Graphs of Sine and Cosine · 正弦与余弦的图像
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| sine/saɪn/ | 正弦 | zhèng xián |
| cosine/ˈkəʊsaɪn/ | 余弦 | yú xián |
| period/ˈpɪərɪəd/ | 周期 | zhōu qī |
| amplitude/ˈæmplɪtjuːd/ | 振幅 | zhèn fú |
| midline/ˈmɪdlaɪn/ | 中线 | zhōng xiàn |
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.
把圆展开成一条波
- 想象那个点绕单位圆旋转,只追踪它随时间变化的高度。
- 把那个高度对角度作图,一条平滑的波就出现了。
- 那条波就是正弦的图像——被展开的圆。
- 余弦是同样的想法,只是追踪的是横向坐标。
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.
正弦图像
- 正弦(sine)图像 $y = \sin x$ 从 $0$ 开始,升到 $1$,经过 $0$ 落到 $-1$,再回来。
- 一个完整循环走过 $2\pi$ 之后,模式才重复。
- 它在水平轴上下摆动固定的距离。
- 结果就是那条经典的、平滑滚动的波。
Trace the sine wave · 描画正弦波
y = a·sin(b(x − c)) + d
This is the parent sine wave — amplitude 1, midline y = 0, and period 2π. Watch it repeat. · 这是母正弦波——振幅 1,中线 y = 0,周期 2π。看它重复。
The parent sine graph $y = \sin x$ starts (at $x = 0$) at height… · 母正弦图像 $y = \sin x$ 在(($x = 0$ 时)从高度……开始。
Since $\sin 0 = 0$, the sine wave begins at the midline and rises to its peak of $1$. · 因为 $\sin 0 = 0$,正弦波从中线开始,升到它的峰值 $1$。
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.
余弦图像
- 余弦(cosine)图像 $y = \cos x$ 有着完全相同的形状。
- 但它从它的峰值 $1$ 开始(因为 $\cos 0 = 1$),比正弦提前四分之一个周期。
- 事实上,$\cos x = \sin\!\left(x + \tfrac{\pi}{2}\right)$——余弦是向左平移的正弦。
- 所以学会一条波,基本上就同时拥有了两条。
The parent cosine graph $y = \cos x$ starts (at $x = 0$) at height… · 母余弦图像 $y = \cos x$ 在(($x = 0$ 时)从高度……开始。
Since $\cos 0 = 1$, cosine starts at its maximum — a quarter-period ahead of sine. · 因为 $\cos 0 = 1$,余弦从它的最大值开始——比正弦提前四分之一个周期。
Select all · 所有 true statements about $y = \sin x$ and $y = \cos x$. · 选出关于 $y = \sin x$ 和 $y = \cos x$ 的所有正确说法。
They oscillate, never increasing forever. The other three are correct. · 它们在振荡,从不永远递增。其余三条正确。
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.
振幅、中线与周期
- 母波的振幅(amplitude)是 $1$:它们向上升 $1$,向下沉 $1$。
- 它们的中线(midline)是 $y = 0$,即它们围绕振荡的水平线。
- 它们的周期(period)是 $2\pi$,即一个循环的水平长度。
- 每一条其他正弦形状的图像,都是这些经过拉伸或平移得到的。
The parent sine and cosine graphs each have a period of ____ radians. · 母正弦和余弦图像的周期各是____弧度。
One full cycle spans $2\pi$ before the wave repeats — the period · 周期 of the parent functions. · 一个完整循环跨越 $2\pi$ 之后波才重复——这是母函数的周期。
The parent sine and cosine graphs have amplitude $1$ and midline $y = 0$. · 母正弦和余弦图像的振幅是 $1$,中线是 $y = 0$。
They swing between $-1$ and $1$: amplitude · 振幅 $1$ above/below the midline $y = 0$. · 它们在 $-1$ 和 $1$ 之间摆动:在中线 $y = 0$ 上下 振幅 为 $1$。
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.
读这条波
- 要画一个循环,标出五个关键点:起点、波峰、中点、波谷、终点。
- 对正弦来说,这些高度在一个周期上是 $0, 1, 0, -1, 0$。
- 对余弦来说,它们是 $1, 0, -1, 0, 1$。
- 用一条平滑的曲线把它们连起来,再向左右永远重复。
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.
正弦和余弦图像从不停止或趋平——它们永远在 $-1$ 和 $1$ 之间振荡。不要把它们画成朝渐近线趋平;与指数和对数不同,它们只是不停地起伏。
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.
$y = \sin x$ 第一次到达最大值和最小值在哪里?
- 最大值 $1$ 出现在 $x = \tfrac{\pi}{2}$(一个循环的四分之一处)。
- 最小值 $-1$ 出现在 $x = \tfrac{3\pi}{2}$(四分之三处)。
- 此后两者都每隔 $2\pi$ 重复。
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 $\cos x = \sin(x + \tfrac{\pi}{2})$.
正弦和余弦图像是被展开的单位圆:平滑的波,振幅 $1$,中线 $y = 0$,周期 $2\pi$。它们形状完全相同,相差四分之一周期,所以 $\cos x = \sin(x + \tfrac{\pi}{2})$。