Trigonometry from Triangles

Trigonometry: The Big Picture

In this course, Professor Thomas Garrity (Williams College) explores the big picture of trigonometry — specifically that of sines and cosines — in the modern world of mathematics and in our daily lives. In the first mini-lecture, we define the sine and cosine functions using triangles and discuss their key properties. In the second mini-lecture, we extend our understanding of sines and cosines by defining them using the unit circle and discussing their periodic nature. In the third mini-lecture, we think about sines and cosines as waves by graphing these functions and discussing the incredible, but not obvious fact that almost all waves are made from sines and cosines. In the fourth mini-lecture, we discuss sines and cosines as infinite polynomials (infinite series), known as Taylor series, and how such series expansions are useful in calculations. In the fifth mini-lecture, we connect sines and cosines to the exponential function using Euler’s formula. Finally, in the sixth mini-lecture, we see how sines and cosines permeate our modern life, particularly in the music and images we get from our phones.

Trigonometry from Triangles

In this mini-lecture, we define sines and cosines using right triangles. In particular, we think about: (i) the big picture of the nature and importance of trigonometry and functions; (ii) historically how trigonometry came from studying triangles; (iii) the angles and sides (adjacent, opposite, hypotenuse) of right triangles; (iv) the definitions of the sine and cosine functions using the right triangle; (v) the four other trigonometric functions that come from sines and cosines (tangent, cotangent, secant, cosecant); (vi) and the key property of sines and cosines given by sin²(θ) + cos²(θ) = 1, as well as a proof of this property using the Pythagorean Theorem.