📚 Negative and Complementary Angle Formulae | 负角与余角公式
In trigonometry, understanding how angles relate to each other is essential for simplifying expressions, solving equations, and modeling periodic behaviour. The negative angle formulae and complementary angle formulae form a fundamental set of identities that allow us to rewrite trigonometric functions of −θ and (90° − θ) in terms of standard acute angles. Mastering these relationships will sharpen your analytical skills and prepare you for more advanced topics such as compound angles, inverse trigonometric functions, and calculus with trigonometric functions.
在三角学中，理解角度之间的关系对于化简表达式、解方程以及建立周期模型至关重要。负角公式和余角公式构成了一组基本恒等式，使我们能够用标准锐角来表示−θ和(90°−θ)的三角函数。掌握这些关系将增强你的分析能力，并为学习复合角、反三角函数以及含三角函数的微积分等高级内容做好准备。
1. The Unit Circle as a Visual Foundation | 单位圆作为直观基础
On a circle of radius 1 centred at the origin, any angle θ is measured from the positive x‑ − θ corresponds to reflecting the terminal point across the line y = x, which swaps the x‑ and y‑coordinates.
所有这些公式都源于单位圆的对称性。在以原点为心、半径为1的圆上，任意角θ都是从x轴正半轴开始度量的。终边点的坐标为(cosθ, sinθ)。负角按顺时针方向度量，为sin(−θ)和cos(−θ)提供了自然的解释。类似地，余角90°−θ相当于将终边点关于直线y = x反射，从而交换了x坐标和y坐标。
2. Negative Angle Formulae for Sine, Cosine and Tangent | 正弦、余弦和正切的负角公式
For any angle θ (in degrees or radians), the negative angle identities are:
对任意角θ（以度或弧度为单位），负角恒等式为：
sin(−θ) = −sinθ
This parity is odd: reflecting the y‑coordinate across the x‑axis changes its sign while the magnitude stays the same.
这是一个奇函数性质：将y坐标关于x轴反射会改变其符号，而大小保持不变。
cos(−θ) = cosθ
Cosine is even: the x‑coordinate remains unchanged when the angle is reversed.
余弦是偶函数：当角度反向时，x坐标保持不变。
tan(−θ) = −tanθ
tan = sin/cos, the negative sign from sine carries through, leaving the cosine unchanged.
由于tan=sin/cos，正弦的负号会传递过来，而余弦不变。
A quick way to remember: only cosine and its reciprocal secant are even; the other four trigonometric functions are odd.
快速记忆法：只有余弦及其倒数正割是偶函数；其他四个三角函数都是奇函数。
3. Complementary Angle (Co‑function) Identities | 余角（互余）恒等式
Two angles are complementary if their sum is 90° (π/2 radians). The co‑function identities express a trigonometric function of (90° − θ) as the co‑function of θ.
如果两个角的和为90°（π/2弧度），它们就互余。互余恒等式将(90°−θ)的三角函数表示为θ的余函数。
sin(90° − θ) = cosθ
cos(90° − θ) = sinθ
tan(90° − θ) = cotθ = 1/tanθ
These identities are visually obvious when you consider a right triangle: the side opposite to θ is adjacent to (90°−θ) and vice versa. On the unit circle, the point for (90°−θ) is (sinθ, cosθ).
当考虑直角三角形时，这些恒等式在视觉上很明显：θ的对边是(90°−θ)的邻边，反之亦然。在单位圆上，(90°−θ)对应的点是(sinθ, cosθ)。
The term “co‑function” itself comes from this complementary relationship – cosine is the sine of the complementary angle, cotangent is the tangent of the complementary angle, and cosecant is the secant of the complementary angle.
“余函数”这一名称本身就来源于这种余角关系——余弦是余角的正弦，余切是余角的正切，余割是余角的正割。
4. Extending to Reciprocal Functions | 扩展到倒数函数
Applying the same logic to secant, cosecant and cotangent gives:
将同样的逻辑应用于正割、余割和余切，可得：
Negative angles:
负角：
csc(−θ) = −cscθ
sec(−θ) = secθ
cot(−θ) = −cotθ
Complementary angles (radians shown for completeness):
余角（同时给出弧度形式以作参考）：
sec(π/2 − θ) = cscθ
csc(π/2 − θ) = secθ
cot(π/2 − θ) = tanθ
c(−θ)=secθ. Always check the parity of the underlying sine or cosine to avoid sign errors.
