Build Your Edge: The Ultimate Prep Course for Advanced Maths

Understand how sine, cosine, and tangent are defined using the unit circle. This lesson introduces the geometric meaning of each ratio and shows how they relate to right-angled triangles. A solid first step into trigonometry!

Learn how degrees and radians relate, and why radians are essential in advanced maths. This lesson explains key angle conversions and helps you memorise exact values of sine, cosine, and tangent — crucial building blocks for everything to come.

Explore the shape, period, and amplitude of sine and cosine graphs. This lesson shows how these functions repeat, what their key features mean, and how to spot identities directly from the graphs — a vital skill for solving trigonometric problems.

Dive into the unique features of the tangent graph — its period, domain restrictions, vertical asymptotes, and symmetry. This lesson explains why tangent behaves differently from sine and cosine and highlights key properties every student should know.

Master the key identities that come from the symmetry of sine, cosine, and tangent. This lesson highlights essential formulas — especially those worth memorising — and introduces the powerful Pythagorean identity used throughout trigonometry.

Learn how to solve trigonometric equations using symmetry, periodicity, and inverse functions. This lesson walks you through key steps using an example, helping you apply known identities and transformations to find all valid solutions within a given range.

Discover how trigonometric graphs change through dilations, shifts, and reflections. This lesson walks you through a worked example of transformation — helping you visualise how each algebraic change affects the graph’s shape and position.

Learn how to solve non-right-angled triangles using the sine and cosine rules. This lesson covers the formulas, step-by-step proofs, and worked examples — giving you the tools to find unknown sides and angles in any triangle.

Learn how to identify and work with parallel lines using angle relationships. This lesson covers alternate angles, corresponding angles, and co-interior angles — and shows how these properties can help prove lines are parallel.

Explore the fundamental rules of triangles, including the triangle inequality theorem, Pythagoras’ theorem for right-angled triangles, and key properties of isosceles triangles. Understand how side lengths and angles relate — essential knowledge for geometric problem-solving.

Discover how to calculate the total interior angle sum of any polygon using a simple formula. This lesson explains how polygons can be divided into triangles, introduces the relationship between interior and exterior angles, and includes worked examples for regular and irregular polygons.

Master key angle and length properties within circles. This lesson explores central and inscribed angles, cyclic quadrilaterals, tangents, and the Alternate Segment Theorem — essential for solving complex geometric problems involving circles.

Understand how sequences are structured, how terms are defined, and how recurrence relations can generate entire sequences. This lesson introduces arithmetic and recursive patterns, including the Fibonacci sequence, and explores how to move from recursive to explicit formulas.

Learn how to identify and calculate terms and sums in both arithmetic and geometric series. This lesson covers key formulas, including nth-term rules and sum formulas, and provides examples showing how linear and exponential patterns arise in these sequences.

Explore what happens when a geometric series goes on forever. This lesson explains when an infinite series converges to a fixed sum, the formula used to calculate it, and why some series diverge instead. Includes worked examples to build understanding.

Learn how the modulus (or absolute value) function behaves algebraically and graphically. This lesson explains key properties, how to solve modulus inequalities, and how to sketch piecewise-defined graphs involving modulus expressions.

Learn how modulus affects function graphs, including transformations like y = |f(x)| and y = f(|x|). This lesson shows how to reflect parts of graphs across axes and interpret how modulus changes the shape and symmetry of curves.

Learn how to derive and apply the equation of a circle on the coordinate plane. This lesson covers circles centered at the origin and at any point (h, k), using distance formula and completing the square to interpret general equations.

Explore the geometry of ellipses on the coordinate plane. This lesson explains how to derive and interpret the standard form of an ellipse, how it relates to circles through transformations, and how to convert general equations by completing the square.

Understand the structure and behavior of hyperbolas on the coordinate plane. This lesson introduces the standard form of a hyperbola, explores asymptotes, rectangular hyperbolas, and conjugate hyperbolas, and provides visual insights into graphing them accurately.

Understand how to express curves using a parameter instead of linking x and y directly. This lesson introduces parametric forms for circles, ellipses, hyperbolas, and parabolas, and shows how to convert between parametric and Cartesian equations using substitution and trigonometric identities.

Step into 3D geometry by exploring the Cartesian coordinate system in three dimensions. This lesson introduces 3D points, axes, and how to calculate distances using the 3D extension of Pythagoras’ Theorem — laying the foundation for spatial reasoning and vector geometry.

Extend your knowledge from circles to spheres in 3D space. This lesson introduces the equation of a sphere centered at the origin or any point (a, b, c), and shows how it parallels the 2D circle equation using 3D distance formulas.

Discover what it means to differentiate a function. This lesson introduces the formal definition of a derivative as a limit, explains its geometric interpretation as the gradient of the tangent line, and links it to the angle formed with the x-axis.

Master the key rules of differentiation, including derivatives of standard functions, and essential rules like the product, quotient, and chain rules. This lesson provides quick-reference tables and helpful patterns to boost confidence when working with calculus problems.

This lesson explores how to differentiate composite functions involving logarithms and trigonometric expressions. Using the example f(x) = ln|sec x|, students learn to apply domain restrictions, rewrite expressions, and use derivative rules to simplify and justify that the final result is f’(x) = tan x.

Learn how to differentiate inverse functions using the chain rule and implicit differentiation. We show why flipping the function can simplify the process, with a clear example using x = y³.

In this lesson, you’ll learn how integration works as the reverse of differentiation. We explore why a single function can have many antiderivatives and what the “+ c” really means. An example and graph help illustrate how all these functions belong to the same family.

This lesson covers essential integration rules, including common results and how they relate to the reverse of differentiation. We focus on patterns (not memorisation), and explain how to handle forms like ax + b using chain rule ideas. A handy summary table and a few key rules wrap it up.

This lesson introduces definite integrals, which help resolve the “+c” ambiguity in antiderivatives. You’ll learn how limits of integration set clear boundaries and how the result represents the net area under a curve. We also explain why the Fundamental Theorem of Calculus makes this work.

This lesson introduces systems of linear equations — multiple equations with shared variables. You’ll learn how to solve them algebraically or geometrically, interpret solutions as intersections of planes, and use parameters when infinite solutions exist. Visuals help clarify all possible outcomes.

When solving systems with more than three variables, the basic methods still apply — but solving by hand becomes impractical. This lesson explains why matrix methods and technology are preferred for larger systems, and notes that these advanced techniques are usually taught in university-level courses.