The 210° Angle: Trigonometric Values & Applications | 210°角：三角函数值与应用

📚 The 210° Angle: Trigonometric Values & Applications | 210°角：三角函数值与应用

In the IGCSE Edexcel Mathematics syllabus, understanding angles beyond 180° is essential for mastering trigonometry. The angle 210° is a classic example that brings together the unit circle, reference angles, and the signs of trigonometric functions in the third quadrant.

在 IGCSE Edexcel 数学大纲中，理解大于 180° 的角度是掌握三角学的关键。210° 角是一个经典例子，它将单位圆、参考角以及第三象限中三角函数的符号结合在了一起。

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1. What Is a 210° Angle? | 什么是210°角？

A 210° angle is measured anticlockwise from the positive x-axis. It lies between 180° and 270°, meaning its terminal arm is in the third quadrant of the Cartesian plane.

210° 角是从正 x 轴开始逆时针测量的角。它位于 180° 和 270° 之间，意味着它的终边落在平面直角坐标系的第三象限中。

If you think of a clock, 210° is equivalent to rotating from the 3 o’clock position to roughly the 7 o’clock position.

如果你想象一个钟表，210° 相当于从 3 点钟方向旋转到大约 7 点钟方向。

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2. Locating 210° on the Unit Circle | 在单位圆上定位210°

On the unit circle, the point corresponding to 210° has coordinates (cos 210°, sin 210°). Because 210° is in the third quadrant, both coordinates are negative.

在单位圆上，与 210° 对应的点的坐标是 (cos 210°, sin 210°)。因为 210° 在第三象限，所以它的两个坐标都是负数。

• The x-coordinate is cos 210°.
• The y-coordinate is sin 210°.

• x 坐标是 cos 210°。
• y 坐标是 sin 210°。

P = (cos 210°, sin 210°)

This point lies on the circle of radius 1, exactly halfway between the negative x-axis and the negative y-axis if you move anticlockwise.

该点位于半径为 1 的圆上，在逆时针方向上恰好位于负 x 轴与负 y 轴的中间位置。

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3. Reference Angle | 参考角

The reference angle is the acute angle between the terminal arm and the x-axis. For any angle in the third quadrant, subtract 180° from the angle.

参考角是终边与 x 轴之间的锐角。对于第三象限的任何角，用该角减去 180° 即可得到参考角。

Reference angle = 210° − 180° = 30°

Therefore, the reference angle for 210° is 30°.

因此，210° 的参考角是 30°。

This is important because the absolute values of sin, cos, and tan for 210° are the same as those for 30°.

这一点很重要，因为 210° 的 sin、cos 和 tan 的绝对值与 30° 的对应值完全相同。

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4. Exact Trigonometric Values | 精确三角函数值

Using the reference angle and the signs of the third quadrant, we can write the exact values for 210°.

利用参考角和第三象限的符号，我们可以写出 210° 的精确三角函数值。

sin 210° = −sin 30° = −½

cos 210° = −cos 30° = −√3/2

tan 210° = tan 30° = 1/√3 = √3/3

Notice that tangent is positive in the third quadrant because both sine and cosine are negative, and a negative divided by a negative is positive.

注意，正切在第三象限是正的，因为正弦和余弦都是负数，负数除以负数结果为正。

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5. Signs in Quadrant III | 第三象限的符号

A useful memory aid is the phrase “All Silly Tom Cats” or the CAST rule. In Quadrant III, only tan is positive.

一个有用的助记法是 “All Silly Tom Cats” 或 CAST 法则。在第三象限中，只有正切为正。

Quadrant	Positive functions	象限	正函数
I	All	一	全部
II	sin	二	sin
III	tan	三	tan
IV	cos	四	cos

For 210°, sine and cosine are negative, while tangent is positive.

对于 210°，正弦和余弦为负，而正切为正。

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6. Radians and Degrees | 弧度与角度

In IGCSE mathematics, you may also need to express 210° in radians. Multiply by π/180°.

在 IGCSE 数学中，你可能还需要将 210° 用弧度表示。乘以 π/180° 即可。

210° × π/180° = 7π/6

So 210° equals 7π/6 radians.

