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. Let us confirm the relationship with a formula.

$\cos x$ is $\sin x$ shifted to the left by $\dfrac{\pi}{2}$. In symbols,

Shifting a graph left by $\dfrac{\pi}{2}$ amounts to replacing $x$ with $x + \dfrac{\pi}{2}$, and the peaks of $\sin$ land exactly on the peaks of $\cos$.

Indeed, $\sin x$ has its maximum $1$ (a peak) at $x = \dfrac{\pi}{2}$, while $\cos x$ has its maximum $1$ at $x = 0$. The peak of $\cos$ is $\dfrac{\pi}{2}$ to the left of the peak of $\sin$, exactly the amount of the shift.

Likewise $\sin x = \cos\left(x - \dfrac{\pi}{2}\right)$, so $\sin$ is $\cos$ shifted right by $\dfrac{\pi}{2}$. Sine and cosine are the same wave, carried onto each other by a shift of $\dfrac{\pi}{2}$. The large dots on the graph are the peak of $\cos$ at $(0, 1)$ and the peak of $\sin$ at $\left(\dfrac{\pi}{2}, 1\right)$.