Is decent. It's really just Pythagoras and basic trig.
Describing a circle as a parametric curve Is one of the simplest examples of parametric curves and hence worth understanding.
So the start/end of the arc are the start/end values of t. So subtract to get the angular width.
sorry i dont think the unit circle is making this clearer could you perphaps link this back to the equaton for angle in circle or an alternative route
At the end of the day, you have to understand how the arc is generated by the parametric curve, and the arc is part of the unit circle, so ..
So for example, look at how is made by drawing a line which is to the x-axis. This line happens to be equal to 1 because it is also the radius of the circle. But using trigonometry, we can also make a right-angled triangle, setting the line as the hypotenuse, using the projection along the x-axis as the "adjacent" and the projection along the y-axis as the "opposite".
So from trig:
This is shown in the figure below:
Fig. 1. https://www.khanacademy.org/math/alg...-circle-review
Similarly for sine:
Now you can vary the angle theta going from the positive x-axis anti-clockwise all the way back around again. A complete revolution is . In radians, . You can build the unit circle (a circle with radius of 1) by effectively making these triangles in each quadrant of the graph. But a pattern emerges because there is a symmtery between the triangles you construct. So you find that when you build a triangle in the top left corner that makes with the negative x-axis. But it revolves counter-clockwise from the positive x-axis and this leads to relationships such as . You can see this more clear when you plot the sine function:
So finally, you are left with why do the two angles subtract. Well imagine all the angles start from the positive x axis and revolve counter-clockwise. They can go around the circle once () or twice () and so fourth - but in this case it just begins to repeat again! If you have an angle in radians of you are in the top left quadrant. You can then figure out where the other angle is and work out what the angle is in between the two.
As an aside, you can therefore immediately derive the relationship:
(we showed this in Fig. 1)
Why is this all useful? Aside from the mathematical interest, well it often comes up in signal work and vibrations analysis. You often have use sine functions to model vibrations on surfaces or to model signals. You may process signals using Fourier analysis which uses this work etc. All things used in the real world to build engineering components or study nature.