Trigonometric Function variables

Solution 1:

It is common to plot multiple graphs on the same $x$-$y$ Cartesian plane and label them, say, $y=\tan 2x$ and $y=\log x$ and $y=\cos x.$ In each of these three graphs, the function’s output $y$ depends on its input $x.$

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.

Solution 2:

The trigonometric circle is commonly drawn with an horizontal and a vertical axis. Let us call them the cosine and sine axis, respectively.

On the trigonometric circle, a point has the coordinates $(\cos(u), \sin(u))$.

Now if you want to study the cosine and sine as functions of an angle, you will plot them in diagrams with the horizontal axis denoting values of $u$ and the vertical axis the value of the cosine or sine.

On the function plot, a point has the coordinates $(u,\cos(u))$ or $(u,\sin(u))$.

Notice that I avoided any reference to $x$ and $y$, on purpose.

Now contemplate the following animation (one of the plots is $(\cos (u),u)$, axis exchanged):

Solution 3:

This is definitely a little bit tricky the first time you're learning about it. The key issue here is that we're using $y$ to represent two different things.

In the context of the unit circle, when $u$ is your angle, you can write $x = \cos u$ and $y = \sin u$, where $x$ and $y$ refer to points on the unit circle.

So, for example, if you let $u = \frac{\pi}{6}$, then the equations above give us $x = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$, and $y = \sin \frac{\pi}{6} = \frac{1}{2}$. This is telling us that when we move $\frac{\pi}{6}$ radians counterclockwise from the positive $x$-axis along the unit circle, we get to the point $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Now, when you see $y = \cos x$ or $y = \sin x$, those $x$ and $y$ variables represent something different. They're not referring to the $x$ and $y$-coordinates on the unit circle. Note, in both of these equations, $x$ is the input to the trig function. So, that $x$ represents the same thing as what we were calling $u$ above - the angle we're moving through along the unit circle from the positive $x$-axis.

The $y$ variable in this case represents the output of the trig function, which can be either of the coordinates on the unit circle, $x$ or $y$, depending on whether the trig function is $\cos$ or $\sin$.

So, if our function is $y = \cos x$, the input $x$ is the angle on the unit circle, and the output $y$ is the $x$-coordinate on the unit circle. If our function is $y = \sin x$, the input $x$ is still the angle on the unit circle, and the output $y$ is the $y$-coordinate on the unit circle.

Using the same example as above, if we want to evaluate $y = \cos \frac{\pi}{6}$, we take angle $\frac{\pi}{6}$ on the unit circle, then since it's cosine, we find the $x$-coordinate, which is $\frac{\sqrt{3}}{2}$, and that's the value of the output $y$. If we want to evaluate $y = \sin \frac{\pi}{6}$, we go to the same angle on the unit circle, look at the $y$-coordinate because we're evaluating sine, and get that $y = \frac{1}{2}$ is our output.