Graphing Sine and Cosine: Amplitude, Period and Midline

Unrolling the circle

On the unit circle, $\sin\theta$ is a height. Walk counterclockwise from $(1, 0)$ and that height changes: up to $1$ at a quarter turn, back to $0$ at a half turn, down to $-1$ at three quarters, back to $0$ at a full turn.

Now plot it. Put the angle turned on the horizontal axis and the height on the vertical axis. The circle unrolls into a wave.

• y = sin x

The horizontal axis is measured in radians, so the numbers on it are plain real numbers. The peak sits at $\tfrac{\pi}{2} \approx 1.57$ and the wave returns to the start at $2\pi \approx 6.28$.

Cosine unrolls the same way, except it records the width instead of the height. Width starts at $1$, so the cosine graph starts at the top.

• y = sin x
• y = cos x

They are one shape, drawn from two starting points. Sliding the sine graph $\tfrac{\pi}{2}$ to the left lands exactly on the cosine graph.

$$\cos x = \sin\left(x + \tfrac{\pi}{2}\right)$$

Three numbers describe any wave

Feature	What it measures	For $y = \sin x$
Midline	the horizontal line the wave oscillates about	$y = 0$
Amplitude	the rise from the midline to a peak	$1$
Period	the horizontal length of one full cycle	$2\pi$

From a graph, read them like this:

$$\text{amplitude} = \frac{\text{max} - \text{min}}{2} \qquad \text{midline} = \frac{\text{max} + \text{min}}{2}$$

Amplitude is a distance, so it is never negative.

The general form

$$y = a\sin\bigl(b(x - h)\bigr) + k$$

Letter	Effect
$a$	amplitude $= \lvert a \rvert$; a negative $a$ flips the wave upside down
$b$	period $= \tfrac{2\pi}{\lvert b \rvert}$
$h$	horizontal shift, called the phase shift
$k$	midline $y = k$

Notice that $a$ and $k$ behave exactly as they do for any function transformation: $a$ stretches vertically and $k$ slides vertically. Only $b$ carries a surprise.

• y = sin x
• y = 2 sin x
• y = sin 2x

Why the period is 2π divided by b

Doubling $b$ halves the period, which runs against the usual reading of a coefficient. The reason sits inside the parentheses.

Sine completes one cycle when its input runs from $0$ to $2\pi$. For $y = \sin(bx)$, the input is $bx$, not $x$. So ask how far $x$ must travel for $bx$ to cover $2\pi$:

$$bx = 2\pi \quad\Longrightarrow\quad x = \frac{2\pi}{b}$$

With $b = 2$, the input runs twice as fast as $x$ does, so $x$ only has to reach $\pi$ — half as far — before a full cycle is done. A bigger $b$ means the wave finishes sooner, so the period shrinks.

The same reasoning explains the phase shift. The graph of $\sin(b(x - h))$ starts its cycle where the inside equals zero, which is at $x = h$.

Modeling something that repeats

Anything that cycles can be modeled this way: a Ferris wheel at a county fair, the tide at a harbor, hours of daylight across the year, the current in a household outlet.

The recipe is short. Find the maximum and the minimum, which give the amplitude and the midline. Find how long one cycle takes, which gives $b$. Then pick the function that matches the start: cosine if the cycle begins at a peak or a trough, sine if it begins on the midline.

Take a Ferris wheel with a radius of $25$ feet whose center sits $30$ feet above the ground. It makes one full turn every $40$ seconds, and a rider boards at the bottom.

The rider’s height runs from $5$ feet to $55$ feet, so the midline is $30$ and the amplitude is $25$. One cycle takes $40$ seconds:

$$\frac{2\pi}{b} = 40 \quad\Longrightarrow\quad b = \frac{\pi}{20}$$

The ride starts at the bottom, which is a trough, so use cosine — flipped, since cosine normally starts at a peak.

$$h(t) = 30 - 25\cos\left(\frac{\pi t}{20}\right)$$

• h(t), feet
• midline h = 30

Check it at the three moments you already know. At $t = 0$ it gives $30 - 25 = 5$ feet, the boarding platform. At $t = 20$ it gives $30 + 25 = 55$ feet, the top. At $t = 40$ it is back to $5$ feet, ready for the next rider.

Worked examples

Common mistakes

Practice problems