I'm currently reading Precalculus with limits and got into chapter 4 of Trigonometry.
I now understood that, an angle $u$ is a real number that correspond to the points $(x,y)$ in the unit circle. So we have now two functions which is $x=\cos u$ and $y=\sin u$ according to the definitions of the right triangle (trigonometric ratios) and unit circle.
My dilemma is, since we have now defined cosine as a function of $x$, so we could call it a function $x=g(u)=\cos u$, when I make a graph, I choose $x$ as vertical axis and $u$ as horizontal axis, because cosine move from right to left in a unit circle. But in the book they choose cosine as $y=\cos x$. But $y$ is already taken as a definition of sine function, $y=f(u)=\sin u$. So how did they interchanged the variables here? This matter is very confusing to me.
Best Answer
It is common to plot multiple graphs on the same $x$-$y$ Cartesian plane and label them, say, $y_1=\tan 2x$ and $y_2=\log x$ and $y_3=\cos x.$ As $x$ varies, the three functions each output a $y$ value.
On the other hand, think of that trigonometric unit circle as a parameterised surface rather than as a graph: here, the circle's $x$- and $y$- coordinates are each expressed as a trigonometric function of parameter $\theta.$
In every case, $x, y$ and $\theta$ are just handles. Whether $y=\cos x,\:$ or $y=\cos\theta,\:$ or $y=\sin\theta,\,$ depends entirely on the context.