Sine, Cosine, and Tangent · 正弦、余弦与正切
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| sine/saɪn/ | 正弦 | zhèng xián |
| cosine/ˈkəʊsaɪn/ | 余弦 | yú xián |
| tangent/ˈtændʒənt/ | 正切 | zhèng qiē |
| unit circle/ˈjuːnɪt ˈsɜːkl/ | 单位圆 | dān wèi yuán |
| radians/ˈreɪdɪənz/ | 弧度 | hú dù |
Three ratios that unlock circles
- Long ago, surveyors and astronomers needed to link angles to lengths.
- The answer was three ratios inside a right triangle: sine, cosine, and tangent.
- They turn an angle into a number, and a number back into an angle.
- These three functions are the doorway to everything periodic.
三个打开圆的比值
- 很久以前,测量员和天文学家需要把角度和长度联系起来。
- 答案是直角三角形里的三个比值:正弦、余弦和正切。
- 它们把角度变成一个数,又把数变回角度。
- 这三个函数是通往一切周期现象的大门。
Right-triangle ratios
- For an acute angle in a right triangle, label the sides opposite, adjacent, and hypotenuse.
- Sine 正弦: $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$.
- Cosine 余弦: $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$.
- Tangent 正切: $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$. Remember SOH-CAH-TOA.
直角三角形的比值
- 对直角三角形里的一个锐角,把各边标为对边、邻边和斜边。
- 正弦(sine):$\sin\theta =$ 对边 ÷ 斜边。
- 余弦(cosine):$\cos\theta =$ 邻边 ÷ 斜边。
- 正切(tangent):$\tan\theta =$ 对边 ÷ 邻边。记住 SOH-CAH-TOA。
In a right triangle, $\sin\theta$ equals… · 在直角三角形中,$\sin\theta$ 等于……
SOH: sine · 正弦 is opposite over hypotenuse. CAH: cosine is adjacent over hypotenuse. TOA: tangent is opposite over adjacent. · SOH:正弦是对边比斜边。CAH:余弦是邻边比斜边。TOA:正切是对边比邻边。
The unit circle
- Draw a circle of radius $1$ centred at the origin — the unit circle 单位圆.
- A point on it at angle $\theta$ has coordinates exactly $(\cos\theta, \sin\theta)$.
- So cosine is the x-coordinate and sine is the y-coordinate.
- This lets angles beyond $90°$ have sine and cosine too — all the way around.
单位圆
- 画一个以原点为圆心、半径为 $1$ 的圆——单位圆(unit circle)。
- 圆上角度为 $\theta$ 的点,坐标恰好是 $(\cos\theta, \sin\theta)$。
- 所以余弦是 x 坐标,正弦是 y 坐标。
- 这让超过 $90°$ 的角也有正弦和余弦——绕圆一整圈都有。
Spin a point around the unit circle · 让一个点绕单位圆旋转
The x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine. · 这个点的 x 坐标是角的余弦,y 坐标是正弦。
On the unit circle, a point at angle $\theta$ has coordinates… · 在单位圆上,角度为 $\theta$ 的点的坐标是……
On a circle of radius 1, the point is $(\cos\theta, \sin\theta)$ — cosine is the x-coordinate, sine the y. · 在半径为 1 的圆上,该点是 $(\cos\theta, \sin\theta)$——余弦是 x 坐标,正弦是 y 坐标。
Select all · 所有 true statements. · 选出所有正确的说法。
A full circle is $360°$ (or $2\pi$ radians), not $100°$. The other three are correct. · 一整圈是 $360°$(或 $2\pi$ 弧度),不是 $100°$。其余三条正确。
Radians and tangent
- Angles can be measured in degrees or in radians 弧度, where a full circle is $2\pi$.
- Radians measure the arc length swept on the unit circle, so $180° = \pi$.
- Tangent connects the other two: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.
- On the unit circle, tangent is the slope of the line from the origin to the point.
弧度与正切
- 角可以用度或弧度(radians)度量,其中一整圈是 $2\pi$。
- 弧度度量的是单位圆上扫过的弧长,所以 $180° = \pi$。
- 正切把另外两个连起来:$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$。
- 在单位圆上,正切是从原点到该点的直线的斜率。
Tangent equals sine divided by ____ (so $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$). · 正切等于正弦除以____(所以 $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$)。
The tangent · 相切 is sine divided by cosine: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. · 正切是正弦除以余弦:$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$。
Angles can be measured in radians, where a full circle is $2\pi$. · 角可以用弧度来度量,其中一整圈是 $2\pi$。
A full turn is $360°$ or, in radians, $2\pi$. Radians measure the arc length on the unit circle. · 一整圈是 $360°$,或用弧度表示是 $2\pi$。弧度度量的是单位圆上的弧长。
Around and around
- As the point travels once around the circle, sine and cosine cycle through all their values.
- Past one full turn, everything repeats — which is why these functions are periodic.
- Cosine starts at $1$ (angle $0$); sine starts at $0$.
- The unit circle is the single picture behind every trig value.
一圈又一圈
- 当点绕圆走一圈时,正弦和余弦循环过它们所有的值。
- 过了一整圈,一切都重复——这就是这些函数为什么是周期性的。
- 余弦从 $1$ 开始(角度 $0$);正弦从 $0$ 开始。
- 单位圆是每一个三角函数值背后那唯一的一幅图。
Keep track of your angle units. $\sin(90)$ is nearly $1$ if $90$ means degrees, but a very different value if $90$ means radians. Always know whether a problem is in degrees or radians before you compute.
要注意你的角度单位。如果 $90$ 表示度,$\sin(90)$ 几乎是 $1$;但如果 $90$ 表示弧度,值就完全不同。计算之前,永远要知道题目用的是度还是弧度。
Find $\sin\theta$ and $\cos\theta$ for the angle whose unit-circle point is $(0.6, 0.8)$.
- The x-coordinate is cosine: $\cos\theta = 0.6$.
- The y-coordinate is sine: $\sin\theta = 0.8$.
- Check: $0.6^2 + 0.8^2 = 0.36 + 0.64 = 1$, as the unit circle requires.
某角在单位圆上的点是 $(0.6, 0.8)$,求它的 $\sin\theta$ 和 $\cos\theta$。
- x 坐标是余弦:$\cos\theta = 0.6$。
- y 坐标是正弦:$\sin\theta = 0.8$。
- 检验:$0.6^2 + 0.8^2 = 0.36 + 0.64 = 1$,正如单位圆所要求的。
Sine, cosine, and tangent are ratios in a right triangle (SOH-CAH-TOA). On the unit circle, a point at angle $\theta$ is $(\cos\theta, \sin\theta)$, which extends the functions to any angle. Angles measure in degrees or radians ($2\pi$ per turn), and $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$.
正弦、余弦和正切是直角三角形里的比值(SOH-CAH-TOA)。在单位圆上,角度为 $\theta$ 的点是 $(\cos\theta, \sin\theta)$,这把这些函数推广到任何角度。角用度或弧度度量(每圈 $2\pi$),且 $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$。