Introduction to Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, appear arbitrary when first encountered in high school. However, with a deeper dive into their origins and applications, we uncover the elegance and necessity of these functions in mathematics and beyond.

Origins and Derivation of Trigonometric Functions

The natural position of trigonometry in the development of mathematics can be traced back to complex numbers and the exponential series. Starting from Euler's famous equation, we derive the core relationship between exponential and trigonometric functions. Let's explore this mathematical journey step by step.

Euler's Exp Expression

Consider Euler's expression for the exponential of an imaginary number:

exp(ix) cos(x) i*sin(x)

Where exp(ix) and exp(-ix) can be separated into real and imaginary parts:

exp(ix) cos(x) i*sin(x)

exp(-ix) cos(x) - i*sin(x)

Deriving Cosine and Sine

To derive the cosine and sine functions, we solve for them from the above expressions:

cos(x) (exp(ix) exp(-ix))/2

sin(x) (exp(ix) - exp(-ix))/(2i)

This derivation provides a profound insight into the nature of trigonometric functions, linking them directly to the exponential function in the complex plane.

The Pythagorean Theorem and Trigonometry

Another key element in understanding trigonometric functions is the Pythagorean Theorem. Applying this theorem in the context of a right triangle, where the hypotenuse is 1, we get:

cos^2(x) sin^2(x) 1

This identity demonstrates that the sine and cosine functions are precisely defined, and their values are constrained by the length of the sides of a right triangle.

The Second Purpose: Applications and Explaining Concepts

Throughout history, trigonometric functions have served two primary purposes. Firstly, they provide a foundational understanding of relationships, starting with right triangles and extending to more complex scenarios, such as the unit circle. Secondly, they are instrumental in explaining phenomena and answering questions across various fields of study.

Real-World Applications

Let's explore a real-world problem. Suppose you are an astronaut orbiting a spherical planet, 500 miles above the surface, and you can see the horizon 1000 miles away. To find the planet's radius, we can use trigonometry:

radius (500^2 1000^2)^(1/2) - 500

This problem highlights the practical application of trigonometric functions in space exploration.

Solving a Simpler Problem

A more straightforward example involves a tall tree and a 40-degree angle of elevation. If you are 100 feet away from the tree, the height of the tree h can be calculated as:

tan(40°) h / 100

Solving for h, you can use the value of the tangent function, which is known from a calculator or table.

The Role of Humans in Defining Trigonometric Functions

Trigonometric functions were not arbitrarily created. They are the result of human ingenuity and the quest to understand and quantify natural phenomena. The process of defining these functions has evolved over centuries, leading to the precise definitions we have today. Cosine, sine, tangent, and their counterparts are well-defined, each serving a specific purpose in mathematics and its applications.

Conclusion

While trigonometric functions may appear arbitrary at first glance, they are in fact deeply rooted in mathematical theory and have significant real-world applications. From satellite navigation to solving problems in astronomy and engineering, these functions play a crucial role in our understanding of the world around us.