📚 Trigonometric Relationships | 三角函数关系
Trigonometric relationships form the backbone of IB Mathematics, linking angles to ratios and enabling the modelling of periodic phenomena. From the fundamental Pythagorean identity to compound-angle formulas and the laws governing general triangles, these relationships are essential for solving equations, proving identities, and analysing real-world oscillations. This article presents a structured walkthrough of the key trigonometric identities, triangle rules, and equation-solving strategies you will encounter in the IB SL and HL courses.
三角函数关系是 IB 数学的基石，它将角度与边长之比联系起来，并可用于对周期现象进行建模。从基本的毕达哥拉斯恒等式到复角公式以及适用于一般三角形的定律，这些关系对于求解方程、证明恒等式以及分析现实世界中的振荡现象都至关重要。本文系统地介绍了 IB SL 和 HL 课程中将遇到的关键三角恒等式、三角形规则以及解三角方程的策略。
1. The Unit Circle and Definitions | 单位圆与定义
Sine and cosine are defined on the unit circle. For an angle θ measured from the positive x‑axis, the coordinates of the point on the circle are (cos θ, sin θ). Tangent is then the ratio sin θ / cos θ, which also corresponds to the slope of the terminal ray. This geometric view extends the definitions beyond acute angles and underlies the periodic nature of the functions.
正弦和余弦在单位圆上定义。对于从 x 轴正方向开始度量的角度 θ，圆周上对应点的坐标为 (cos θ, sin θ)。正切则定义为 sin θ / cos θ 之比，这也对应于终边的斜率。这种几何视角将定义扩展到锐角之外，并揭示了函数的周期性本质。
cos²θ + sin²θ = 1
tan θ = sin θ / cos θ, θ ≠ π/2 + kπ
The quadrant of the angle determines the sign of each ratio. CAST diagrams (starting with "All" in QI, then "Sine" in QII, "Tangent" in QIII, "Cosine" in QIV) help recall which ratios are positive.
角所在的象限决定了每个比值的符号。CAST 图（第一象限所有值均为正，第二象限正弦为正，第三象限正切为正，第四象限余弦为正）有助于记忆哪些比值为正。
2. Fundamental Identities | 基本恒等式
reciprocal identities complete the set: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
毕达哥拉斯恒等式及其变形构成了核心工具包。由 cos²θ + sin²θ = 1 分别除以 cos²θ 或 sin²θ，可得 1 + tan²θ = sec²θ 和 1 + cot²θ = csc²θ。倒数恒等式则完整了这组关系：csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。
| sin²θ + cos²θ = 1 | 1 + tan²θ = sec²θ |
| 1 + cot²θ = csc²θ | tan θ = sin θ / cos θ |
These identities are indispensable for simplifying expressions and proving more complex relationships.
这些恒等式对于化简表达式和证明更复杂的关系不可或缺。
3. Complementary and Supplementary Angle Identities | 余角与补角恒等式
The co‑ nd tan(π − θ) = −tan θ. Odd/even properties are also crucial: sine is odd, cosine is even.
余函数恒等式将互余角的正弦和余弦联系起来：sin(π/2 − θ) = cos θ，cos(π/2 − θ) = sin θ。类似地，tan(π/2 − θ) = cot θ。补角关系反映了单位圆的对称性：sin(π − θ) = sin θ，cos(π − θ) = −cos θ，tan(π − θ) = −tan θ。奇偶特性同样关键：正弦是奇函数，余弦是偶函数。
ear in different quadrants. For example, sin(180° + θ) = −sin θ and cos(360° − θ) = cos θ.
这些对称性质能够快速化简出现在不同象限的角度。例如，sin(180° + θ) = −sin θ，cos(360° − θ) = cos θ。
4. Addition and Subtraction Formulas | 和差公式
The compound‑ anding sin(A ± B), cos(A ± B), and tan(A ± B). They also underpin the derivation of double‑angle and half‑angle identities.
复角公式对于展开 sin(A ± B)、cos(A ± B) 和 tan(A ± B) 至关重要，它们也是推导倍角公式和半角公式的基础。
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
Note the sign flip in the denominator for tan(A + B) compared to tan(A − B). Memorising these formulas is less important than recognising their structure and knowing how to apply them to exact‑value questions and identity proofs.
