Q. 105 - Precalculus Enhanced with Graphing Utilities

Question

Open the Trace Sine 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 y-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 $y=\sin \alpha$.

Step 1. Use the Sine 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 zero at the origin.

So it is a graph of sine function.

Step 3. Use the sine definition

The sine of an angle is the ratio of the opposite side in a triangle to the hypotenuse.

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

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

$$\begin{gathered} \sin \alpha =\frac{opposite side}{hypotenuse} \\ \sin \alpha =\frac{y}{1} \\ \sin \alpha =y \end{gathered}$$