Trigonometry

Welcome to the fifth episode of our Geometry and Trigonometry course! This episode introduces the fascinating world of Trigonometry, the branch of mathematics that explores the relationships between the angles and side lengths of triangles. Discover its historical roots in ancient astronomy and surveying and understand why the right-angled triangle is so fundamental. We will define the key sides—hypotenuse, opposite, and adjacent—and explore the core principle that the ratios of these sides are constant for any given angle. You'll be introduced to your first trigonometric ratio, the tangent, and see how it can be used to measure heights and distances that seem impossible to reach. This episode lays the essential groundwork for future explorations of sine and cosine.

Check your understanding

These are the same multiple-choice questions you will see in the Quiz section after you listen to the episode. Use them here to preview or review the answers.

What is the literal meaning of the word 'Trigonometry', based on its Greek roots?

  1. The study of circles
  2. The measurement of triangles
  3. The properties of angles
  4. The science of three-dimensional shapes

In a right-angled triangle, which side is referred to as the 'hypotenuse'?

  1. The side next to the angle of interest.
  2. The shortest side of the triangle.
  3. The side opposite the 90-degree angle.
  4. The side opposite the angle of interest.

If you are focused on a specific angle (not the right angle) in a right-angled triangle, how are the 'opposite' and 'adjacent' sides determined?

  1. They are always the two shortest sides.
  2. The 'opposite' side is across from the angle, and the 'adjacent' side is next to it (and is not the hypotenuse).
  3. Their positions are fixed regardless of the angle you choose.
  4. The 'adjacent' side is always longer than the 'opposite' side.
  5. The 'opposite' side is next to the angle, and the 'adjacent' side is across from it.

What is the fundamental principle that makes trigonometry so powerful?

  1. All right-angled triangles have the same side lengths.
  2. In any right-angled triangle, the sum of two sides equals the third.
  3. For a given angle in a right-angled triangle, the ratios of the side lengths are constant, regardless of the triangle's size.
  4. Triangles can be used to create circles.

How is the 'tangent' of an angle in a right-angled triangle defined?

  1. The ratio of the adjacent side to the hypotenuse.
  2. The ratio of the opposite side to the hypotenuse.
  3. The product of the opposite and adjacent sides.
  4. The ratio of the opposite side to the adjacent side.
  5. The sum of the opposite and adjacent sides.

Suggested next

Related episodes that are a natural follow-on.

  • Sine

    Welcome to the sixth episode of our Geometry and Trigonometry course! Building on our understanding of triangles and circles, we dive into the heart of trigonometry by introducing the Sine function. In this episode, you will learn how to define sine … Welcome to the sixth episode of our Geometry and Trigonometry course! Building on our understanding of triangles and circles, we dive into the heart of trigonometry by introducing the Sine function. In this episode, you will learn how to define sine as a ratio of side lengths in a right-angled triangle, specifically the 'opposite over hypotenuse' relationship. We'll then expand this concept beyond triangles by exploring the unit circle, providing a more powerful and general definition. Discover the fundamental properties of sine, its range of values, and its behavior at key angles like 0, 90, and 180 degrees. This foundational knowledge will pave the way for solving complex geometric problems and understanding the periodic nature of the world around us.

  • Marine engineering

    Welcome to the fifth episode of our course. Having explored the iron arteries of the land with railway engineering, we now set sail into the vast world of **Marine Engineering**. This episode serves as your introduction to the discipline responsible … Welcome to the fifth episode of our course. Having explored the iron arteries of the land with railway engineering, we now set sail into the vast world of **Marine Engineering**. This episode serves as your introduction to the discipline responsible for conquering the world's oceans. You will learn what marine engineering encompasses, from designing colossal ships to building offshore structures. We'll explore the core scientific principles that govern everything that floats, the complex challenges posed by the harsh marine environment, and the iterative design process that turns a concept into a seaworthy vessel. This foundational knowledge will prepare you for our upcoming deep dives into specific topics like ship design, propulsion, and port operations.

  • Aircraft

    Welcome to the third episode of our Aerospace Engineering course! Building on our understanding of aerodynamics, we now explore the aircraft itself. This episode dissects the anatomy of an airplane, from the fuselage that houses passengers to the win… Welcome to the third episode of our Aerospace Engineering course! Building on our understanding of aerodynamics, we now explore the aircraft itself. This episode dissects the anatomy of an airplane, from the fuselage that houses passengers to the wings that generate lift and the tail that ensures stability. We will examine the function of each major component and how they work in harmony. You'll also learn how aircraft are classified based on their design, propulsion, and purpose, such as fixed-wing, rotary-wing, civil, and military types. By the end, you'll understand how these components manage the four fundamental forces of flight, setting the stage for our next discussions on propulsion and control.

  • Ship

    Welcome to the sixth episode of Railway and Marine Engineering. Having explored the fundamentals of land-based transport like railway tracks and locomotives, we now turn our attention to the vast world of the sea. This episode introduces the ship, a … Welcome to the sixth episode of Railway and Marine Engineering. Having explored the fundamentals of land-based transport like railway tracks and locomotives, we now turn our attention to the vast world of the sea. This episode introduces the ship, a cornerstone of global logistics and human history. We will define what constitutes a ship, explore its basic anatomy, and trace its remarkable evolution from ancient vessels to the steel giants of today. You'll learn about the primary classifications of modern ships based on their purpose, setting the stage for future discussions on specific components like hulls, propulsion systems, and specialized vessels such as container ships. This foundational knowledge will bridge our understanding from railways to the complex domain of marine engineering.

