Unit III: Lec 1 | MIT Calculus Revisited: Single Variable Calculus

The professor begins the lecture by addressing the importance of understanding circular functions, which are essentially trigonometric functions without the need for triangles. He shares a personal anecdote about his initial skepticism towards the practicality of trigonometry, which was dispelled when he encountered it in calculus and physics. The lecture aims to explore the connection between circular functions and traditional trigonometry, emphasizing the relevance of these mathematical tools beyond surveying. The concept is introduced through the visualization of a circle with a radius of 1, laying the groundwork for defining sine and cosine in terms of real numbers rather than angles.

The professor explains how to define sine and cosine functions using a unit circle. By considering a real number 't' as a length along the circle's circumference, the sine of 't' is defined as the y-coordinate of the termination point 'P', and the cosine as the x-coordinate. This method allows for the trigonometric functions to map real numbers to real numbers, independent of angles. The explanation includes the traditional trigonometric conventions and how they translate into this new framework, including the definitions of sine and cosine for positive and negative values of 't'.

The lecture delves into the concept of radian measure, addressing the ambiguity that arises when using degrees versus radians in trigonometry. The professor illustrates how defining an angle in radians—where the angle's measure is equal to the length of the arc it subtends on a unit circle—eliminates confusion between the sine of an angle and the sine of a real number. This section also touches on the historical and mathematical reasons for adopting radians as the standard measure in trigonometry, emphasizing the clarity it brings to the definitions of trigonometric functions.

The professor discusses the flexibility in defining trigonometric-like functions using different geometric figures, such as hyperbolas, and how these definitions can vary based on the chosen method of measuring lengths along the figure. He also considers alternative ways of defining trigonometric functions on a circle, such as using radii of different lengths, and the implications these choices have on the domain of the functions. The section highlights the arbitrariness in the choice of the circle for defining trigonometric functions and the importance of careful definition to avoid ambiguity.

The lecture moves into the realm of calculus, specifically the computation of derivatives of sine and cosine functions. The professor outlines the process of finding the derivative of sine with respect to a real variable 'x', using the definition of the derivative and the properties of trigonometric functions. The section covers the computation of limits necessary for the derivative, such as the limit of 'sine t' over 't' as 't' approaches 0, and how these computations lead to the conclusion that the derivative of sine is cosine.

Building on the previous discussion, the professor derives the derivative of the cosine function and touches on the physical interpretation of these derivatives in the context of simple harmonic motion. The lecture demonstrates how the second derivative of a particle's displacement in simple harmonic motion, which represents acceleration, is proportional and opposite to the displacement itself. This section reinforces the idea that trigonometric functions can be understood and applied without reference to angles, highlighting their utility in physics and engineering.

In the concluding part of the lecture, the professor summarizes the key points covered in the session, emphasizing the importance of understanding the calculus of circular functions and their physical applications. He also previews the next topic, which will be the inverse circular functions and their significance. The lecture ends with acknowledgment of the funding provided by the Gabriella and Paul Rosenbaum Foundation for the publication of the video, and a call to action for donations to support MIT OpenCourseWare.