The unit circle is a fundamental concept in trigonometry. It's a circle with radius 1 centered at the origin. Any point on this circle can be represented using coordinates cos theta and sin theta, where theta is the angle measured from the positive x-axis. The key points are at 1,0, 0,1, negative 1,0, and 0, negative 1.
Now let's see how a point moves around the unit circle as the angle theta increases. The angle is measured in radians from the positive x-axis. As the point moves, notice how its coordinates change. The x-coordinate represents cosine theta, and the y-coordinate represents sine theta. Watch as the point completes one full revolution from 0 to 2 pi radians.
The sine of an angle is simply the y-coordinate of the corresponding point on the unit circle. Watch as the horizontal dashed line connects the point to the y-axis, clearly showing this relationship. At 0 degrees, sine equals 0. At 90 degrees, sine reaches its maximum value of 1. At 180 degrees, sine returns to 0. And at 270 degrees, sine reaches its minimum value of negative 1.
Now let's see how the circular motion creates the familiar sine wave. As the point moves around the unit circle, we trace its y-coordinate over time to generate the sine wave. One complete revolution from 0 to 2 pi creates one complete cycle of the sine wave. Notice the amplitude is 1, which is the radius of our unit circle, and the period is 2 pi radians.
Let's analyze the key properties of the sine function. The amplitude is 1, representing the maximum distance from the center line. The period is 2 pi, meaning the pattern repeats every 2 pi units. The domain includes all real numbers, while the range is limited to values between negative 1 and 1. Key points include sine of pi over 2 equals 1, sine of pi equals 0, sine of 3 pi over 2 equals negative 1, and sine of 2 pi equals 0. This periodic pattern continues indefinitely.