Trigonometry - Mathematics (Undergraduate Foundation)

This course provides a complete guide to trigonometry, from the principles of circular measure to functions of any angle. It covers the unit circle definitions and culminates in a full treatment of the addition and factor formulae. This is the mathematical language of cycles, waves, and angles. A command of trigonometry is essential for any field that models periodic or geometric systems. It is used extensively in physics to analyse waves and oscillations, in engineering for structural analysis and signal processing, and in computer graphics for rotations and 3D modelling. By the end of this course, you will be able to convert between degrees and radians, define the six trigonometric functions for any angle using the unit circle, and apply the addition and factor formulae to manipulate and simplify trigonometric expressions and solve equations. This course is a mandatory prerequisite for any student of mathematics, physics, or engineering. It provides the necessary foundation for the study of calculus, complex numbers, and vector analysis.

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

[NUC Core] MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the mathematics every scientist, engineer, and computer scientist must know. This track covers the NUC Core syllabus for MTH 101: Elementary Mathematics I. From set theory and sequences to trigonometry and complex numbers, you will gain the exact tools used in higher mathematics, algorithm design, data analysis, physics, and engineering. This programme is for first-year university students in mathematics, computer science, engineering, or physics, as well as serious learners preparing for undergraduate study. It also serves professionals who need a rigorous mathematical foundation for technical fields. By the end, you will handle formal set notation, progressions, quadratic models, combinatorics, induction proofs, binomial expansions, trigonometric analysis, and complex arithmetic. You will be equipped to solve real problems in science, engineering, computing, and finance, and ready for advanced study in calculus, statistics, and applied mathematics.

Course Chapters

1. Introduction

This chapter introduces the field of trigonometry. It defines its purpose as the study of relationships between the angles and side lengths of triangles and outlines its broader application to periodic phenomena, such as waves. Key objectives include understanding the course structure and the fundamental scope of trigonometry as a critical branch of mathematics for science and engineering.

Chapter lessons

1-1. Welcome

A direct statement of the course's purpose and structure. This lesson outlines the progression from circular measure and angles to advanced trigonometric identities.

1-2. What is trigonometry?

Defines trigonometry as the study of the relationships between angles and lengths. It establishes its importance in geometry, physics, and engineering.

2. Angles and Circular Measure

This chapter covers the measurement of angles, moving from the familiar system of degrees to the fundamental mathematical concept of circular measure, or radians. Radian measure is essential for calculus and advanced trigonometry. Topics include defining radians, converting between degrees and radians, and applying radian measure to calculate the length of a circular arc and the area of a sector.

Chapter lessons

2-1. Measuring angles in degrees

A brief review of the degree as a unit of angular measure. It covers the conventions for positive and negative angles and angles greater than 360 degrees.

2-2. Circular measure and radians

Introduces the radian as the standard unit of angular measure in mathematics. It provides the formal definition of a radian based on the radius and arc length of a circle.

3. Trigonometric Functions

This chapter extends trigonometry beyond right-angled triangles. It uses the unit circle to define the six trigonometric functions for any angle, a concept critical for analysing periodic phenomena. Key topics include the unit circle, defining functions for angles of any magnitude, and the graphical representation of the sine, cosine, and tangent functions.

Chapter lessons

3-1. The unit circle

Introduces the unit circle as the foundational tool for modern trigonometry. It establishes the coordinate system used to define trigonometric functions for all real numbers.

3-2. Functions of any angle

Defines the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) for any angle based on the coordinates of a point on the unit circle.

3-3. Graphs of sine, cosine, and tangent

Covers the graphical representation of the three primary trigonometric functions. This lesson details their periodic nature, amplitude, and key features.

4. Trigonometric Identities

This chapter covers the manipulation of trigonometric expressions using standard identities. A command of these identities is essential for solving complex trigonometric equations and for simplification in calculus. Key topics covered are the addition formulae (for sin(A±B), cos(A±B)) and the factor formulae (product-to-sum and sum-to-product).

Chapter lessons

4-1. The addition formulae

Introduces the addition and subtraction formulae for sine and cosine. These identities are fundamental tools for expanding expressions like cos(A - B).

4-2. The factor formulae

Covers the two sets of factor formulae: product-to-sum and sum-to-product. These identities are used to convert between different forms of trigonometric expressions.

5. Conclusion

This chapter consolidates the core concepts of trigonometry. It provides a structured summary of circular measure, trigonometric functions, and key identities, reinforcing the principles required for progression. The conclusion summarises the unit circle definitions and standard formulae. It also provides a forward look to how these concepts are applied in calculus and complex numbers.

Chapter lessons

5-1. Summary of trigonometric principles

A concise review of the key definitions, graphs, and identities covered in the course. This lesson ensures all foundational trigonometric material has been consolidated.

5-2. Next steps in mathematics

Explains how trigonometry is a direct prerequisite for calculus (derivatives of trig functions) and for the polar form of complex numbers.