请注意 tan(A + B) 与 tan(A − B) 分母中的符号变化。熟记这些公式不如识别其结构并知道如何将它们应用于精确值问题和恒等式证明来得重要。
5. Double-Angle Formulas | 倍角公式
Setting B = A in the addition formulas yields the double‑angle identities. The sine double‑ ormula is sin 2A = 2 sin A cos A. For cosine there are three useful forms, all equivalent:
在加法公式中令 B = A 即可得到倍角恒等式。正弦的倍角公式为 sin 2A = 2 sin A cos A。余弦则有三种有用的等价形式：
cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A
The tangent double‑angle formula is tan 2A = 2 tan A / (1 − tan²A), valid when tan A ≠ ±1 and A ≠ π/4 + kπ/2.
正切的倍角公式为 tan 2A = 2 tan A / (1 − tan²A)，在 tan A ≠ ±1 且 A ≠ π/4 + kπ/2 时成立。
These are particularly powerful for solving trigonometric equations where the same angle appears both in single and double form. The cosine forms also allow us to express sin²A and cos²A in terms of cos 2A, a technique used heavily in integration.
这些公式在求解同时包含单角和倍角的三角方程时特别强大。余弦的几种形式还允许我们用 cos 2A 来表示 sin²A 和 cos²A，这一技巧在积分中被大量使用。
6. Half-Angle Formulas | 半角公式
By rearranging the double‑angle forms of cosine, we obtain the half‑angle identities:
通过重新整理余弦的倍角形式，我们得到半角恒等式：
sin²(θ/2) = (1 − cos θ) / 2
cos²(θ/2) = (1 + cos θ) / 2
tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ
The sign of the square root when solving for sin(θ/2) or cos(θ/2) depends on the quadrant of θ/2. These formulas are less frequently memorised at SL but are useful for exact values and in HL calculus when dealing with rational parametrisations.
在求解 sin(θ/2) 或 cos(θ/2) 时平方根的符号取决于 θ/2 所在的象限。在 SL 中这些公式较少要求记忆，但对于精确值计算以及 HL 微积分中有理化参数表示时非常有用。
7. Product-to-Sum and Sum-to-Product Formulas | 积化和差与和差化积公式
These identities convert products of sine and cosine into sums, and vice‑versa. They stem directly from the addition formulas. The product‑to‑sum set includes:
这些恒等式将正弦和余弦的乘积转换为和差，反之亦然。它们直接源自加法公式。积化和差组包括：
sin A cos B = ½[sin(A + B) + sin(A − B)]
cos A sin B = ½[sin(A + B) − sin(A − B)]
cos A cos B = ½[cos(A − B) + cos(A + B)]
sin A sin B = ½[cos(A − B) − cos(A + B)]
The sum‑to‑product versions are equally handy when solving equations or proving identities:
和差化积的版本在求解方程或证明恒等式时同样方便：
sin P + sin Q = 2 sin((P + Q)/2) cos((P − Q)/2)
sin P − sin Q = 2 cos((P + Q)/2) sin((P − Q)/2)
cos P + cos Q = 2 cos((P + Q)/2) cos((P − Q)/2)
cos P − cos Q = −2 sin((P + Q)/2) sin((P − Q)/2)
While these formulas are provided in the IB formula booklet, being able to recognise when they are applicable saves time and prevents algebraic errors.
尽管这些公式在 IB 公式手册中提供，但能够识别其适用场合可以节省时间并避免代数错误。
8. Solving Trigonometric Equations | 解三角方程
Trigonometric equations often have infinitely many solutions due to periodicity. The general approach is to isolate a single trigonometric function, use identities to obtain a factorised form, and then find all solutions within a specified interval. Key steps:
- Rearrange to one side, aiming for a product equal to zero.
- Use algebraic substitution (e.g., let u = sin x) if helpful.
- he basic equation, e.g., sin x = k, keeping in mind the CAST rule to locate all quadrants.
- Add the appropriate period (sin/cos: 2πk; tan: πk) for the general solution.
三角方程由于周期性通常有无穷多个解。一般方法是分离出单个三角函数，利用恒等式得到因式分解形式，然后求出指定区间内的所有解。关键步骤包括：将其移项到一边，目标为乘积等于零；如果合适，使用代数换元（例如令 u = sin x）；求解基本方程，如 sin x = k，牢记 CAST 规则以定位所有象限的解；再根据周期加上通解项（sin/cos 为 2πk；tan 为 πk）。
n x − 1) = 0, giving sin x = −½ or sin x = 1. Then find all x in [0, 2π) satisfying these.
