The sine and cosine curves

Turn the angle $\theta$ into the horizontal axis and plot the terminal point's y-coordinate against it: that is $y = \sin\theta$. At $\theta = 0$ the point is $(1, 0)$, height 0, so sine starts at 0 and rises. It peaks at 1 when $\theta = \pi/2$ (top of the circle), returns to 0 at $\pi$, bottoms out at $-1$ at $3\pi/2$, and closes the loop at $2\pi$.

Plot the x-coordinate instead and you get $y = \cos\theta$: it starts at its maximum, 1, because the ride begins at $(1, 0)$ where the point is as far right as it will ever be. Cosine falls to 0 at $\pi/2$, reaches $-1$ at $\pi$, and climbs back to 1 at $2\pi$. Same wave, different starting phase.

The two parent curves have identical measurements:

1. Period $2\pi$: one lap of the circle is one full wave. The graph from $2\pi$ to $4\pi$ is a perfect copy of the graph from 0 to $2\pi$.
2. Amplitude 1 and midline $y = 0$: the circle's radius is 1, so the coordinates swing 1 above and 1 below their center.
3. Range $[-1, 1]$, endpoints included: the values $\pm 1$ are genuinely reached, at the axis crossings of the circle.

Locations differ: sine has zeros at $0, \pi, 2\pi, \ldots$ (multiples of $\pi$) with peaks at $\pi/2 + 2\pi k$; cosine has zeros at $\pi/2, 3\pi/2, \ldots$ (odd multiples of $\pi/2$) with peaks at $0, 2\pi, 4\pi, \ldots$. When you cite a feature, name the curve.

The two curves are congruent: slide the sine curve left by $\pi/2$ and it lands exactly on cosine. In symbols, $\cos x = \sin(x + \pi/2)$. The circle explains it: the x-coordinate of the moving point does now what the y-coordinate will do a quarter turn later.

Direction matters. Sliding sine right by $\pi/2$ gives $\sin(x - \pi/2)$, which is $-\cos x$, the cosine curve flipped upside down. Check with one point: sine's peak sits at $\pi/2$; sliding left by $\pi/2$ moves that peak to 0, exactly where cosine peaks. Sliding right moves it to $\pi$, where cosine is at its minimum.

Read behavior in quarter-circle chunks. Sine increases on $(-\pi/2, \pi/2)$ and decreases on $(\pi/2, 3\pi/2)$: the height of the circling point grows on the right half of the circle and shrinks on the left. Cosine decreases all the way from 0 to $\pi$ and increases from $\pi$ to $2\pi$.

Concavity follows the midline: each curve is concave down while above its midline (arching over a peak) and concave up while below (cupping under a trough). The bend switches exactly at the midline crossings, so sine's inflection points sit at $0, \pi, 2\pi, \ldots$, the same places as its zeros. A peak is a place where the curve turns, not where it bends the other way.