Graphs of sine and cosine

On the unit circle, the $y$-coordinate of the terminal point is $\sin\theta$. As $\theta$ increases from $0$ to $2\pi$, that $y$-coordinate rises to $1$, falls back through zero to $-1$, and returns to zero, tracing out a smooth wave. The graph of $y = \sin\theta$ records this motion on a standard set of axes.

Five key points mark out one complete cycle:

Two numbers measure the wave itself: how long one full cycle takes, and how far it reaches above and below the centre line. We call these the period and the amplitude.

Because the unit circle has no start or end, the wave repeats identically in both directions. The portion from $0$ to $2\pi$ is one copy of a pattern that extends forever in each direction.

The sine graph tracks the $y$-coordinate of the unit-circle point. The cosine graph tracks the $x$-coordinate instead. As $\theta$ increases from $0$ to $2\pi$, $\cos\theta$ starts at $1$, falls through zero, reaches $-1$, and returns to $1$. The result is a wave with the same shape as sine, shifted along the horizontal axis.

Cosine starts at its maximum ($1$), crosses zero at $\dfrac{\pi}{2}$, hits its minimum ($-1$) at $\pi$, crosses zero again at $\dfrac{3\pi}{2}$, and returns to $1$ at $2\pi$. Like sine, it has period $2\pi$ and amplitude $1$.

Comparing the two curves, every feature of cosine appears $\dfrac{\pi}{2}$ earlier than the same feature of sine. The cosine peak is at $0$ while the sine peak is at $\dfrac{\pi}{2}$; cosine crosses zero at $\dfrac{\pi}{2}$ while sine crosses zero at $\pi$; and so on. This relationship is captured by the identity:

Sine and cosine accept any real number as input - the wave keeps oscillating no matter how large or negative $\theta$ becomes. Their outputs, however, are tightly bounded: a unit-circle point has coordinates $(\cos\theta, \sin\theta)$ on a circle of radius $1$, so neither coordinate can exceed $1$ in size or fall below $-1$.

To tell the curves apart without labels, check the starting value: $\sin(0) = 0$ (the curve starts on the axis), while $\cos(0) = 1$ (the curve starts at the top).

Suppose we want to find every $\theta$ in $[0, 2\pi]$ that satisfies $\sin\theta = c$ for some target value $c$. The number of solutions is determined entirely by where $c$ sits relative to the range $[-1, 1]$.

Visually, each value of $c$ corresponds to a horizontal line $y = c$. The number of solutions is the number of places that line meets the sine curve.

The same three cases apply equally to both functions. Switching the toggle changes where the solutions sit, not how many there are.