The relationship between sine and cosine

The graphs of $y = \sin x$ and $y = \cos x$ have exactly the same shape and differ only by a horizontal shift. Here we confirm that relationship with formulas.

The translation

The cosine is the sine shifted left by $\dfrac{\pi}{2}$ .

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

Shifting a graph left by $\dfrac{\pi}{2}$ amounts to replacing $x$ with $x + \dfrac{\pi}{2}$ , and the crests of the sine then land exactly on the crests of the cosine. In general $y = f(x - a)$ is the graph shifted right by $a$ , so a $+\dfrac{\pi}{2}$ produces a shift to the left. The sign is easy to read backwards, so it is worth some care.

Checking with the crests

Function	Where the maximum falls	Zeros
$y = \sin x$	$\dfrac{\pi}{2} + 2n\pi$	$n\pi$
$y = \cos x$	$2n\pi$	$\dfrac{\pi}{2} + n\pi$

The crest of the cosine sits $\dfrac{\pi}{2}$ to the left of the crest of the sine, which is exactly the amount of the shift. The zeros are displaced by the same amount.

Seeing it on the unit circle

The relationship is clearest on the unit circle. For a point on that circle the cosine is the horizontal coordinate and the sine is the vertical one. Advancing the angle by $\dfrac{\pi}{2}$ rotates the point by $90^\circ$ , so the present vertical coordinate becomes the next horizontal one.

\sin\left(\theta + \frac{\pi}{2}\right) = \cos\theta

The shift that differentiation produces

Since $(\sin x)' = \cos x$ , differentiation can be read as an operation that shifts the wave left by $\dfrac{\pi}{2}$ .

Times differentiated	Result	Total shift
$1$	$\cos x$	$\dfrac{\pi}{2}$
$2$	$-\sin x$	$\pi$
$3$	$-\cos x$	$\dfrac{3\pi}{2}$
$4$	$\sin x$	$2\pi$

That four differentiations return the original fits this reading: the total shift is a full $2\pi$ .

The relation the other way round

Equally, the sine is the cosine shifted right by $\dfrac{\pi}{2}$ .

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

Phase difference

This displacement between two waves is called the phase difference. It is a basic idea in applications, such as the current through an inductor or a capacitor in an alternating-current circuit running $\dfrac{\pi}{2}$ out of step with the voltage.

The large dots on the graph are the crest of the cosine at $(0, 1)$ and the crest of the sine at $\left( \dfrac{\pi}{2}, 1 \right)$ .