Is a formula sheet provided in the DSE M2 exam?

The M2 question paper prints a short list of trigonometric formulas for reference (compound angle, product-to-sum and sum-to-product). Derivatives, integrals, matrix and vector formulas are not given, so they need to be memorised.

Which M2 formulas should I memorise first?

The derivative and integral tables, integration by parts, the 2 × 2 inverse, the cross product determinant, and the projection and area formulas for vectors. They are used in almost every paper.

How do I remember the sum-to-product formulas?

Derive them from the compound angle formulas: add or subtract sin(A + B) and sin(A − B), then rename the angles. Understanding where they come from is more reliable than memorising eight lines.

DSE M2 Formula Sheet: Every Formula by Topic

This page collects the formulas the DSE M2 curriculum expects you to know, grouped by topic. Each heading links to the full notes for that topic, with worked examples and an interactive graph.

Ask how to use a formula

The unit circle behind the trigonometric formulas: for the angle π/3 radians, the point on the circle is (cos θ, sin θ) = (½, √3/2). Try it yourself →

Trigonometry Formulas

π rad = 180°; 1 + tan² θ = sec² θ; 1 + cot² θ = cosec² θ
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
sin 2A = 2 sin A cos A; cos 2A = cos² A − sin² A = 1 − 2 sin² A = 2 cos² A − 1; tan 2A = 2 tan A / (1 − tan² A)
sin² A = ½(1 − cos 2A); cos² A = ½(1 + cos 2A)
2 sin A cos B = sin(A + B) + sin(A − B); 2 cos A cos B = cos(A + B) + cos(A − B); 2 sin A sin B = cos(A − B) − cos(A + B)
sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2); sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2)
cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2); cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)

Notes: trigonometry in M2 →

Binomial Theorem, e and Limits

(a + b)ⁿ = Σ C(n, r) aⁿ⁻ʳ bʳ; general term T_r+1 = C(n, r) aⁿ⁻ʳ bʳ
e = lim_n→∞ (1 + 1/n)ⁿ; eˣ = 1 + x + x²/2! + x³/3! + …
lim_θ→0 (sin θ)/θ = 1; lim_x→0 (eˣ − 1)/x = 1

Binomial theorem → · Limits →

Differentiation Formulas

f′(x) = lim_h→0 [f(x + h) − f(x)] / h
(xⁿ)′ = nxⁿ⁻¹; (sin x)′ = cos x; (cos x)′ = −sin x; (tan x)′ = sec² x; (eˣ)′ = eˣ; (ln x)′ = 1/x
(uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²; dy/dx = (dy/du)(du/dx)
Tangent: y − f(x₀) = f′(x₀)(x − x₀)

Differentiation → · Applications →

Integration Formulas

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1); ∫ (1/x) dx = ln|x| + C; ∫ eˣ dx = eˣ + C
∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C
∫ u dv = uv − ∫ v du
∫_−a^a f(x) dx = 0 (odd); = 2∫₀^a f(x) dx (even)
Area between curves: ∫_a^b [f(x) − g(x)] dx
Volume about the x-axis: π∫_a^b y² dx; about the y-axis: π∫_c^d x² dy

Indefinite integration → · Which method? → · Area and volume →

Matrices, Determinants and Linear Systems

|a b; c d| = ad − bc; order 3 by expansion along a row with signs + − +
A⁻¹ = (1/|A|) adj A; [a b; c d]⁻¹ = (1/(ad − bc)) [d −b; −c a]
(AB)⁻¹ = B⁻¹A⁻¹; |AB| = |A||B|; (Aᵀ)⁻¹ = (A⁻¹)ᵀ
Cramer’s rule: x = Δₓ/Δ, y = Δᵧ/Δ, z = Δ_z/Δ when Δ ≠ 0
Homogeneous system: non-trivial solutions ⟺ |A| = 0

Matrices → · Linear systems →

Vector Formulas

AB = OB − OA; |xi + yj + zk| = √(x² + y² + z²)
a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃
a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k; |a × b| = |a||b| sin θ
Projection of b onto a: ((a · b)/|a|²)a
Area of triangle: ½|AB × AC|; section formula: OP = (na + mb)/(m + n)

Dot and cross product → · Applications of vectors →