MISSION · Geometry
Graphs of Sine and Cosine
The shape of anything that repeats
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Sine and cosine graphs are waves, and four numbers in the equation control their height, their width, and where they sit.
\[y = a\sin(b(x - c)) + d\]The basic shapes
\(y = \sin x\) starts at 0, rises to 1 at \(90°\), returns to 0 at \(180°\), drops to \(-1\) at \(270°\), and closes the loop at \(360°\). Then it does it again, forever.
\(y = \cos x\) is the identical wave, started a quarter turn earlier — it begins at its maximum of 1. That’s the only difference: \(\cos x = \sin(x + 90°)\).
Both come straight from the unit circle. The wave is the height of a point going round, plotted against the angle it has turned. Which is why they repeat: the point comes back round.
The four controls
\(a\) — amplitude. How far the wave reaches above and below its middle. \(y = 3\sin x\) runs between \(-3\) and 3. A negative \(a\) flips the wave upside down.
\(b\) — how many cycles fit. The period is \(\frac{360°}{b}\), so \(y = \sin(2x)\) completes two full waves in the space the plain one takes for one. Bigger \(b\) means a squashed, faster wave — the inside-the-bracket inversion again.
\(c\) — phase shift. Slides the wave sideways by \(c\).
\(d\) — vertical shift. Raises the whole wave, moving its centre line from \(y = 0\) to \(y = d\).
Two of those, \(a\) and \(d\), are the same stretch-and-shift moves that work on any function. The other two are the same moves applied inside the bracket, which is why they run backwards.
Reading a wave backwards
Given a graph, recover the equation by measuring:
- Midline — halfway between the peak and the trough. That’s \(d\).
- Amplitude — from the midline up to the peak. That’s \(a\).
- Period — peak to the next peak. Then \(b = \frac{360°}{\text{period}}\).
Do those three and only the phase shift remains, which you read from where a peak or a zero-crossing has moved to.
Why anyone cares
Because periodic behaviour is everywhere, and all of it is described by these two functions: sound and light, alternating current, tides, daylight hours through the year, a heartbeat, a pendulum, the vibration of a bridge.
An audio wave is a sum of sines. Compressing it, filtering it, or recognising speech in it are all operations on amplitudes and frequencies — which are the \(a\) and \(b\) on this page.
Common mix-ups
- Confusing period and frequency. Bigger \(b\) gives a shorter period.
- Reading \(a\) as the height above zero when \(d\) has moved the midline.
- Getting the phase shift backwards. \(\sin(x - 30°)\) moves the wave right.
- Mixing degrees and radians, which changes every number here.
Check yourself
Describe \(y = 2\cos(3x) - 1\): its amplitude, period, and midline.
Amplitude 2, so it swings 2 either side. Period \(\frac{360}{3} = 120°\). Midline \(y = -1\), so the wave runs between \(-3\) and \(1\).
LEVEL 3 · REFLECT
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