The Trigonometric Unit Circle is a pretty common thing taught in High-School, yet it doesn't cover some questions I had.
There have been questions in the past about why trigonometric functions can be negative, but none detailing exactly why the centre of the circle has to be on the Origin.
Surely, I can re-draw the same circle alone in the 1st quadrant of a cartesian plane, without overstepping the boundaries. Everything would then simply be positive - yet that would surely break down calculations that yield negative values?
It is very confusing to me why the circle is on the origin, presenting needless problems regarding signs. But Supposing if we do shift the circle, then the values would be different and incorrect.
Any ideas?
Notice that the meaning of negative values in the coordinate system essentially is 'left to' a point (on the x-axis) and 'down to' a point (on the y-axis). So for example $\cos\left(\frac{3\pi}{4}\right)=-\frac{\sqrt{2}}{2}$ means that on a unit circle, regardless where that centre is, the x-coordinate of the point of the triangle on the circle is $\frac{\sqrt{2}}{2}$ units left to the centre of the circle. The only reason why we use the centre as the origin of a unit circle because it allows for easier reading (i.e. the x-coordinate essentially dictates how far away in the horizontal direction it is from the origin, same argument applies for y-coordinate). It does not depend on the location of the unit circle.