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 by coordinates (x, y). When we measure an angle theta from the positive x-axis, we can find the corresponding point on the circle. Positive angles are measured counterclockwise, while negative angles go clockwise.
The sine function is defined using the unit circle. For any angle theta, sine of theta equals the y-coordinate of the point where the terminal side intersects the circle. Let's see this for key angles: sine of 0 degrees is 0, sine of 30 degrees is one half, sine of 45 degrees is square root of 2 over 2, sine of 60 degrees is square root of 3 over 2, and sine of 90 degrees is 1.
The cosine function is defined as the x-coordinate of a point on the unit circle. Now we can see both functions together: cosine of theta equals x, and sine of theta equals y. Let's examine key values: cosine of 0 degrees is 1, cosine of 30 degrees is square root of 3 over 2, cosine of 45 degrees is square root of 2 over 2, cosine of 60 degrees is one half, and cosine of 90 degrees is 0. Notice how both coordinates change as the point moves around the circle.
Now let's see how the sine wave is generated from the unit circle. As the radius rotates around the circle, we track the y-coordinate and plot it against the angle. The result is the characteristic sine wave. Key points include: at 0 the value is 0, at pi over 2 it reaches 1, at pi it returns to 0, at 3 pi over 2 it drops to negative 1, and at 2 pi it completes the cycle back to 0.
Now let's generate the cosine wave using the same method. As the radius rotates, we track the x-coordinate instead of the y-coordinate. The cosine wave has the same shape as the sine wave but is shifted. Key points for cosine: at 0 the value is 1, at pi over 2 it's 0, at pi it's negative 1, at 3 pi over 2 it's 0 again, and at 2 pi it returns to 1. Notice that cosine leads sine by pi over 2 radians or 90 degrees.