Taking the Unit Circle into another dimension!

The graph of the Unit Circle is generated in two ways. In one sense, it is the equation \(x^2+y^2=1\), which is to say, the collection of all the points whose squared sum is exactly one. However, in another sense, we know that the Unit Circle is the coordinate pair of \(\left(\cos(\theta),\sin(\theta)\right)\) where \(\theta\) is the angle formed between the \(x\)-axis and the terminal side of the angle. This poses an interesting observation: the input in this scenario is the angle \(\theta\). However, we are currently plotting two outputs, both Sine and Cosine, which results in the Unit Circle. However, what if we instead consider just the Sine function and let the input of \(\theta\) become the horizontal axis? If we do this point by point, we can generate the well known Sinusoidal Wave Graph. You can slide the points in the interactive demo below to see how each angle of \(\theta\) results in a new point on the Sine graph:

Some different observations we may make about this graph:

• When \(\theta=0\), the output is \(0\) (which is to say that the Sine graph has a vertical intercept at \(0\)).
• As \(\theta\) increases just above \(0\), the Sine graph increases to a maximum value of  \(1\) when \(\theta=\frac{\pi}{{2}}\).
• From that point, the graph decreases to a minimum value of \(1\), before returning to \(0\).
• If you keep going past \(\theta=2\pi\approx 6.2\), the graph will repeat itself.

With these observations in hand, let's detail a few of the features of the Sine graph:

• Domain: All real numbers (you can move the starting value to the left)
• Range : \([-1,1]\)
• Vertical Intercept: \((0,0)\)
• Horizontal Intercepts: Any multiple of \(\pi\) (can you answer why this is true?)
• The graph of Sine is periodic with a period of \(2\pi\).

Let's define those last two terms: a periodic graph will repeat itself exactly whenever the input is shifted by the period (either to the left or to the right).

Since the Sine graph is \(2\pi\) periodic, we can say with certainty that \(\sin(x) = \sin(x\pm2\pi)\).