x) = -tan(x)
- sin(pi - x) = sin(x), cos(pi - x) = -cos(x)
- Double angle: sin(2x) = 2 sin(x) cos(x); cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1 = 1 - 2sin^2(x)
Logarithm and exponential rules
- log_a(xy) = log_a(x) + log_a(y)
- log_a(x/y) = log_a(x) - log_a(y)
- log_a(x^n) = n log_a(x)
- log_a(b) = ln(b) / ln(a) (change of base)
- e^(ln(x)) = x and ln(e^x) = x
- a^x a^y = a^(x+y)
- (a^x)^y = a^(xy)
Probability and discrete random variables
- P(A and B) = P(A) P(B) when A and B are independent.
- P(A | B) = P(A and B) / P(B) (conditional probability).
- P(A or B) = P(A) + P(B) - P(A and B).
- Expected value E(X) = sum of x P(x) for discrete random variables.
- Variance Var(X) = E(X^2) - (E(X))^2.
- Binomial distribution X ~ Bin(n, p): P(X = k) = nCk p^k (1-p)^(n-k); E(X) = np; Var(X) = np(1-p).
Continuous random variables
- P(a <= X <= b) = integral from a to b of f(x) dx where f is the probability density function.
- E(X) = integral of x f(x) dx (over the support).
- Total area under f(x) = 1.
- Normal distribution Z ~ N(0, 1) is symmetric about 0; use the standard normal table or CAS.
- For X ~ N(mu, sigma^2): standardise as Z = (X - mu) / sigma.
Statistical inference
- Sample mean: (sum of x_i) / n.
- Confidence interval for population mean (large n): xbar +/- z (sigma / sqrt(n)).
- Margin of error: z (sigma / sqrt(n)) at the chosen confidence level.
- Hypothesis test for proportion: z = (phat - p0) / sqrt(p0 (1 - p0) / n).
Print and pin this
Print this page. Stick it inside the cover of your Methods folder. Refer to it before every test, every past paper, and every revision session. By exam day, you should know every entry from memory.
For the weekly study cadence that builds these formulas into long-term memory, see our how to study Methods post. For calculator-free practice protocols, see the calculator-free section explained.
If you want a Methods tutor who drills these formulas through past paper practice every week, book a free trial class. The first lesson is free.