Graphs of Sine and Cosine
Introduced| English |
|---|
| sine/saɪn/ |
| cosine/ˈkəʊsaɪn/ |
| period/ˈpɪərɪəd/ |
| amplitude/ˈæmplɪtjuːd/ |
| midline/ˈmɪdlaɪn/ |
Unrolling the circle into a wave
- Imagine the point spinning around the unit circle, and track just its height over time.
- Plot that height against the angle, and a smooth wave appears.
- That wave is the graph of sine — the circle unrolled.
- Cosine is the same idea, tracking the horizontal coordinate instead.
The sine graph
- The sine 正弦 graph $y = \sin x$ starts at $0$, rises to $1$, falls through $0$ to $-1$, then returns.
- One full cycle takes $2\pi$ before the pattern repeats.
- It swings a fixed distance above and below the horizontal axis.
- The result is the classic smooth, rolling wave.
Trace the sine wave
y = a·sin(b(x − c)) + d
This is the parent sine wave — amplitude 1, midline y = 0, and period 2π. Watch it repeat.
The parent sine graph $y = \sin x$ starts (at $x = 0$) at height…
Since $\sin 0 = 0$, the sine wave begins at the midline and rises to its peak of $1$.
The cosine graph
- The cosine 余弦 graph $y = \cos x$ has the exact same shape.
- But it starts at its peak $1$ (since $\cos 0 = 1$), a quarter-period ahead of sine.
- In fact, $\cos x = \sin\!\left(x + \tfrac{\pi}{2}\right)$ — cosine is sine shifted left.
- So learning one wave essentially gives you both.
The parent cosine graph $y = \cos x$ starts (at $x = 0$) at height…
Since $\cos 0 = 1$, cosine starts at its maximum — a quarter-period ahead of sine.
Select all true statements about $y = \sin x$ and $y = \cos x$.
They oscillate, never increasing forever. The other three are correct.
Amplitude, midline, and period
- The parent waves have amplitude 振幅 $1$: they rise $1$ above and dip $1$ below.
- Their midline 中线 is $y = 0$, the level they oscillate around.
- Their period 周期 is $2\pi$, the horizontal length of one cycle.
- Every other sine-shaped graph is one of these, stretched or shifted.
The parent sine and cosine graphs each have a period of ____ radians.
One full cycle spans $2\pi$ before the wave repeats — the period of the parent functions.
The parent sine and cosine graphs have amplitude $1$ and midline $y = 0$.
They swing between $-1$ and $1$: amplitude $1$ above/below the midline $y = 0$.
Reading the wave
- To sketch a cycle, mark five key points: start, peak, middle, trough, end.
- For sine those heights are $0, 1, 0, -1, 0$ across one period.
- For cosine they are $1, 0, -1, 0, 1$.
- Join them with a smooth curve and repeat left and right forever.
Sine and cosine graphs never stop or level off — they oscillate forever between $-1$ and $1$. Do not draw them flattening toward an asymptote; unlike exponentials and logs, they simply keep waving.
Where does $y = \sin x$ first reach its maximum, and its minimum?
- Maximum $1$ occurs at $x = \tfrac{\pi}{2}$ (a quarter of the way through a cycle).
- Minimum $-1$ occurs at $x = \tfrac{3\pi}{2}$ (three-quarters through).
- Both repeat every $2\pi$ thereafter.
The sine and cosine graphs are the unit circle unrolled: smooth waves with amplitude $1$, midline $y = 0$, and period $2\pi$. They are identical in shape, offset by a quarter period, so $\cos x = \sin(x + \tfrac{\pi}{2})$.