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.

MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science. This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions. Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Course Chapters

1. Introduction

This chapter provides the fundamental language of trigonometry by defining how we measure rotation and relate triangle sides. These basics are the bedrock for all calculations in engineering, surveying, and physics. Without mastering these definitions, advanced analysis of waves and structures is impossible. You will master converting between degrees, radians, and gradians; defining primary and reciprocal ratios; and using inverse functions to calculate unknown angles in practical structures.

Concept Overviews

5 Lessons

Problem Walkthroughs

5 Lessons

2. Circular Measure

This chapter moves from triangles to circles to cover arc geometry and the unit circle. These tools are vital for modelling periodic motion in physics and engineering. You will learn to calculate arc lengths and sector areas; define functions for any angle using the unit circle; and identify ratio signs across all quadrants.

Concept Overviews

5 Lessons

Problem Walkthroughs

3 Lessons

3. General Angles

This chapter extends trigonometry to any angle and identifies ratios for specific values. It introduces reference angles as the key to finding values in any quadrant. You will learn to evaluate exact ratios for special angles; apply the reference angle concept to large rotations; and use negative angle identities for simplification.

Concept Overviews

4 Lessons

Problem Walkthroughs

3 Lessons

4. Graphs

Trigonometric functions repeat over time and are best understood through their graphs. Visualising these waves is essential for students studying periodic systems like sound and light. You will learn to plot sine, cosine, and tangent graphs; determine the period and amplitude of wave functions; and identify values where functions become undefined.

Concept Overviews

3 Lessons

Problem Walkthroughs

3 Lessons

5. Fundamental Identities

Identities allow for the algebraic manipulation of trigonometric expressions. This chapter provides the tools to simplify complex equations and prove mathematical relationships. You will learn to derive and apply the Pythagorean identity; use quotient relationships to rewrite expressions; and substitute values to solve multi-step problems.

Concept Overviews

2 Lessons

Problem Walkthroughs

2 Lessons

6. Addition Formulae

Addition formulae enable the calculation of angles that are not special. These rules are fundamental for vector analysis and rotating objects in three dimensions. You will learn to calculate exact values for compound angles; apply double angle formulae to reduce expressions; and use half-angle rules for precise derivations.

Concept Overviews

3 Lessons

Problem Walkthroughs

1 Lesson

7. Multiple Angle Formulae

This chapter covers double, triple, and half-angle identities, including t-substitutions. These tools reduce complex powers and multiples into simpler forms, which is vital for integration in calculus and analysing wave harmonics. You will master double and triple-angle formulae; apply t-substitution to solve complex equations; and simplify expressions by converting trigonometric functions into algebraic terms.

Concept Overviews

4 Lessons

Problem Walkthroughs

2 Lessons

8. Factor Formulae

This chapter covers the transformation of products into sums and sums into products. These techniques are crucial for solving trigonometric equations and analysing wave interference. You will learn to transform products into sum or difference forms; express sums as products for easier simplification; and solve complex quotient proofs.

Concept Overviews

2 Lessons

Problem Walkthroughs

3 Lessons

9. Equations and R-Formula

This chapter covers methods for solving trigonometric equations and using the R-formula to simplify expressions. These tools are essential for finding unknown angles and combining multiple wave functions into a single form for analysis in physics and engineering. You will master finding general and particular solutions to equations, applying the R-formula to combine sine and cosine terms, and determining the maximum and minimum values of trigonometric expressions.

Concept Overviews

3 Lessons

10. Solving Triangles

This chapter details the methods for finding missing lengths and angles in any triangle. These rules are essential for practical applications in surveying, navigation, and engineering where right angles are rare. You will master the sine and cosine rules; calculate triangle areas using trigonometric formulae; and solve practical problems involving bearings, elevation, and depression.

Concept Overviews

2 Lessons

11. Conclusion

The final chapter reviews all techniques to ensure you can choose the right tool for any problem. It bridges the gap between basic trigonometry and advanced calculus. You will summarise all core formulae and identities; identify the logical hierarchy of trigonometric concepts; and prepare for higher-level mathematics.

Concept Overviews

1 Lesson