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.

# 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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