Sine calculation by hand?

Question

I know sine represents a ratio in a triangle, but when you're doing things like $\sin(x)$, when you plug that into a calculator, the output is a completely separate number, without it knowing anything else about the triangle. That would mean that you don't need the actual triangle to calculate $\sin(x)$ otherwise you'd need to plug in the measurements to your calculator when solving for $\sin(x)$, right?

My general question is what equation is sin? It obviously represents some complicated function/equation/whatever that the calculator does to actually find what $\sin(x)$ equals, right? What is the equation used? How could I go about plugging in variables to possibly calculate $\sin(x)$ by hand without using any actual triangles/measurements?

If you have an answer, please explain like I'm an idiot who barely knows anything, I'm just some autistic freshman in highschool who's WAY too into mathematical stuff. I probably used incorrect math jargon in this question but still, I'm really curious

Welcome! The only incorrectly used word I can see is idiot. It's a very good question and there's no reason to demean yourself for wondering. — Matthew Leingang, Jun 14 '21 at 16:53
One definition of sine is in terms of right-triangle trigonometry, but another way to think about it is in terms of the unit circle, and it's very closely related to the right triangles you're familiar with. The unit circle gives you a more general way to define the function. There are several different approximations of the sine function, and your calculator uses one of them. Take a look at the Taylor series for the function. — Amaan M, Jun 14 '21 at 16:53
Does this answer your question? How does a calculator calculate the sine, cosine, tangent using just a number? — Sathvik, Jun 14 '21 at 16:56
One more note - the reason you don't need to know the details of the triangle to calculate the sine of an angle, and just need the measure of the angle, is because all right triangles with that specific angle are similar to each other. So, they'll always have the same ratios of corresponding sides, and that's what the sine (and cosine and tangent) are calculating. — Amaan M, Jun 14 '21 at 17:01
The sine of $x$ degrees can be computed as: $$\sin(x^{\large{\circ}})=\frac{\pi}{180}x-\left(\frac{\pi}{180}x\right)^3+\left(\frac{\pi}{180}x\right)^5-\left(\frac{\pi}{180}x\right)^7+\dots$$ This is an infinite series. You can approximate $\sin(x^{\large{\circ}})$ by truncating the series, so if $x$ is close to $0$, then $$ \sin(x^{\large{\circ}})\approx \frac{\pi}{180}x-\left(\frac{\pi}{180}x\right)^3 $$ — Joe, Jun 14 '21 at 17:02
If you are familiar with radians, then sine of $x$ radians is $$ \sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots $$ — Joe, Jun 14 '21 at 17:02
We also have the following approximation for sine of $x$ degrees: $$\sin(x^{\large{\circ}})=\frac{4x(180-x)}{40,500-x(180-x)} , .$$ — Joe, Jun 14 '21 at 17:12
See How did people calculate numerical values of transcendental and trigonometric functions? AND How to evaluate trigonometric functions by pen and paper? AND How were the sine, cosine and tangent tables originally calculated? — Dave L. Renfro, Jun 14 '21 at 17:44

Sine calculation by hand?

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