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Investigation 2 – Negative and Complementary Angles | 探究2：负角与互余角

2026-08-02 22:21 · 本文部分内容由AI辅助创作

📚 Investigation 2 – Negative and Complementary Angles | 探究2：负角与互余角

In this investigation, we will explore how trigonometric functions behave when we change the sign of an angle or when we replace an angle with its complement. Mastering these relationships gives you powerful tools for simplifying expressions, evaluating trigonometric values without a calculator, and solving equations efficiently. The ideas of negative angles (clockwise rotation) and complementary angles (summing to 90° or π/2 rad) are connected by elegant symmetries on the unit circle.

在本探究中，我们将探讨当改变角的符号或用它的余角替换时，三角函数是如何变化的。掌握这些关系能为你提供强大的工具，用以化简表达式、不用计算器就求值以及高效地解方程。负角（顺时针旋转）和互余角（和为90°或π/2弧度）的思想通过单位圆上优美的对称性联系在一起。

1. Review of Angle Measurement | 角度度量回顾

In IB Mathematics, angles are measured from the positive x‑axis. An angle is positive when measured anticlockwise and negative when measured clockwise. Both degree and radian measures are used. A full revolution is 360° or 2π radians, so a negative angle –θ simply represents an equal‑sized rotation in the opposite direction.

在IB数学中，角从正x轴开始度量。逆时针方向度量的角为正，顺时针方向度量的角为负。度数和弧度两种单位都会用到。一整圈是360°或2π弧度，因此一个负角 –θ 仅仅表示大小相同但方向相反的旋转。

Because the terminal side of a negative angle –θ coincides with a positive angle (360° – θ or 2π – θ), any trigonometric function can be evaluated by relating the negative angle to its positive counterpart. This link is the foundation of the negative‑angle identities.

由于负角 –θ 的终边与正角 (360° – θ 或 2π – θ) 重合，任何三角函数都可以通过将负角与其正角对应联系起来进行求值。这种联系就是负角恒等式的基础。

2. Negative Angles on the Unit Circle | 单位圆上的负角

sin θ). If we rotate the terminal arm clockwise by the same amount, we obtain the point Q(–θ). Because the x‑axis acts as a mirror, Q has the same x‑coordinate as P but the opposite y‑coordinate: Q = (cos θ, –sin θ).

把角 θ 放在单位圆上得到点 P(θ) = (cos θ, sin θ)。如果顺时针把终边旋转同样的量，就得到点 Q(–θ)。因为x轴起到镜面作用，Q 的x坐标与 P 相同，但y坐标相反：Q = (cos θ, –sin θ)。

This geometric symmetry immediately reveals the even‑odd nature of the sine and cosine functions and explains why the negative sign “pops out” of sine but not cosine.

这种几何对称立刻揭示了正弦函数和余弦函数的奇偶性，并说明了为什么负号会从正弦中“跳出来”，而不会从余弦中跳出来。

3. Trigonometric Functions of Negative Angles | 负角的三角函数

By reading the coordinates of Q(–θ) directly from the unit circle, we obtain the fundamental negative‑angle identities:

sin(–θ) = –sin θ

cos(–θ) = cos θ

tan(–θ) = –tan θ

从单位圆上直接读取 Q(–θ) 的坐标，我们得到基本的负角恒等式：

sin(–θ) = –sin θ

cos(–θ) = cos θ

tan(–θ) = –tan θ

ine and secant are even functions. The reciprocal identities follow the same pattern: csc(–θ) = –csc θ, sec(–θ) = sec θ, cot(–θ) = –cot θ.

正弦、正切、余割和余切是奇函数，而余弦和正割是偶函数。倒数恒等式遵循相同的模式：csc(–θ) = –csc θ，sec(–θ) = sec θ，cot(–θ) = –cot θ。

FunctionNegative‑Angle IdentityParity
sin θsin(–θ) = –sin θOdd
cos θcos(–θ) = cos θEven
tan θtan(–θ) = –tan θOdd
csc θcsc(–θ) = –csc θOdd
sec θsec(–θ) = sec θEven
cot θcot(–θ) = –cot θOdd

4. Proof of Negative Angle Identities | 负角恒等式的证明

angle. The point at angle θ is (x, y) = (cos θ, sin θ). Reflecting this point across the x‑ egative angle is a clockwise rotation, which mirrors the anticlockwise position) gives (x, –y). Since the reflected point lies on the unit circle at angle –θ, its coordinates must equal (cos(–θ), sin(–θ)). Equating coordinates yields cos(–θ) = cos θ and sin(–θ) = –sin θ.

