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How do trig functions actually work?

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It sounds like you are looking for a formula to calculate the output of a trig function for a given input. While there are methods for doing this approximately, it’s important to note that a function doesn’t need an equation in order to be a function. A function is just a special collection of inputs and outputs. How you define the function is up to you.

The key thing that allows us to define trig functions how we do is a fact from geometry. Namely, if you draw two right triangles that have the same angles the ratios of their corresponding side lengths will always be the same. In other words the ratios “opposite/hypotenuse”, “adjacent/hypotenuse”, and “opposite/adjacent” only depend on the angle, not the size of the triangle. Since these ratios only depend on the angle, we can define a function that associates one of these ratios to each angle. This defines a function. This *is* what the function is “doing.”

Actually calculating these things is interesting, but it’s a whole other story. Traditionally, these things were actually measured and tabulated in look-up tables.
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I'm not sure what you're asking. By "what is the actual value" are you asking how we calculate the numerical value?

Or are you asking "what are these good for"?
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I see you've gotten an answer that explains the calculation side of things.    But we can talk about functions without having any idea how to calculate them.   

You're correct that the the sin(x) function takes an angle as input and outputs a value.   But you can completely understand what the output value is without having any idea how to calculate it.

Given an angle x, if you draw any right triangle with that angle between one of the sides and the hypoteneuse, measure the lengths of the side opposite that angle and the hypoteneuse, and then divide those lengths, you'll end up with the same number no matter how large you made the triangle.    That value you ended up with is exactly what sin(x) is.

Even if we never figured out a way to calculate it, sin(x) would still be a valid function and by making our drawings really accurate we could approximate sin(x) for any angle x.

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