Mathematics: Geometry and Trigonometry

Mathematics: Geometry and Trigonometry

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

• Pythagorean theorem for a right triangle: a² + b² = c², where c is the hypotenuse and a, b are the two legs.
• Pythagorean identity: sin²θ + cos²θ = 1 for every angle θ.
• Law of sines: a/sin A = b/sin B = c/sin C; law of cosines: c² = a² + b² − 2ab·cos C. Use these on non-right triangles only.
• Area of a triangle with two sides and the included angle: A = ½·a·b·sin C.
• Sum of interior angles of an n-sided polygon: (n − 2)·180°. Sum of exterior angles is always 360°.
• Unit circle values to memorise: sin 0° = 0, sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1; cosines mirror these in reverse order.
• Arc length s = rθ and sector area A = ½r²θ use radians (π rad = 180°), not degrees.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Angle and Triangle Foundations

• Complementary angles sum to 90°, supplementary to 180°, and vertically opposite angles are equal. When a transversal cuts two parallel lines, corresponding, alternate, and co-interior angles obey fixed relationships (e.g. alternate interior angles are equal).
• Triangle congruence rules: SSS, SAS, ASA, RHS. Similarity follows from equal angles (AAA) or proportional sides (SSS similarity), with the scale factor squared affecting areas and cubed affecting volumes.

Polygons and Circles

• For an n-sided polygon, sum of interior angles = (n − 2)·180°; each interior angle of a regular polygon = (n − 2)·180°/n.
• Circle theorems frequently tested in NAT-I: the inscribed angle equals half the central angle subtending the same arc; the angle in a semicircle is 90°; a tangent is perpendicular to the radius at the point of contact; the tangent–chord angle equals the inscribed angle in the alternate segment.

Trigonometric Ratios and Identities

• In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Reciprocals are cosecant, secant, cotangent.
• For non-right triangles use the law of sines to find a side given an opposite angle pair, and the law of cosines to find a side from the other two sides and the included angle (or to find an angle from three sides).
• Heights and distances problems: draw the right triangle, label the angle of elevation or depression, then apply tan θ = height/base or sin θ = height/hypotenuse.

Mensuration Reference

Shape	Area / Volume	Key variables
Triangle	A = ½·b·h (or ½·a·b·sin C)	base b, height h
Circle / sector	A = πr², A_sector = ½r²θ (rad)	radius r, angle θ
Cylinder	V = πr²h, LSA = 2πrh	radius r, height h
Cone	V = ⅓πr²h, CSA = πrℓ	radius r, height h, slant ℓ = √(r²+h²)
Sphere	V = 4/3 πr³, SA = 4πr²	radius r

NAT-I Question Patterns

• 5–8 MCQs of this combined topic per test, mixing one-step area/volume with two-step trig problems.
• Convert all linear units before computing areas (cm² vs m²) and volumes (cm³ vs m³).

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Quadrant Behaviour and the ASTC Rule

On the unit circle, signs of the three primary functions follow ASTC: in All (Quadrant I) all are positive; Sine positive in II; Tangent positive in III; Cosine positive in IV. The reference angle is the acute angle made with the x-axis. For example, sin 150° = sin 30° = ½ (QII), while cos 150° = −cos 30° = −√3/2. Compound identities worth committing:

• tan θ = sin θ / cos θ
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ
• sin(90° − θ) = cos θ, cos(90° − θ) = sin θ

Radians vs Degrees

θ (rad) = θ° × π/180. A common NAT-I trap: using s = rθ with degrees — the answer comes out 180/π times too large. Always convert first when the angle is given in degrees but the formula expects radians.

Heights, Distances, and Bearings

• Angle of elevation is measured upward from the horizontal at the observer; angle of depression is measured downward from the horizontal at the top of an object. These two angles are equal when measured on the same line of sight.
• A bearing is measured clockwise from North (e.g. N 60° E or simply 060°). Translate bearings into a right triangle by drawing the North line and decomposing motion into N–S and E–W components.

Common Mistakes to Eliminate

1. Applying Pythagoras to a triangle that is not right-angled — switch to the law of cosines.
2. Using ½r²θ for the segment area; that formula gives the sector. Subtract the triangular portion ½r²sin θ for the segment.
3. Confusing slant height ℓ with vertical height h of a cone: ℓ = √(r² + h²), and CSA = πrℓ while V = ⅓πr²h.
4. Dropping units when converting cm² to m² — divide by 10 000, not 100.
5. Mixing up sin, cos, tan because the problem asks for the opposite side but the diagram is rotated; always re-label the angle.

Worked Micro-Example

A right triangle has legs 5 cm and 12 cm. Find the hypotenuse and the area.

• Hypotenuse: c = √(5² + 12²) = √169 = 13 cm.
• Area: A = ½·5·12 = 30 cm².
• Trig ratios at the angle opposite the 5 cm side: sin θ = 5/13, cos θ = 12/13, tan θ = 5/12. Verify: sin²θ + cos²θ = 25/169 + 144/169 = 1 ✓

Practice Prompts

1. A triangle has sides 7, 9, and an included angle of 60°. Use the law of cosines to find the third side, then compute its area using ½·a·b·sin C.
2. A sector of a circle with radius 10 cm subtends an angle of 1.2 radians. Find its arc length and area, then state what fraction of the full circle it represents.

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