一个严格的证明从单位圆出发。设 θ 为任意角。在角 θ 处的点是 (x, y) = (cos θ, sin θ)。将这个点关于x轴反射（因为负角是顺时针旋转，恰好是逆时针位置的镜像），得到 (x, –y)。由于反射后的点位于单位圆上角 –θ 处，它的坐标必须等于 (cos(–θ), sin(–θ))。比较坐标即得 cos(–θ) = cos θ 以及 sin(–θ) = –sin θ。

The tangent identity follows from the definition tan(–θ) = sin(–θ) / cos(–θ) = –sin θ / cos θ = –tan θ. The reciprocal functions are proved similarly, confirming that the negative‑angle identities hold for all values where the functions are defined.

正切恒等式可由定义直接推出：tan(–θ) = sin(–θ) / cos(–θ) = –sin θ / cos θ = –tan θ。倒数函数可类似证明，从而确认负角恒等式对所有有定义的值都成立。

5. Complementary Angles Defined | 互余角定义

Two angles are called complementary if their measures add up to 90° (π/2 radians). For any angle θ, its complement is (90° – θ) or (π/2 – θ). This simple definition lies at the heart of many trigonometric transformations and can be visualised in a right‑angled triangle where the two acute angles are complements of each other.

如果两个角的度数之和为90°（π/2弧度），它们就称为互余角。对任意角 θ，它的余角是 (90° – θ) 或 (π/2 – θ)。这个简单的定义是许多三角变换的核心，并且可以在直角三角形中直观地体现出来——直角三角形的两个锐角互为余角。

Because the sum is a right angle, swapping the roles of the two acute angles in a triangle interchanges the opposite and adjacent sides relative to each angle. This geometric fact is precisely why sine and cosine, as well as tangent and cotangent, exchange roles for complementary angles.

由于总和为直角，在三角形中交换两个锐角的角色就会相对每个角交换对边和邻边。这一几何事实正是正弦与余弦、正切与余切在互余角关系中相互转换的原因。

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6. Trigonometric Identities for Complementary Angles | 互余角的三角恒等式

The fundamental complementary‑angle identities are:

sin(π/2 – θ) = cos θ

cos(π/2 – θ) = sin θ

tan(π/2 – θ) = cot θ

基本的互余角恒等式为：

sin(π/2 – θ) = cos θ

cos(π/2 – θ) = sin θ

tan(π/2 – θ) = cot θ

Using these identities, we can transform any trigonometric function of an angle into a cofunction of its complement. For example, sin 20° is exactly cos 70°, and tan 15° equals cot 75°. This co‑function relationship is extremely useful when working with angles that are not standard but are linked by a 90° difference.

使用这些恒等式，我们可以把任意角的三角函数转化为它的余角的余函数。例如，sin 20° 恰好等于 cos 70°，tan 15° 等于 cot 75°。当遇到并非标准角但彼此相差90°的角时，这种余函数关系极其有用。

7. Visualising Complementary Angles on the Unit Circle | 单位圆上互余角的可视化

On the unit circle, the points corresponding to θ and (π/2 – o the line y = x. If P = (cos θ, sin θ), then the point for the complement is Q = (cos(π/2 – θ), sin(π/2 – θ)) = (sin θ, cos θ). The x‑ and y‑coordinates are simply swapped.

在单位圆上，对应 θ 和 (π/2 – θ) 的点关于直线 y = x 对称。如果 P = (cos θ, sin θ)，那么余角的点为 Q = (cos(π/2 – θ), sin(π/2 – θ)) = (sin θ, cos θ)。x坐标和y坐标恰好互换。

This swap visually explains why cos(π/2 – θ) becomes sin θ and sin(π/2 – θ) becomes cos θ. The tangent identity follows because tan(π/2 – θ) = sin(π/2 – θ) / cos(π/2 – θ) = cos θ / sin θ = cot θ. The symmetry line y = x neatly captures the entire set of co‑function identities.

这种坐标互换直观地解释了为何 cos(π/2 – θ) 变成 sin θ，sin(π/2 – θ) 变成 cos θ。正切恒等式随之可得，因为 tan(π/2 – θ) = sin(π/2 – θ) / cos(π/2 – θ) = cos θ / sin θ = cot θ。对称轴 y = x 精炼地概括了整个余函数恒等式集合。

8. Using Negative and Complementary Identities Together | 联合使用负角和互余恒等式

In many problems you will need to combine negative‑angle and complementary‑angle identities. A common technique is to first absorb any negative sign inside the trigonometric function using the odd‑even identities, then convert the resulting angle to a co‑function using complementary relationships. This allows you to express any trigonometric value in terms of an acute angle, often between 0° and 45°.