这些是定义的直接结果，例如sec=1/cos，而cos(−θ)=cosθ，所以sec(−θ)=secθ。始终检查底层正弦或余弦的奇偶性，以避免出现符号错误。
5. Derivation Using Right‑angled Triangles | 利用直角三角形推导
ine rotating the terminal side clockwise; the reference triangle is congruent to that for θ, but the signed y‑coordinate flips sign, giving sin(−θ)=−sinθ while the x‑coordinate stays positive.
给一个直角三角形标上锐角θ和90°−θ。若斜边为1，则两直角边分别为sinθ和cosθ。角90°−θ的对边是cosθ，邻边是sinθ，立刻得出sin(90°−θ)=cosθ及cos(90°−θ)=sinθ。对于负角，想象将终边顺时针旋转，得到的参考三角形与θ的全等，但有符号的y坐标反号，因此sin(−θ)=−sinθ，而x坐标保持不变。
This geometric approach reinforces that these identities are not arbitrary rules but emerge naturally from the geometry of the circle and triangle.
这种几何方法强调了这些恒等式并非任意规则，而是从圆和三角形的几何中自然产生的。
6. Converting Between Functions with Example Angles | 利用示例角度进行函数转换
60°=90°−30°, so cos(60°)=sin30°=1/2. This matches the known value and demonstrates the co‑function identity in action.
利用这些恒等式，无需计算器就可以求出三角函数值。例如，求sin(−30°)：因为sin(−θ)=−sinθ，sin(−30°)=−sin30°=−1/2。要求cos(60°)，注意到60°=90°−30°，因此cos(60°)=sin30°=1/2。这与已知值吻合，并演示了互余恒等式的运用。
/2, cos(π/3) = sin(π/6) = 1/2. Practice converting between degrees and radians seamlessly.
弧度制下的例子：sin(−π/6)=−1/2，cos(π/3)=sin(π/6)=1/2。要练习在度和弧度之间灵活转换。
7. Simplifying Expressions Using the Identities | 利用恒等式化简表达式
The real power of these formulae appears when simplifying complex trigonometric expressions.
当化简复杂的三角表达式时，这些公式的真正威力就显现出来了。
Example: Simplify sin(90° − x)·cos(−x) + cos(90° − x)·sin(−x).
示例： 化简 sin(90° − x)·cos(−x) + cos(90° − x)·sin(−x)。
Step‑by‑step:
逐步进行：
sin(90° − x) = cos x
cos(−x) = cos x
cos(90° − x) = sin x
sin(−x) = −sin x
Substituting: (cos x)(cos x) + (sin x)(−sin x) = cos²x − sin²x = cos(2x) using the double‑angle identity.
代入得：(cos x)(cos x)+(sin x)(−sin x) = cos²x − sin²x = cos(2x)，利用了二倍角公式。
This illustrates how negative and complementary identities often serve as the first step before applying more advanced formulae.
这说明了负角和余角恒等式常常是应用更高级公式之前的第一步。
8. Solving Equations That Involve −θ or 90°−θ | 解含有−θ或90°−θ的方程
solving equations such as sin(90° − θ) = cos(−θ), replace both sides using the identities.
当解诸如 sin(90° − θ) = cos(−θ) 的方程时，用恒等式替换两边。
θ, so the equation reduces to cosθ = cosθ, which is true for all θ. Thus the solution is θ ∈ ℝ ll real numbers). Although trivial, this check confirms the consistency of the identities.
sin(90° − θ)=cosθ，且cos(−θ)=cosθ，所以方程简化为cosθ=cosθ，这对所有θ成立。因此解为θ∈ℝ（全体实数）。虽然平凡，但这一检验确认了恒等式的一致性。
A more practical equation: 2 sin(−x) = √3 . Replace sin(−x) with −sin x and cos(90° − x) with sin x. The equation becomes −2 sin x = √3 sin x, so 0 = (√3 + 2) sin x, hence sin x = 0, giving x = nπ, n ∈ ℤ.
一个更实际的方程：2 sin(−x) = √3 cos(90° − x)。将sin(−x)替换为−sin x，cos(90° − x)替换为sin x。方程变为−2 sin x = √3 sin x，因此0=(√3+2) sin x，故sin x=0，解得x=nπ, n∈ℤ。
9. Graphical Interpretation and Symmetry | 图形解释与对称性
gin. The graph of y = cos(−x) is identical to y = cos x, showing even symmetry about the y‑ phs of y = sin(90° − x) and y = cos x are the same, as are y = cos(90° − x) and y = sin x.