  • Aerospace engineering

    Welcome to the first episode of our Aerospace Engineering course! This introduction will define the vast and exciting field of aerospace engineering, exploring its two primary branches: aeronautical and astronautical engineering. We'll journey throug… Welcome to the first episode of our Aerospace Engineering course! This introduction will define the vast and exciting field of aerospace engineering, exploring its two primary branches: aeronautical and astronautical engineering. We'll journey through a brief history of flight, from early myths to the Wright brothers' groundbreaking achievement and the dawn of the space age. You'll be introduced to the fundamental principles that govern flight, such as the four key forces acting on any vehicle in motion through the air or space. This foundational episode sets the stage for our deep dive into more specific topics like aerodynamics, propulsion, and spacecraft design in future lessons.

Often studied before

Episodes that tend to come earlier on similar paths.

  • Coordinate geometry

    Welcome to the world of Coordinate Geometry, also known as Analytic Geometry! This episode forges a powerful connection between the abstract shapes of Euclidean geometry and the concrete numbers of algebra. We'll introduce the Cartesian coordinate sy… Welcome to the world of Coordinate Geometry, also known as Analytic Geometry! This episode forges a powerful connection between the abstract shapes of Euclidean geometry and the concrete numbers of algebra. We'll introduce the Cartesian coordinate system, a brilliant invention that allows us to place any point, line, or shape onto a two-dimensional grid. You will learn how to use coordinates to calculate the distance between two points and find the exact middle of a line segment. We will then explore how to translate geometric figures like lines and circles into the language of algebraic equations, unlocking a new way to analyze and solve geometric problems. This episode provides the foundational tools for describing geometry with the precision of algebra.

  • Euclidean geometry

    Welcome to the first episode of Geometry and Trigonometry! This episode introduces the foundational concepts of Euclidean geometry, the system that describes the space we experience every day. We will start with the basic, undefined building blocks: … Welcome to the first episode of Geometry and Trigonometry! This episode introduces the foundational concepts of Euclidean geometry, the system that describes the space we experience every day. We will start with the basic, undefined building blocks: points, lines, and planes. From there, we'll explore angles and their relationships. The core of our discussion will be the five fundamental postulates laid down by the ancient Greek mathematician Euclid, which serve as the unbreakable rules for this entire field of study. This episode lays the essential groundwork for understanding more complex shapes and concepts, such as triangles, circles, and trigonometry, which will be covered in future lessons. Get ready to build your geometric understanding from the ground up!

  • Triangle

    Welcome to the second episode of our Geometry and Trigonometry course! Building on our introduction to Euclidean geometry, we now focus on the simplest and most fundamental polygon: the triangle. In this episode, you will discover the essential prope… Welcome to the second episode of our Geometry and Trigonometry course! Building on our introduction to Euclidean geometry, we now focus on the simplest and most fundamental polygon: the triangle. In this episode, you will discover the essential properties of this three-sided shape. We will explore how triangles are classified based on their side lengths (equilateral, isosceles, scalene) and their internal angles (acute, obtuse, right-angled). A key concept we'll uncover is that the sum of the angles in any triangle is always 180 degrees. Understanding triangles is crucial as they form the building blocks for more complex shapes and are the foundation for the upcoming topic of trigonometry. Get ready to explore the world of triangles!

  • Vector space

    Welcome to the seventh episode of Algebra and Number Theory! This episode introduces the fundamental concept of a Vector Space. Building on your knowledge of linear algebra and matrices, we will move beyond the familiar idea of vectors as arrows in s… Welcome to the seventh episode of Algebra and Number Theory! This episode introduces the fundamental concept of a Vector Space. Building on your knowledge of linear algebra and matrices, we will move beyond the familiar idea of vectors as arrows in space. We will explore the abstract definition of a vector space through a specific set of rules, or axioms, that govern vector addition and scalar multiplication. You will discover that many different mathematical objects, including polynomials and matrices, can be considered vectors in their own right. This episode provides a crucial abstract foundation that unifies many areas of mathematics and prepares you for more advanced algebraic structures.

  • Geometry

    Welcome to the fifth episode of Foundations of Mathematics! In this installment, we journey into the world of **Geometry**, the study of shape, size, and space. Building upon your understanding of Number, Arithmetic, and Algebra, we will explore the … Welcome to the fifth episode of Foundations of Mathematics! In this installment, we journey into the world of **Geometry**, the study of shape, size, and space. Building upon your understanding of Number, Arithmetic, and Algebra, we will explore the elegant logical system developed by the ancient Greeks, most notably Euclid. You will learn about the fundamental building blocks—points, lines, and planes—and see how they are used to construct complex figures. We'll examine the properties of familiar shapes like triangles and circles and even venture beyond the flat plane into the fascinating realms of non-Euclidean geometries that shape our modern understanding of the universe. This episode will provide you with a foundational appreciation for the spatial relationships that define the world around us, from art and architecture to the very fabric of spacetime.