The sine function is one of the most important functions in mathematics. It describes the relationship between angles and side ratios in right triangles. In a right triangle, the sine of an angle theta is defined as the ratio of the opposite side to the hypotenuse.
The sine function can also be understood using the unit circle. On a unit circle with radius one, when we draw an angle theta from the positive x-axis, the sine of theta equals the y-coordinate of the point where the angle intersects the circle. As the angle changes, we can see how the sine value varies.
When we plot the sine function on a coordinate system, we get a beautiful wave pattern. The graph shows that sine starts at zero, rises to one at pi over two, returns to zero at pi, drops to negative one at three pi over two, and completes the cycle at two pi. This pattern repeats indefinitely with a period of two pi.
There are several key values of the sine function that are important to memorize. At zero radians, sine equals zero. At pi over six, sine equals one half. At pi over four, sine equals square root of two over two. At pi over three, sine equals square root of three over two. At pi over two, sine reaches its maximum value of one. These values form the foundation for understanding trigonometric calculations.
To summarize what we have learned about the sine function: It is a fundamental trigonometric function that relates angles to side ratios in right triangles. On the unit circle, sine represents the y-coordinate. The sine graph creates a beautiful periodic wave pattern. Understanding key values and properties of sine is essential for mathematics, physics, and engineering applications.