例如，解方程 2 sin²x − sin x − 1 = 0：将其视为关于 sin x 的二次方程，因式分解为 (2 sin x + 1)(sin x − 1) = 0，得到 sin x = −½ 或 sin x = 1，然后在 [0, 2π) 中找出所有满足条件的 x。
9. Sine Rule and Cosine Rule | 正弦定理与余弦定理
For non‑right‑angled triangles, the sine rule and cosine rule relate sides and angles. The sine rule states:
对于非直角三角形，正弦定理和余弦定理将边与角联系起来。正弦定理表述为：
a / sin A = b / sin B = c / sin C = 2R
where R is the circumradius. It is used when two angles and a side are known (AAS or ASA), or when two sides and a non‑included angle are given (SSA), though the latter requires care due to the ambiguous case.
其中 R 为外接圆半径。当已知两角与一边（AAS 或 ASA），或已知两边及一个非夹角（SSA）时使用该定理，但后者需注意由于可能产生歧义的情况。
The cosine rule generalises the Pythagorean theorem:
余弦定理是毕达哥拉斯定理的推广：
a² = b² + c² − 2bc cos A
cos A = (b² + c² − a²) / (2bc)
It is applied when three sides are known (SSS) or when two sides and the included angle are known (SAS).
当已知三边（SSS）或已知两边及其夹角（SAS）时使用余弦定理。
10. Area of a Triangle | 三角形面积公式
The area of a triangle can be expressed in trigonometric form:
三角形的面积可以用三角形式表示：
Area = ½ ab sin C = ½ bc sin A = ½ ac sin B
e formula Area = ½ a² (sin B sin C)/sin A and other variations. For IB questions, the half‑product form is the most common starting point.
当已知两边及其夹角时，该公式非常有用。结合正弦定理，它还可以推导出 Area = ½ a² (sin B sin C)/sin A 等其他变体。在 IB 试题中，半乘积形式是最常见的起点。
her option when only the three sides are given, though the cosine rule plus the sine area formula often suffices.
当仅已知三边时，赫伦公式 Area = √[s(s−a)(s−b)(s−c)]（其中 s = (a+b+c)/2）是另一种选择，不过余弦定理加正弦面积公式通常足以解决问题。
11. Graphs and Transformations | 图像与变换
a sin(b(x − c)) + d and y = a cos(b(x − c)) + d model periodic behaviour. The amplitude is |a|, the period is 2π/|b| (or 360°/|b| cognising the effect of each parameter is essential for sketching and interpreting graphs.
正弦型函数 y = a sin(b(x − c)) + d 和 y = a cos(b(x − c)) + d 可用于模拟周期性行为。振幅为 |a|，周期为 2π/|b|（或使用角度制时为 360°/|b|），相移为 c，垂直平移为 d。识别每个参数的影响对于绘制和理解函数图像至关重要。
The tangent graph y = a tan(b(x − c)) + d has period π/|b| etric graphs appear frequently in modelling temperature, tides, and ferris wheel problems.
正切图像 y = a tan(b(x − c)) + d 的周期为 π/|b|，且在函数未定义处存在垂直渐近线。三角图像的变换经常出现在温度、潮汐和摩天轮等建模问题中。
12. Modelling with Trigonometric Functions | 三角函数的建模应用
IB exam questions often embed trigonometric relationships in real‑world contexts. A typical task is to fit a sine or cosine model to data such as daylight hours or the height of a point on a rotating wheel. The steps involve determining the amplitude, vertical shift, period, and phase shift from given information, then using the model to predict values or solve for specific times.
IB 考试题常将三角函数关系嵌入实际情境中。典型任务是使用正弦或余弦模型拟合数据，如日照时长或轮子上某点的高度。步骤包括根据给定信息确定振幅、垂直平移、周期和相移，然后使用模型进行预测或求解特定时间点。
Solving modelling equations often requires using inverse trigonometric functions and understanding the principal range. Always check whether the solution should be reported in radians or degrees and whether rounding instructions apply.
求解建模方程通常需要使用反三角函数并理解其主值范围。务必检查答案应以弧度还是角度制给出，以及是否有四舍五入的要求。
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