Trig graphs

As the point keeps traveling around the unit circle, its sine value repeats at regular intervals. Plotted against the angle, this creates an even wave. Such waves describe an astonishing amount in nature: sound, light, tides, the turn of the seasons.

From circle to wave

Plot the angle on the horizontal axis and the sine value on the vertical, and you get the sine curve. It starts at 0, rises to 1 at 90 degrees, falls back through 0 at 180 degrees, drops to −1 at 270 degrees, and returns to 0 at 360 degrees. The cosine curve has exactly the same shape but is shifted 90 degrees to the left, which is why it already starts at 1.

The period

After 360 degrees the basic curve repeats completely. This length of one full swing is called the period. Writing $y = \sin(b \cdot x)$ with a positive number $b$ changes that length: with $b = 2$ a full swing already fits into 180 degrees, so the period is $\frac{360°}{b}$.

The amplitude

A factor in front of the sine changes the height. For $y = a \cdot \sin x$ with a positive $a$ the curve swings between $-a$ and $+a$. This greatest displacement is called the amplitude. So $y = 3 \cdot \sin x$ runs from −3 to 3, its amplitude is 3. Amplitude and period together pin down the shape of a wave $y = a \cdot \sin(b \cdot x)$ completely.