Geometry and Trigonometry

Introduction to Geometry and Trigonometry

Geometry and trigonometry in Math AA is not a memorisation exercise for area formulas — it is the study of space, shape, and periodic behaviour, grounded in precise algebraic reasoning. You begin concretely: distances in 3D, volumes of solids, and the sine and cosine rules that liberate you from right-angled triangles. From there, the unit circle reframes sine and cosine as coordinates of a point — a geometric insight that extends trigonometry to any angle and makes the periodic nature of these functions immediately visible. At HL, vectors complete the picture by giving you an algebraic system for describing lines, planes, and their intersections in space. The trigonometric functions you study here reappear throughout calculus (their derivatives, their integrals, their Maclaurin series) and in the modelling of oscillatory phenomena, making this section as much a foundation for later work as it is a destination in itself.

Section Map — What Is Covered

• 3D Geometry and Mensuration (3.1): distance and midpoint in space, volumes and surface areas of cones, spheres, and pyramids, and finding angles between lines and planes in solids
• The Sine Rule and Cosine Rule (3.2): solving any triangle — plus the trigonometric area formula $\\frac12 ab\\sin C$ — and handling the ambiguous case (SSA)
• Radian Measure and the Unit Circle (3.3): why radians make arc length $l=r\\theta$ and sector area $A=\\frac12 r^2\\theta$ elegantly simple, and the unit circle as the geometric definition of sine and cosine for any angle
• Trigonometric Identities and Graphs (3.4): the Pythagorean identity $\\sin^2\\theta+\\cos^2\\theta=1$ and the amplitude-period-phase-shift analysis of the general form $y=a\\sin(b(x-c))+d$
• Solving Trigonometric Equations (3.5): finding all solutions in a finite interval — not just the principal value your calculator returns — including equations with multiplied angles and quadratics in sin or cos
• Further Trigonometric Identities (3.6, HL): compound-angle formulas, double-angle identities including the three equivalent forms of $\\cos(2\\theta)$, and the reciprocal functions sec, csc, and cot
• Inverse Trigonometric Functions (3.7, HL): arcsin, arccos, arctan — their restricted domains and ranges, and why $\\arcsin(\\sin 150\\degree)$ equals 30°, not 150°
• Vectors (3.8–3.10, HL): from representation and basic operations through the scalar product (angles, perpendicularity) and vector product (normal vectors, areas), to the vector equations of lines and planes in 3D