画出y=sin(−x)与y=−sin x的图像：二者完全重合，反映了关于原点的奇对称性。y=cos(−x)的图像与y=cos x相同，显示出关于y轴的偶对称性。对于余角，y=sin(90° − x)与y=cos x的图像一样，y=cos(90° − x)与y=sin x的图像一样。
Understanding these graphical connections helps you predict the behaviour of functions without algebraic manipulation, a skill tested in both calculator and non‑calculator IB exam papers.
理解这些图形联系有助于你无需代数推导即可预测函数行为，这一技能在IB数学的允许和不允许使用计算器的试卷中都会考查。
10. Common Errors and How to Avoid Them | 常见错误及其避免方法
Error 1: . Many students incorrectly write sin(−θ)=sinθ because they mistakenly treat sine as even. Fix: Recall that sine is odd; use the unit circle to visualise the flip in y‑coordinate.
错误1：忘记sin(−θ)的符号。许多学生错误地写成sin(−θ)=sinθ，因为他们错误地将正弦当作偶函数。纠正：牢记正弦是奇函数；用单位圆来想象y坐标的反转。
Error 2: Applying co‑ (90° − θ)=cosθ hold for all θ, but you must be careful with signs when evaluating further. Fix: Treat the identity as exact; then use ASTC or the unit circle to determine the sign of the result if needed.
错误2：对不在第一象限的角使用互余恒等式。恒等式sin(90°−θ)=cosθ对所有θ成立，但在进一步求值时必须留意符号。纠正：将恒等式视为精确关系；若需确定结果的符号，再使用ASTC法则或单位圆。
Error 3: The supplementary identities, such as sin(180°−θ)=sinθ, are different. Keep them separate.
错误3：将互余（和为90°）与互补（和为180°）混淆。补角恒等式，如sin(180°−θ)=sinθ，是不同的。要把它们区分开来。
11. Summary Table of Key Formulae | 关键公式汇总表
Keep this reference table handy when practising.
练习时将此参考表放在手边。
| Function | Negative Angle | Complementary Angle (90° − θ) |
|---|---|---|
| sin | −sinθ | cosθ |
| cos | cosθ | sinθ |
| tan | −tanθ | cotθ |
| csc | −cscθ | secθ |
| sec | secθ | cscθ |
| cot | −cotθ | tanθ |
For radian measure, simply replace 90° with π/2. The identities remain structurally identical.
对于弧度制，只需将90°替换为π/2。恒等式在结构上保持不变。
12. Practice Problems and Quick Solutions | 练习题与快速解答
1. Evaluate sin(−45°) and cos(−45°).
1. 求 sin(−45°) 和 cos(−45°)。
sin(−45°) = −sin45° = −√2/2; cos(−45°) = cos45° = √2/2.
sin(−45°) = −sin45° = −√2/2；cos(−45°) = cos45° = √2/2。
2. Simplify tan(90° − θ)·tan(−θ).
2. 化简 tan(90° − θ)·tan(−θ)。
tan(90° − θ) = cotθ = 1/tanθ; tan(−θ) = −tanθ. Product = −1.
tan(90° − θ) = cotθ = 1/tanθ；tan(−θ) = −tanθ。乘积为−1。
3. If cos(−x) = 0.6, find sin(90° − x) and sin x.
3. 若 cos(−x) = 0.6，求 sin(90° − x) 和 sin x。
cos(−x) = cos x = 0.6. Then sin(90° − x) = cos x = 0.6. And sin x = ±√1 − 0.6² = ±0.8; the sign depends on the quadrant of x.
cos(−x) = cos x = 0.6。那么 sin(90° − x) = cos x = 0.6。而 sin x = ±√1−0.6²=±0.8；符号取决于x所在的象限。
4. Solve 3 sin(−θ) = 2 cos(90° − θ) for 0 ≤ θ < 360°.
4. 在 0 ≤ θ < 360° 内解方程 3 sin(−θ) = 2 cos(90° − θ)。
3(−sinθ) = 2 sinθ → −3 sinθ = 2 sinθ → 5 sinθ = 0 → sinθ = 0 → θ = 0°, 180°.
3(−sinθ) = 2 sinθ → −3 sinθ = 2 sinθ → 5 sinθ = 0 → sinθ = 0 → θ = 0°, 180°。
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