在许多问题中，你将需要联合使用负角恒等式和互余角恒等式。一种常见的技巧是先用奇偶恒等式把函数内部的负号吸收掉，然后再用互余关系将得到的角转化为余函数。这样做就能把任意三角函数值表示成一个锐角（通常在0°到45°之间）的三角函数值。

For instance, to simplify sin(–110°), write sin(–110°) = –sin 110°. Since 110° = 180° – 70°, sin 110° = sin 70°. Combining with a complement gives sin 70° = cos 20°. Hence sin(–110°) = – trong foundation for solving equations and proving more complex trigonometric expressions.

例如，要化简 sin(–110°)，可写成 sin(–110°) = –sin 110°。因为 110° = 180° – 70°，所以 sin 110° = sin 70°。再结合互余关系得 sin 70° = cos 20°。因此 sin(–110°) = –cos 20°。系统地使用这些恒等式，可以为解方程和证明更复杂的三角表达式奠定扎实的基础。

9. Worked Example 1 – Simplifying an Expression | 例题1 – 化简表达式

Simplify the expression (sin(–x) tan(π/2 – x)) / cos(–x).

化简表达式 (sin(–x) tan(π/2 – x)) / cos(–x)。

First, apply negative‑angle identities: sin(–x) = –sin x, cos(–x) = cos x. Next, use the complementary identity tan(π/2 – x) = cot x = cos x / sin x. Substitute these into the original expression:

首先应用负角恒等式：sin(–x) = –sin x，cos(–x) = cos x。接着使用互余恒等式 tan(π/2 – x) = cot x = cos x / sin x。将这些代入原表达式：

(–sin x × (cos x / sin x)) / cos x = –cos x / cos x = –1

The expression simplifies to –1 for all x where it is defined, demonstrating how neatly negative and complementary identities work together.

该表达式对所有有定义的 x 化简为 –1，展示了负角和互余恒等式如何精妙地协同运作。

10. Worked Example 2 – Solving a Trigonometric Equation | 例题2 – 解三角方程

Solve the equation cos(π/2 + θ) = sin(–θ) for 0 ≤ θ < 2π.

在 0 ≤ θ < 2π 范围内解方程 cos(π/2 + θ) = sin(–θ)。

First recognise that cos(π/2 + θ) = –sin θ (using the shift identity, which is related to complements). Meanwhile, sin(–θ) = –sin θ. Thus the equation reduces to –sin θ = – always check domain restrictions; here all real θ work, so the solution set is 0 ≤ θ < 2π.

首先认识到 cos(π/2 + θ) = –sin θ（利用平移恒等式，它与互余关系相关）。同时 sin(–θ) = –sin θ。因此方程简化为 –sin θ = –sin θ，这是一个恒等式，意味着定义域内的每个 θ 都满足该方程。然而，始终要检查定义域限制；这里所有实数 θ 都成立，因此解集为 0 ≤ θ < 2π。

Sometimes the combination of negative and complementary angles can disguise an identity. Learning to spot these simplifies the solving process dramatically.

有时负角和互余角的组合可能会隐藏一个恒等式。学会识别这一点能大大简化求解过程。

11. Investigation Extension – General Solutions and Periodicity | 探究拓展 – 通解与周期性

Consider the equation sin(2x) = cos x. Using the complementary identity, cos x = sin(π/2 – x). Therefore sin(2x) = sin(π/2 – x). By the general solution for sine, we have:

考虑方程 sin(2x) = cos x。使用互余恒等式，cos x = sin(π/2 – x)。因此 sin(2x) = sin(π/2 – x)。根据正弦的通解公式，我们有：

2x = π/2 – x + 2πk    or    2x = π – (π/2 – x) + 2πk,   k ∈ ℤ

en be included by allowing k to be negative, showing that all solutions can be expressed within the principal domain by adjusting the integer parameter.

解这些线性方程得到两组解：3x = π/2 + 2πk → x = π/6 + 2πk/3，以及 2x = π/2 + x + 2πk → x = π/2 + 2πk。然后可以通过让 k 取负数把负角也包括进来，这表明通过调整整数参数，所有解都能在主值区域内表示出来。

Extension tasks like this highlight how negative and complementary angles work inside general solution formulae, linking the investigation to periodic behaviour and the infinite nature of trigonometric equations.

像这样的拓展任务突显了负角和互余角如何在通解公式中发挥作用，将探究与周期性和三角方程的无穷性联系起来。

12. Summary and Key Takeaways | 总结与要点

In this investigation we have established that negative angles generate odd and even function properties: sine, tangent and their reciprocals change sign, while cosine and secant remain unchanged. Complementary angles swap sine with cosine and tangent with cotangent. These two sets of identities, summarised below, form an essential toolkit for IB Mathematics.

在本探究中，我们建立了负角带来的奇偶函数性质：正弦、正切及其倒数改变符号，而余弦和正割保持不变。互余角将正弦与余弦互换，正切与余切互换。这两组恒等式（总结如下）构成了IB数学必不可少的一个工具箱。