Derivatives of Trigonometric Functions

Greek letters being usual to denote angles, we will take as the usual letter for any variable angle the letter  (“theta”). In this chapter, is measured in radians.¹

Derivative of Sine

Let us consider the function

What we have to investigate is the value of ; or, in other words, if the angle  varies, we have to find the relation between the increment of the sine and the increment of the angle, both increments being indefinitely small in themselves. Examine the next figure, wherein, if the radius of the circle is unity, the height of  is the sine, and is the angle. Now, if is supposed to increase by the addition to it of the small angle —an element of angle—the height of , the sine, will be increased by a small element . The new height  will be the sine of the new angle , 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 and  are two different angles,

If, then, we put for one angle, and for the other, we may write or,

But if we regard as indefinitely small, then in the limit we may neglect  by comparison with , and may also take as being the same as . The equation then becomes: and, finally, [Notice that the approximation is true only when is measured in radians.]

The accompanying curves in the next two figures show, plotted to scale, the values of , and , for the corresponding values of .

\tan(A+B)=\frac{\tan_A+\tan_B}{1-\tan_A\tan_B}$" transform="translate(0,1330)"> whence

Now remember that if is indefinitely diminished, the value of  becomes identical with , and is negligibly small compared with , so that the expression reduces to so that or ↩︎