Q. 106 - Precalculus Enhanced with Graphing Utilities

Question

Open the Trace Cosine Curve applet. On the screen, you will see the graph of the unit circle with a point C labeled. Use your mouse and move point C around the unit circle in the counterclockwise direction. What do you notice? In particular, what is the relation between the angle and the x-coordinate of point C? 

Step-by-Step Solution

Answer

We notice that by tracing we get a graph of a sinusoidal function.

The relation is $x=\cos \alpha$.

Step 1. Use the Cosine Curve Applet

The applet is used, the point is moved around the unit circle and the result can be seen as 

Step 2. Observation

We can see that the point when moved traced a sinusoidal curve with amplitude of $1$ and period of $2\mathrm{\pi }$.  Also, the value is maximum at the origin.

So it is a graph of cosine function.

Step 3. Use the cosine definition

The cosine of an angle is the ratio of the adjacent side in a triangle to the hypotenuse.

For the point C  let the coordinates be $(x,y)$ then the hypotenuse is $1$ and the adjacent side is $x$.

So the cosine of the angle $\alpha$ is given as

$$\begin{gathered} \cos \alpha =\frac{adjacent side}{hypotenuse} \\ \cos \alpha =\frac{x}{1} \\ \cos \alpha =x \end{gathered}$$