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 now we are mapping a gradient to a point on the \(y\) axis. However, try this:
Draw the graph \(y=\sin x\) on a set of axes marked in degrees from \(-540^\circ\) to \(540^\circ\).
Draw the graph \(y=\sin x\) on a set of axes marked in degrees from \(-540^\circ\) to \(540^\circ\).
On the same axes, draw the graphs \(y=\sin (-x)\) and \(y=-\sin x\).
On the same axes, draw the graphs \(y=\sin (-x)\) and \(y=-\sin x\).
Really, we only need one well-chosen point on the new graphs to see immediately what they will look like. For example, when \(x=90^\circ\) on the graph \(y=\sin(-x)\), then \(y=\sin(-90^\circ)=-1\). With this information, you can probably draw the graph \(y=\sin(-x)\) straight away. You might like to check a few other points just to be sure.
You can also draw \(y=-\sin x\) the same way. And yes, it turns out to be the same as \(y=\sin(-x)\).
Describe the transformation of the graph \(y=\sin x\) to the graph \(y=\sin (-x)\) or \(y=-\sin x\) in as many ways as you can. Remember you can use reflections or translations!
Describe the transformation of the graph \(y=\sin x\) to the graph \(y=\sin (-x)\) or \(y=-\sin x\) in as many ways as you can. Remember you can use reflections or translations!
Describe the transformation of the graph \(y=\sin x\) to the graph \(y=\sin (-x)\) or \(y=-\sin x\) in as many ways as you can. Remember you can use reflections or translations!
We can reflect in the \(x\) axis, reflect in the \(y\) axis, translate left or right by \(180^\circ\) or \(540^\circ\) or add multiples of \(360^\circ\) to either of these.
Draw the graph \(y=\cos x\) on a set of axes marked in degrees from \(-540^\circ\) to \(540^\circ\).
Draw the graph \(y=\cos x\) on a set of axes marked in degrees from \(-540^\circ\) to \(540^\circ\).
On the same axes, draw the graphs \(y=\cos (-x)\) and \(y=-\cos x\).
On the same axes, draw the graphs \(y=\cos (-x)\) and \(y=-\cos x\).
Really, we only need one well-chosen point on the new graphs to see immediately what they will look like. For example, when \(x=180^\circ\) on the graph \(y=\cos(-x)\), then \(y=\cos(-180^\circ)=-1\). With this information, you can probably draw the graph \(y=\cos(-x)\) straight away. You might like to check a few other points just to be sure.
You can also draw \(y=-\cos x\) the same way.
Describe the transformation of the graph \(y=\sin x\) to the graph \(y=\cos (-x)\) or \(y=-\cos x\) in as many ways as you can. Remember you can use reflections or translations!
Describe the transformation of the graph \(y=\cos x\) to the graph \(y=\cos (-x)\) or \(y=-\cos x\) in as many ways as you can. Remember you can use reflections or translations!
Describe the transformation of the graph \(y=\cos x\) to the graph \(y=\cos (-x)\) or \(y=-\cos x\) in as many ways as you can. Remember you can use reflections or translations!
Describe the transformation of the graph \(y=\cos x\) to the graph \(y=\cos (-x)\) or \(y=-\cos x\) in as many ways as you can. Remember you can use reflections or translations!
Draw the graph \(y=\tan x\) on a set of axes marked in degrees from \(-540^\circ\) to \(540^\circ\).
On the same axes, draw the graphs \(y=\tan (-x)\) and \(y=-\tan x\).
On the same axes, draw the graphs \(y=\sin x\) and \(y=\cos x\).
Describe in as many ways as you can the transformations that take one graph to the other.
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