As the angle $x$ grows, the point on the unit circle goes around and around, and its height rises and falls. Plot that height against $x$ and you get the graph of $y = \sin x$: the unit circle unrolled into a wave. The symmetry and periodicity you found on the circle become features of the graph.
Key idea
$y = \sin x$ starts at $0$, rises to $1$ at $\frac{\pi}{2}$, returns to $0$ at $\pi$, falls to $-1$ at $\frac{3\pi}{2}$ and is back at $0$ at $2\pi$. $y = \cos x$ follows the same shape but starts at its maximum, $1$. Both have domain all real numbers, range $[-1, 1]$ and period $2\pi$.
Why it works
$\sin x$ is the height ($y$-coordinate) of the point at angle $x$, and $\cos x$ is how far across it is ($x$-coordinate). A full turn brings the point back to the start, so each graph repeats every $2\pi$. The cosine wave is the sine wave shifted left by a quarter turn: $\cos x = \sin\left(x + \frac{\pi}{2}\right)$, because the point at angle $x + \frac{\pi}{2}$ is a quarter turn ahead.
| $x$ | $0$ | $\frac{\pi}{2}$ | $\pi$ | $\frac{3\pi}{2}$ | $2\pi$ |
|---|---|---|---|---|---|
| $\sin x$ | $0$ | $1$ | $0$ | $-1$ | $0$ |
| $\cos x$ | $1$ | $0$ | $-1$ | $0$ | $1$ |
Worked examples
Worked example: Zeros
Where does $y = \sin x$ cross the $x$-axis on $[0, 2\pi]$?
- The graph is at height $0$ where the point on the unit circle is on the $x$-axis.
- That happens at angles $0$, $\pi$ and $2\pi$.
Answer: $x = 0, \pi, 2\pi$
Worked example: Rising and falling
On $[0, 2\pi]$, where is $y = \cos x$ decreasing?
- $\cos x$ is the $x$-coordinate. It starts at $1$ when $x = 0$ and moves left as the point travels over the top of the circle.
- It reaches $-1$ at $x = \pi$, then moves right again.
Answer: Decreasing on $[0, \pi]$, increasing on $[\pi, 2\pi]$
Worked example: Sketching quickly
Sketch $y = \sin x$ on $[-2\pi, 2\pi]$.
- Plot the five key points on $[0, 2\pi]$: $(0, 0)$, $\left(\frac{\pi}{2}, 1\right)$, $(\pi, 0)$, $\left(\frac{3\pi}{2}, -1\right)$, $(2\pi, 0)$.
- Sine is odd, so the part on $[-2\pi, 0]$ is the same wave turned upside down through the origin. (Or use the period: copy the cycle one period to the left.)
- Join everything with a smooth curve that is rounded at the peaks.
Answer: Two full waves, crossing the axis at every multiple of $\pi$
Common mistake
Mixing up which wave starts where. A quick check at $x = 0$: $\sin 0 = 0$, so the sine graph passes through the origin; $\cos 0 = 1$, so the cosine graph starts at the top. Another slip is drawing pointed peaks: the graph is rounded at the top because the point moves almost sideways there.