Sine and Cosine Function Values · 正弦与余弦的函数值
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| sine/saɪn/ | 正弦 | zhèng xián |
| cosine/ˈkəʊsaɪn/ | 余弦 | yú xián |
| unit circle/ˈjuːnɪt ˈsɜːkl/ | 单位圆 | dān wèi yuán |
| radians/ˈreɪdɪənz/ | 弧度 | hú dù |
| period/ˈpɪərɪəd/ | 周期 | zhōu qī |
Values you can read off a circle
- Every angle has a sine and a cosine — but where do the numbers come from?
- They are just coordinates on the unit circle, read straight off.
- A handful of "special" angles give clean, memorable values.
- Learn those, and the symmetry of the circle hands you the rest.
能从圆上读出的值
- 每个角都有一个正弦和一个余弦——但这些数从哪来?
- 它们不过是单位圆上的坐标,直接读出来。
- 少数几个"特殊"角给出干净、好记的值。
- 学会那些,圆的对称性就把其余的都交给你。
Special-angle values
- At $\theta = 0$: the point is $(1, 0)$, so $\cos 0 = 1$ and $\sin 0 = 0$.
- At $\theta = \tfrac{\pi}{2}$ ($90°$): the point is $(0, 1)$, so $\cos = 0$, $\sin = 1$.
- The angles $30°, 45°, 60°$ give the familiar values with $\tfrac12$, $\tfrac{\sqrt2}{2}$, $\tfrac{\sqrt3}{2}$.
- Reading a sine 正弦 or cosine 余弦 is just reading a coordinate.
特殊角的值
- 在 $\theta = 0$:点是 $(1, 0)$,所以 $\cos 0 = 1$,$\sin 0 = 0$。
- 在 $\theta = \tfrac{\pi}{2}$($90°$):点是 $(0, 1)$,所以 $\cos = 0$,$\sin = 1$。
- 角 $30°, 45°, 60°$ 给出熟悉的值 $\tfrac12$、$\tfrac{\sqrt2}{2}$、$\tfrac{\sqrt3}{2}$。
- 读一个正弦(sine)或余弦(cosine),就是读一个坐标。
Read sine and cosine at any angle · 在任意角度读出正弦和余弦
Move the angle and watch the sine (height) and cosine (across) values change and repeat. · 移动角度,看正弦(高度)和余弦(横向)的值变化并重复。
What is $\sin 0$? · $\sin 0$ 是多少?
At angle $0$, the unit-circle point is $(1, 0)$, so the y-coordinate $\sin 0 = 0$. · 在角度 $0$ 时,单位圆上的点是 $(1, 0)$,所以 y 坐标 $\sin 0 = 0$。
What is $\cos 0$? · $\cos 0$ 是多少?
At angle $0$, the point is $(1, 0)$, so the x-coordinate $\cos 0 = 1$. · 在角度 $0$ 时,点是 $(1, 0)$,所以 x 坐标 $\cos 0 = 1$。
Signs by quadrant
- In the top-right quadrant both coordinates are positive, so sine and cosine are positive.
- Cross into other quadrants and one or both turn negative.
- "All, Sine, Tangent, Cosine" (ASTC) records which are positive, quadrant by quadrant.
- The unit circle 单位圆 makes every sign obvious — just look at the coordinate.
各象限的正负号
- 在右上象限,两个坐标都为正,所以正弦和余弦都为正。
- 跨到其他象限,其中一个或两个变负。
- "全、正弦、正切、余弦"(ASTC)记录了每个象限里哪些为正。
- 单位圆(unit circle)让每个正负号一目了然——只看坐标即可。
The output of $\sin\theta$ always lies between… · $\sin\theta$ 的输出总是介于……之间。
On the unit circle the y-coordinate never leaves $[-1, 1]$, so $-1 \le \sin\theta \le 1$. · 在单位圆上 y 坐标从不离开 $[-1, 1]$,所以 $-1 \le \sin\theta \le 1$。
Select all · 所有 true statements about sine and cosine values. · 选出关于正弦和余弦值的所有正确说法。
Sine never reaches $2$ — it is capped at $1$. The other three are correct. · 正弦永远到不了 $2$——它的上限是 $1$。其余三条正确。
Repeating every full turn
- After one full revolution the point lands exactly where it began.
- So the values repeat every $2\pi$ radians 弧度 (or $360°$).
- That regular repeat is the period 周期 of sine and cosine.
- Adding any multiple of $2\pi$ to an angle leaves its sine and cosine unchanged.
每整圈重复一次
- 转过一整圈后,点正好落回它开始的地方。
- 所以值每隔 $2\pi$ 弧度(radians)(或 $360°$)重复一次。
- 那个规律的重复就是正弦和余弦的周期(period)。
- 给一个角加上任意 $2\pi$ 的倍数,它的正弦和余弦都不变。
Sine and cosine values repeat every ____ radians (one full turn). · 正弦和余弦的值每隔____弧度(一整圈)重复一次。
After a full revolution of $2\pi$ (or $360°$), the point returns to where it started, so the values repeat. · 转过一整圈 $2\pi$(或 $360°$)后,点回到起点,所以值开始重复。
Bounded outputs
- Because the circle has radius $1$, no coordinate ever exceeds $1$ or drops below $-1$.
- So both sine and cosine always satisfy $-1 \le \text{value} \le 1$.
- The maximum, $1$, and minimum, $-1$, occur at the top/bottom or left/right of the circle.
- This bound is why sinusoidal waves have a fixed height.
有界的输出
- 因为圆的半径是 $1$,任何坐标都不会超过 $1$ 或低于 $-1$。
- 所以正弦和余弦的值都始终满足 $-1 \le y \le 1$。
- 最大值 $1$ 和最小值 $-1$ 出现在圆的顶/底或左/右。
- 这个界限正是正弦型波有固定高度的原因。
Sine and cosine can never be $2$ or $-5$. Their outputs are locked into $[-1, 1]$ by the unit circle. If a calculation gives $\sin\theta = 1.4$, you have made an error somewhere.
正弦和余弦永远不会是 $2$ 或 $-5$。它们的输出被单位圆锁定在 $[-1, 1]$ 内。如果一个计算给出 $\sin\theta = 1.4$,那么你在某处出错了。
Find $\sin\tfrac{\pi}{2}$, $\cos\pi$, and $\sin 3\pi$.
- $\sin\tfrac{\pi}{2}$: the top of the circle $(0,1)$, so $\sin = 1$.
- $\cos\pi$: the left point $(-1, 0)$, so $\cos = -1$.
- $\sin 3\pi = \sin(\pi + 2\pi) = \sin\pi = 0$ — the extra $2\pi$ changes nothing.
求 $\sin\tfrac{\pi}{2}$、$\cos\pi$ 和 $\sin 3\pi$。
- $\sin\tfrac{\pi}{2}$:圆的顶部 $(0,1)$,所以 $\sin = 1$。
- $\cos\pi$:左边的点 $(-1, 0)$,所以 $\cos = -1$。
- $\sin 3\pi = \sin(\pi + 2\pi) = \sin\pi = 0$——多出的 $2\pi$ 什么都不改变。
Sine and cosine values are coordinates on the unit circle. Special angles give clean values, and the quadrant fixes the sign. The values repeat every $2\pi$ radians (the period), and both always stay within $[-1, 1]$.
正弦和余弦的值是单位圆上的坐标。特殊角给出干净的值,象限确定正负号。值每隔 $2\pi$ 弧度重复(即周期),而两者都始终保持在 $[-1, 1]$ 内。