Graphs of the circular functions
So far, we have explored circular functions through their representations on a unit circle. Now we turn to their graphs. The question is: what is the relationship between these two representations?
Take your time with each video. Pause, rewind, and think before moving on!
Dr Brian Brooks
Mathematics InSight
Graphs of the circular functions
You know only too well that we can draw graphs like \(y=\sin x\), \(y=\cos x\), and \(y=\tan x\). The question is, what is the relationship between these graphs and the unit circle? You can watch a fancy animation like this
and it looks very convincing, but it's easy to miss the details. The next few pages build up each graph very gradually to make the relationship crystal clear.
This video builds up the relationship between the unit circle and the graph \(y=\sin x\) very gradually.
The relationship between the unit circle and the graph \(y=\cos x\) is slightly harder to visualize, but this video builds it up in exactly the same way.
Tan is even harder, because this time, we have to make the \(y\) coordinate on the graph match the gradient of the yellow radius.
How does the position of the blue cross on the \(y\) axis relate to the yellow radius?
The \(y\) coordinate of the blue cross is the gradient of the yellow radius.
How does the position of the blue cross on the \(y\) axis relate to the pink angle, and what is the equation of the curve?
The \(y\) coordinate of the blue cross is the tan of the pink angle, and the equation is
\[y=\tan x\]On the same axes, draw the graphs \(y=\sin x\) and \(y=-\sin x\).
The transformation from \(y=\sin x\) to \(y=-\sin x\) is a reflection in the \(x\) axis. Watch this happening in the video.
Now draw the graph \(y=\sin (-x)\).
The transformation from \(y=\sin x\) to \(y=\sin (-x)\) is a reflection in the \(y\) axis. Watch this happening in the video.
Describe the transformation of the graph \(y=\sin x\) to the graph \(y=\sin (-x)\) or \(y=-\sin x\) using translations instead of reflections
We can translate left or right by \(180^\circ\) or \(540^\circ\) or add multiples of \(360^\circ\) to either of these. Watch these happening in the video.
On the same axes, draw the graphs \(y=\cos x\) and \(y=-\cos x\).
The transformation from \(y=\cos x\) to \(y=-\cos x\) is a reflection in the \(x\) axis. Watch this happening in the video.
Now draw the graph \(y=\cos (-x)\).
The transformation from \(y=\cos x\) to \(y=\cos (-x)\) is a reflection in the \(y\) axis. But \(\cos x\) is already symmetric about the \(y\) axis, so this reflection leaves the graph unchanged. Watch this in the video.
Describe the transformation of the graph \(y=\cos x\) to the graph \(y=-\cos x\) using translations instead of reflections
We can translate left or right by \(180^\circ\) or \(540^\circ\) or add multiples of \(360^\circ\) to either of these. Watch these happening in the video.
On the same axes, draw the graphs \(y=\tan x\) and \(y=-\tan x\).
The transformation from \(y=\tan x\) to \(y=-\tan x\) is a reflection in the \(x\) axis. Watch this happening in the video.
Now draw the graph \(y=\tan (-x)\).
The transformation from \(y=\tan x\) to \(y=\tan (-x)\) is a reflection in the \(y\) axis. Again, watch the video.
Well done!
You've now seen how the graphs of \(y=\sin x\), \(y=\cos x\), and \(y=\tan x\) arise directly from the unit circle. This connection is fundamental to everything that follows.
Dr Brian Brooks
Mathematics InSight