Page:Calculus Made Easy.pdf/189

Now remember that if $d\theta$ is indefinitely diminished, the value of $\tan d\theta$ becomes identical with $d\theta$, and $\tan \cdot d\theta$ is negligibly small compared with $1$, so that the expression reduces to

${\begin{aligned}dy&={\frac {(1+\tan ^{2}\theta )\,d\theta }{1}},\\{\text{so that}}\quad \quad \quad {\frac {dy}{d\theta }}&=1+\tan ^{2}\theta ,\\{\text{or}}\quad \quad \quad {\frac {dy}{d\theta }}&=\sec ^{2}\theta .\end{aligned}}$

Collecting these results, we have:

$y$	${\frac {dy}{d\theta }}$
$\sin \theta$	$\cos \theta$
$\cos \theta$	$-\sin \theta$
$\tan \theta$	$\sec ^{2}\theta$

Sometimes, in mechanical and physical questions, as, for example, in simple harmonic motion and in wave-motions, we have to deal with angles that increase in proportion to the time. Thus, if $T$ be the time of one complete period, or movement round the circle, then, since the angle all round the circle is $2\pi$ radians, or $360^{\circ }$, the amount of angle moved through in time $t$, will be

${\begin{aligned}\theta &=2\pi {\frac {t}{T}},\quad {\text{in radians,}}\\{\text{or}}\quad \quad \quad \theta &=360{\frac {t}{T}},\quad {\text{in degrees.}}\end{aligned}}$