Graphs of Sine and Cosine

From Circle to Wave

We know sine and cosine as coordinates on the unit circle. But what do they look like as graphs?

Watch what happens when we let the angle keep growing and plot the height of the point.

The circle on the left spins. The height of the point (its y-coordinate, which is sine) gets carried across to the right and traced out. The circle unrolls into a wave.

Why It Repeats

Going around the circle once is 360°, or $2\pi$ radians. After that, you are back where you started.

So the graph repeats every $2\pi$. This repeating distance is called the period.

Sine and cosine are periodic: their graphs repeat forever, one identical cycle after another.

Reading the Sine Graph

Start at angle 0 and follow the point around:

The wave rises to 1, falls through 0 down to -1, and climbs back. That is one full cycle.

The Cosine Graph

Cosine is the x-coordinate instead of the y-coordinate.

It has the exact same wave shape. The scan line makes the pattern clear: when sine sits on the midline, cosine is at a peak, and cosine reaches each peak a quarter-cycle before sine does.

The only difference between them is where they start:

Sine starts on the midline. Cosine starts at the peak.

Cosine is just sine shifted to the left by 90°. Same wave, different starting point.

We can write that exactly:

$\cos\theta = \sin(\theta + 90°)$

The Shape in Numbers

Both graphs live between -1 and 1, forever. They never reach higher or lower.

• The highest value is 1
• The lowest value is -1
• The midline (the center they wave around) is 0
• One full cycle takes $2\pi$ (360°)

Everything about a sine or cosine graph is a change to these four numbers. That is the next lesson.