Welcome to our exploration of the sine function. The sine function is one of the most fundamental trigonometric functions in mathematics. It originates from the study of right triangles, where sine of an angle theta is defined as the ratio of the opposite side to the hypotenuse. This simple ratio forms the foundation for understanding how sine behaves as a function.
Now let's see how the sine function emerges from the unit circle. In the unit circle, any point can be described by coordinates cosine theta and sine theta. As the angle theta increases and the point moves around the circle, the y-coordinate gives us the sine value. Watch how the sine value changes as we rotate around the circle - this vertical projection creates the foundation for the sine graph.
Here we see the complete sine graph over one full period from 0 to 2π. The sine function starts at zero, rises to its maximum value of 1 at π/2, returns to zero at π, reaches its minimum value of negative 1 at 3π/2, and completes the cycle back at zero at 2π. The amplitude is 1, representing the maximum distance from the center line, and the period is 2π, meaning the pattern repeats every 2π units.
The sine function can be transformed in several ways. First, we can change the amplitude by multiplying by a constant - here 2 sine x has twice the height. Second, we can change the frequency by multiplying the input - sine of 2x completes two full cycles in the same space. Third, we can shift the phase by adding to the input - sine of x plus π/4 shifts the entire wave to the left. These transformations allow us to model many different periodic phenomena in nature and engineering.