Derivatives of the trigonometric functions

Differentiating $\sin$ gives $\cos$ , and differentiating $\cos$ gives $-\sin$ ¹. Four differentiations return the original.

\begin{align*} \left( \sin x \right)' &= \cos x \\ \left( \cos x \right)' &= -\sin x \end{align*}

Checking on the graph

Each differentiation shifts the wave left by $\dfrac{\pi}{2}$ .

$x$	$\sin x$	Slope	$\cos x$
$0$	$0$	$1$	$1$
$\dfrac{\pi}{2}$	$1$	$0$	$0$
$\pi$	$0$	$-1$	$-1$

The table also shows the slope vanishing where $\sin$ attains its maximum.

Back after four

Times differentiated	Result
$1$	$\cos x$
$2$	$-\sin x$
$3$	$-\cos x$
$4$	$\sin x$

Two differentiations reverse the sign, so the following holds.

y'' = -y

That the vibration of a spring and the swing of a pendulum are written with this equation is because $\sin$ and $\cos$ have that property.

Proving the formula

The proof uses $\lim_{h \to 0} \dfrac{\sin h}{h} = 1$ . The sum-to-product formula rewrites the difference.

\sin(x + h) - \sin x = 2\cos\left( x + \frac{h}{2} \right)\sin\frac{h}{2}

Dividing by $h$ and using $\dfrac{\sin(h/2)}{h/2} \to 1$ leaves $\cos x$ .

Why radians are used

That limit equals $1$ only when the angle is measured in radians. In degrees it becomes $\dfrac{\pi}{180} \approx 0.01745$ , and the factor would multiply in at every differentiation. Radians are used in calculus precisely to make that factor disappear.

The tangent and the inverse functions

The quotient rule gives the derivative of $\tan x$ .

\left( \tan x \right)' = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x}

It is always positive, so $\tan$ increases monotonically on each of its branches. That the slope grows without bound near $\cos x = 0$ matches the way the curve clings to its asymptote.

The inverse trigonometric functions follow too. Writing $y = \arcsin x$ as $\sin y = x$ and differentiating both sides gives $\cos y \cdot y' = 1$ , and $\cos y = \sqrt{1 - x^2}$ leads to the following.

\left( \arcsin x \right)' = \frac{1}{\sqrt{1 - x^2}}

The wave through the origin is $y = \sin x$ , its derivative is $y = \cos x$ , the further derivative is $y = -\sin x$ , and the large dots are $(0, 0)$ , $(0, 1)$ and $\left( \dfrac{\pi}{2}, 0 \right)$ .

Differentiation of trigonometric functions, Wikipedia