what are you actually doing when you do sin(x), cos(x), or tan(x), or the inverses outside of a triangle. Like if you plug sin(7) into a calculator what is actually happening.

In: Mathematics

Well, when you plug sin(7) into a calculator, it’s still kinda about a triangle. sin(7) means: if you have a right triangle where the hypotenuse (the diagonal side of a right triangle) is length 1, and the angle between the horizontal side and the hypotenuse is 7, how long is the vertical side?

You can also think of it like the [angles within a circle of radius 1.](https://cdn.kastatic.org/googleusercontent/TSZaYHIv-JZrb7IgquFrsO1b4Ie104YD3TkNCaZQ06I8RsonAVD_ON5L8t6q9DIQspl6s1yEwUTmrd1E2PFm87oCuQ) This diagram uses θ instead of x. As θ changes, the vertical part of the triangle inside the circle changes too, and the length of that vertical part is sin(θ). The length of the horizontal part is cos(θ)

>what are you actually doing when you do sin(x), cos(x), or tan(x), or the inverses outside of a triangle. Like if you plug sin(7) into a calculator what is actually happening.

There is no exact calculation path for the result. The machine looks up the result in a predefined table or approximates the it with something like the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series).