Page:Calculus Made Easy.pdf/186

 supposed to increase by the addition to it of the small angle $$d \theta$$–an element of angle–the height of $$y$$, the sine, will be increased by a small element $$dy$$. The new height $$y+dy$$ will be the sine of the new angle $$\theta + d \theta$$, or, stating it as an equation,

and subtracting from this the first equation gives

The quantity on the right-hand side is the difference between two sines, and books on trigonometry tell us how to work this out. For they tell us that if $$M$$ and $$N$$ are two different angles,

If, then, we put $$M= \theta + d \theta$$ for one angle, and $$N=\theta$$ for the other, we may write

But if we regard $$d \theta$$ as indefinitely small, then in the limit we may neglect $$\frac{1}{2} d \theta$$ by comparison with $$\theta$$, and may also take $$\sin\frac{1}{2} d \theta$$ as being the same as $$\frac{1}{2} d \theta$$. The equation then becomes: