BIG IDEARecognise each graph by its shape. Then use the graph to read values and solve equations.
y = ax³ (a>0)
y = ax³ (a<0)
y = a/x
y = a/x²
y = kaˣ (a>1)
y = ka⁻ˣ
y = x² − 1
y = x + 1/x (sum)
Graph
Key feature
y = a/x
two branches, never touches axes (asymptotes x = 0, y = 0)
y = a/x²
both branches above x-axis when a > 0
y = kaˣ
cuts y-axis at k, never touches x-axis
y = ax³
S-shape through origin
Gradient of a curve — draw a tangent
tangent touches at one point
Steps
① draw a straight line that just touches the curve ② pick 2 far-apart points on the tangent ③ gradient = rise ÷ run here: 4 ÷ 2 = 2
🧠 HOOKSolve f(x) = 0 graphically: read where the curve cuts y = 0. Solve f(x) = g(x): draw both, read the x-values where they meet.
E Math · Sec 3
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6
Sine & Cosine Rules
G4 · obtuse angles, area ½ab sin C
BIG IDEAFor any triangle: side a is opposite angle A. Choose sine rule or cosine rule by what you are given.
small letter = side opposite capital angle
Formulas (given in exam)
a/sin A = b/sin B = c/sin C a² = b² + c² − 2bc cos A Area = ½ab sin C
You know…
Use
Find
2 angles + 1 side
sine rule
another side
2 sides + non-included angle
sine rule
an angle
2 sides + included angle
cosine rule
3rd side
all 3 sides
cosine rule
cos A = (b² + c² − a²)/2bc
Cosine example
b = 7, c = 5, A = 40° a² = 49 + 25 − 70 cos 40° = 20.38 a = 4.51 · area = ½(7)(5) sin 40° = 11.2
Sine example
A = 50°, B = 60°, a = 8 b = 8 sin 60° ÷ sin 50° = 9.04
Obtuse angles (90° – 180°)
Rules
sin(180° − θ) = sin θ cos(180° − θ) = −cos θ cos is negative for obtuse angles → cosine rule gives obtuse angles automatically
⚠ TRAPsin⁻¹ only gives the acute angle. If the angle could be obtuse, also check 180° − answer.
E Math · Sec 3
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7
Bearings, Elevation & 3D
G4 · applications of trigonometry
BIG IDEABearing = angle measured from North, clockwise, written with 3 digits (e.g. 065°).
bearing of B from A = 051° · A from B = 051° + 180° = 231°
Rules
draw a North line at every point back bearing = bearing ± 180° North lines are parallel → use alternate / co-interior angles
Elevation & depression
both measured from the horizontal — they are equal (alt. ∠s)
🧊 3D problems
① find the right-angled triangle that contains the angle ② you often need a line on the base first (Pythagoras) ③ angle between a line and a plane = angle with its projection on the plane
⚠ TRAPAngle of depression is measured down from the horizontal, never from the vertical. Always draw the North arrow before measuring.
E Math · Sec 3
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8
Radians, Arcs & Sectors
G5 · arc length, sector & segment area
BIG IDEAπ rad = 180°. With θ in radians: arc s = rθ and sector A = ½r²θ.
sector (green) · arc (red) · segment = between chord and arc
Formulas
degrees → rad: × π/180 rad → degrees: × 180/π arc length s = rθ sector area = ½r²θ segment = ½r²(θ − sin θ)
Example
r = 6 cm, θ = 1.2 rad s = 6(1.2) = 7.2 cm sector = ½(36)(1.2) = 21.6 cm² segment = 18(1.2 − sin 1.2) = 4.82 cm²
Perimeter of sector
= 2r + s = 12 + 7.2 = 19.2 cm
⚠ TRAPCalculator in RAD mode for sin 1.2. Using degrees? then use (θ/360) × 2πr and (θ/360) × πr².
BIG IDEA8 circle facts. In every proof, write the reason next to each step.
⊥ from centre bisects chord
equal chords are equidistant from centre
tangent ⊥ radius
∠ in semicircle = 90°
∠ at centre = 2 × ∠ at circumference
∠s in same segment are equal
cyclic quad: opposite ∠s = 180°
tangents from a point are equal
✍ REASONSUse short reasons: "∠ in semicircle", "∠ at centre = 2∠ at circumference", "∠s in same segment", "∠s in opp. segment" (cyclic quad), "tan ⊥ rad", "tangents from ext. pt".
⚠ TRAPLook for hidden isosceles triangles: two radii from the centre are equal → base angles equal.
E Math · Sec 3
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10
Congruence & Similarity Tests
G2 · tests, proofs, bisectors, scale drawings
BIG IDEAProve congruence/similarity by listing the matching parts with reasons, then state the test.
≡ Congruence tests
SSSSASASA / AASRHS RHS = right angle, hypotenuse, side AAA and SSA do NOT prove congruence
~ Similarity tests
AASSS (all in ratio)SAS (ratio + included ∠) AA is the easiest: 2 equal angles
Proof layout
In △ABC and △DEF: AB = DE (given) · ∠B = ∠E (alt. ∠s, AB ∥ DE) · BC = EF (given) ∴ △ABC ≡ △DEF (SAS)
Bisectors & scale drawings
⊥ bisector: points equidistant from A and B
∠ bisector: points equidistant from both lines
🧠 HOOKScale drawing: write the scale, use a sharp pencil, ruler and protractor, keep construction arcs. Measure to the nearest mm / degree.
E Math · Sec 3
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11
Areas & Volumes of Similar Figures
G2
BIG IDEAIf lengths are in ratio k, areas are in ratio k² and volumes (and masses) in ratio k³.
double the length → area ×4, volume ×8
Ratio of…
Rule
lengths
l₁ : l₂ = k
areas
A₁/A₂ = (l₁/l₂)²
volumes / masses
V₁/V₂ = (l₁/l₂)³
Example
Similar cones, heights 3 : 5. Small volume = 54 cm³ V₂ = 54 × (5/3)³ = 250 cm³ area ratio = 9 : 25