The Incredible Sine Wave and its Uses

The video begins with an introduction to the sine wave, a mathematical function that oscillates between -1 and 1. The presenter discusses the historical journey from the concept of sine in right-angled triangles to its use in describing waves. The ancient Greek astronomer Hipparchus is noted as a key figure in the development of trigonometry, using chords of circles in his calculations. The presenter also touches on the various applications of sine waves, including their connection to the workings of the human ear, archery, and banned books, setting the stage for a deep dive into the subject.

This paragraph delves into the transition of the sine concept from being associated with the chords of a circle to a trigonometric function. It explains how the ancient Greeks, and later the Indians, contributed to this evolution. The presenter illustrates how the sine of an angle (sin-theta) can be derived from the ratio of the half-chord length to the radius of a circle. The convenience of using sine over chord lengths is highlighted, as it removes the need to know the circle's radius. The etymology of the word 'sine' is explored, from its Sanskrit origin to its Arabic and Latin translations, showing how mathematical knowledge has been transferred across cultures.

The presenter explains how sine waves can represent angles greater than 90 degrees by using the unit circle concept. It is shown that the sine of an angle can be visualized as the height of a point on the circle's circumference above the horizontal diameter. This leads to an understanding of how the sine function can take both positive and negative values, and how it can be used to calculate the sine of angles beyond the confines of right-angled triangles. The video also touches on the mundane nature of degrees as a unit of measurement, hinting at the mathematically elegant alternative of radians.

The concept of radians as a measure of angles is introduced, highlighting its mathematical advantages over degrees. A radian is defined as the distance traveled around the circumference of a unit circle corresponding to a given angle. The presenter demonstrates how radians provide a natural fit for the sine and cosine functions, allowing for a straightforward representation of points on the circle's circumference. The idea of the sine curve as an infinite undulation is also discussed, showing how it can extend indefinitely in both directions.

This paragraph explores the various practical applications of trigonometric functions, such as astronomy, navigation, and surveying. The presenter discusses the historical use of trigonometry in astronomical observations at the Istanbul Observatory during the Ottoman Empire. The principles of navigation using triangles and trigonometric calculations to determine bearings are explained. Additionally, the technique of triangulation in surveying is described, emphasizing its importance in map-making, especially in challenging terrains.

The presenter discusses the Great Trigonometrical Survey of India, which was conducted over 70 years and completed in 1878. This survey faced the challenge of accounting for the Earth's curvature when dealing with large triangles, leading to the use of spherical trigonometry. The paragraph explains how the angles in a spherical triangle can exceed 180 degrees and how this was managed using a modified sine rule that considers the triangle's area.

The necessity of accurate trigonometric tables for navigation, astronomy, and surveying is emphasized. The presenter introduces Regiomontanus, who authored one of the first trigonometry textbooks and produced early trigonometric tables. Georg Rheticus's highly accurate tables are also highlighted, along with the historical context of their creation and use. The challenges of producing these tables before the advent of calculators and computers are discussed.

The paragraph introduces prosthaphaeresis, a technique used before the invention of logarithms to simplify the multiplication of long numbers. The presenter explains how this method leveraged trigonometric tables to transform multiplication into a series of additions, thereby reducing the complexity of the calculation. A simple example is provided to illustrate how prosthaphaeresis works, showcasing its clever use of existing mathematical infrastructure.

The presenter explores the emergence of sine waves in physical phenomena, particularly in pendulum swings and spring oscillations. It is explained how a force that pulls towards an equilibrium position, proportional to the distance from that position, results in a sine wave pattern. Examples include the motion of a pendulum and Hooke's Law of springs, both of which result in sine functions when mathematically modeled for small oscillations.

The video segment discusses the sine wave patterns found in light waves, including electromagnetic waves like radio and X-rays. Fourier's contribution to the understanding of sine waves is highlighted, with his series allowing for the breakdown of any periodic function into a combination of sine waves. The presenter visually demonstrates how adding different sine waves can create complex periodic waves, including square and sawtooth waves, through the use of Fourier series.

The presenter connects the concept of sine waves to music, explaining the relationship between string length, frequency, and pitch. The principle of harmonics and the pleasing nature of sounds with frequencies in small whole number ratios are discussed. The inner workings of the cochlea in the human ear are described, showing how it performs a Fourier analysis to break down incoming sound into its sine wave components, allowing the brain to interpret the sound.

The video concludes with a summary of the importance of sine waves in various fields and a teaser for an upcoming lecture on the mathematics of gyroscopes and boomerangs by Hugh Hunt. The presenter encourages the audience to attend future lectures and join the mailing list for notifications. The lecture ends with a Q&A session, addressing questions about the origin of gradians and the reason behind the use of sine waves for mains electricity.