Trigonometric functions over obtuse or arbitrary angles doesn't make sense. We can only imagine for eg. sin(x) for angles < 90 degrees because it represents the ratio of the opposite and hypotenuse. Now, from nowhere, we define it for arbitrary angles. I know that these are functions and for an input they produce an output and we can define them as we want. Were is the freedom from to define them the way they're defined(we all know that) and they can be still useful? I know my questions is a little confusing but, this really doesn't make sense if we stick to the intuition.
2026-05-14 17:47:17.1778780837
Trigonometric functions over arbitrary angles
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The standard definitions of sin and cos for angles outside the range $0^\circ \le \theta \le 90^\circ$ make really pretty graphs that provide a good model for things like sound waves and alternating current. Also, they provide analytic functions which have such great things like a Taylor's series. Also they satisfy Euler's formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. And then there are Fourier series which are absolutely indispensable to a great deal of science, engineering and math, which totally rely on these definitions of sin and cos.