所以 210° 等于 7π/6 弧度。

This means that on the unit circle, the angle 7π/6 represents exactly the same position as 210°.

这意味着在单位圆上，7π/6 与 210° 表示完全相同的位置。

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7. Graphs of Trigonometric Functions | 三角函数图像

The value at 210° helps us understand the shapes of sine, cosine, and tangent graphs.

210° 处的值有助于我们理解正弦、余弦和正切图像的形状。

On the sine graph, 210° = 7π/6 is a point below the x-axis, corresponding to −½.

在正弦图像上，210° = 7π/6 是 x 轴下方的一点，对应 −½。

On the cosine graph, 210° gives −√3/2, which is slightly less than −1? Actually −√3/2 ≈ −0.866.

在余弦图像上，210° 给出 −√3/2，约等于 −0.866。

On the tangent graph, the value is √3/3 ≈ 0.577, and the graph is positive at this point.

在正切图像上，值为 √3/3 ≈ 0.577，并且该点处图像为正。

Knowing exact values allows you to sketch these graphs accurately without a calculator.

知道精确值可以让你在不用计算器的情况下准确勾画这些图像。

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8. Real-World Applications | 实际应用

Angles larger than 180° appear in navigation, physics, and engineering. For example, a ship changing course by 210° is turning more than half a full rotation, ending up heading in a south-west direction.

大于 180° 的角出现在航海、物理和工程中。例如，一艘船改变航向 210° 相当于转过超过半圈，最终朝向西南方向。

In physics, wave motion and alternating current often use phase angles beyond 180°. A 210° phase shift tells us how much a wave is delayed relative to another wave.

在物理学中，波动和交流电常使用超过 180° 的相位角。210° 的相位差告诉我们一个波相对于另一个波延迟了多少。

Understanding 210° also helps solve problems involving bearings, where angles are measured clockwise from north. A bearing of 210° points directly south-west.

理解 210° 还有助于解决涉及方位角的问题，方位角从北方向顺时针测量。210° 的方位角直接指向西南。

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9. Common Mistakes | 常见错误

• Using the value of sin 30° directly without applying the negative sign.
• Forgetting that tan 210° is positive.
• Confusing the reference angle with the original angle.
• Writing √3/2 instead of −√3/2 for cos 210°.

• 直接使用 sin 30° 的值而忘记加负号。
• 忘记 tan 210° 是正的。
• 将参考角与原角混淆。
• 将 cos 210° 写成 √3/2 而不是 −√3/2。

To avoid these errors, always draw a quick unit circle and identify the quadrant first.

为避免这些错误，务必先画一个简单的单位圆并确定象限。

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10. Practice Questions | 练习题

Try these IGCSE-style questions to test your understanding.

尝试这些 IGCSE 风格的题目来测试你的理解。

1. Write down the exact value of cos 210°.
2. Express 210° in radians.
3. Find tan 210° without using a calculator.
4. State the reference angle for 210°.

1. 写出 cos 210° 的精确值。
2. 将 210° 用弧度表示。
3. 不使用计算器求 tan 210°。
4. 写出 210° 的参考角。

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11. Solutions | 答案

Check your answers below.

在下方核对你的答案。

1. cos 210° = −√3/2
2. 210° = 7π/6 radians
3. tan 210° = √3/3
4. Reference angle = 30°

1. cos 210° = −√3/2
2. 210° = 7π/6 弧度
3. tan 210° = √3/3
4. 参考角 = 30°

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12. Key Takeaways | 关键要点

The angle 210° is an important benchmark in IGCSE trigonometry. It lives in the third quadrant, has a reference angle of 30°, and its sine and cosine are negative while its tangent is positive.

210° 角是 IGCSE 三角学中的一个重要基准角。它位于第三象限，参考角为 30°，正弦和余弦为负，正切为正。

The exact values — sin 210° = −½, cos 210° = −√3/2, tan 210° = √3/3 — must be memorised and understood using the unit circle.

这些精确值—— sin 210° = −½、cos 210° = −√3/2、tan 210° = √3/3——必须通过单位圆加以记忆和理解。

Once you master 210°, you can apply the same reasoning to any angle in any quadrant.

一旦你掌握了 210°，你就可以将同样的推理方法应用到任何象限中的任何角度。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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