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The history of trigonometric functions
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The history of trigonometric functions

From right triangles to Taylor series

Tivadar Danka's avatar
Tivadar Danka
Dec 19, 2022
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The Palindrome
The Palindrome
The history of trigonometric functions
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This is not a trick: the cosine of the imaginary number i is (e⁻¹ + e)/2.

How on Earth does this follow from the definition of the cosine? No matter how hard you try, you cannot construct a right triangle with an angle i. What kind of sorcery is this?

Behind the scenes, sine and cosine are much more than a simple ratio of sides. They are the building blocks of science and engineering, and we can extend them from angles of right triangles to arbitrary complex numbers.

In this post, we’ll undertake this journey.

If you enjoy my work and get value from it, support me with a paid subscription! Mathematics is neither dull nor dry; it’s beautiful, mesmerizing, and useful. Your support helps me show this to everyone.

Right triangles and their sides

The history of trigonometric functions goes back almost two thousand years. In their original form, as we first encounter them in school, sine and cosine are defined in terms of right triangles.

For an acute angle α, its sine is defined by the ratio of t…

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