Exploring lengths of chords
The radius of the circles in these apps is 1 unit.
Note how the chord length changes as the points travel around the circumference.
Angle at the center
We say a chord subtends an angle at the center of a circle.
Note that as the chord length increases, the angle at the center increases.
While chord lengths (and half chord lengths) have been studied since Hipparchus in 150 BCE, it is only recently that mathematicians have parked the circle on the Cartesian plane to think of all possible chords through vertical/horizontal chords.
The sine curve is the measurement of half of the vertical half chord, using half of the angle at the center.
Using half chords and half angles is useful. For one, we see that they generate all right angle triangles.
Below we see that at 30 degrees, the length of half of the vertical chord is 0.5.
The graph shows the point (30,0.5).
A scientific calculator in degree mode will say sin(30)=0.5
The scientific calculator will also say sin-1(0.5)=30 (read, sin 'inverse' 0.5 equals 30).