IB Maths: Summary of Trigonometric Relationships | IB数学：三角函数关系式总结

📚 IB Maths: Summary of Trigonometric Relationships | IB数学：三角函数关系式总结

Trigonometric relationships are among the most frequently tested topics in IB Maths AA and AI. This summary organizes the identities you need into one clear reference, with definitions, formulas, and exam-style notes.

三角函数关系式是 IB 数学 AA 和 AI 中最高频的考点之一。本文把必备恒等式整理为一份清晰参考，包含定义、公式与考试提醒。

---

1. Basic Definitions from the Unit Circle | 1. 单位圆中的基本定义

Every trigonometric identity can be traced back to the unit circle. For an angle θ in standard position, a point P(x, y) on the unit circle satisfies x² + y² = 1. The sine and cosine of θ are simply the coordinates of P, while the tangent is the ratio y/x.

所有三角恒等式都可以追溯到单位圆。设角 θ 在标准位置，其终边与单位圆交于点 P(x, y)，则有 x² + y² = 1。角 θ 的正弦和余弦分别是点 P 的纵坐标和横坐标，正切则是 y 与 x 的比值。

sinθ = y, cosθ = x, tanθ = y/x (x ≠ 0)

In the right-triangle model, the same functions are defined using opposite, adjacent and hypotenuse. Both models are connected, and the unit circle model is more powerful because it works for angles larger than 90° and for negative angles.

在直角三角形模型中，三角函数用对边、邻边和斜边定义。两种模型是相通的，但单位圆模型更强大，因为它能处理大于 90° 的角和负角。

---

2. Reciprocal Identities | 2. 倒数关系

The reciprocal identities define secant, cosecant and cotangent. In IB, the notation cosec is more common, though csc also appears in some textbooks. These identities are valid whenever the denominator is non-zero.

倒数关系定义正割、余割和余切。在 IB 考试中，cosec 更常见，但部分教材也使用 csc。只要分母不为零，这些恒等式都成立。

secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = 1/tanθ

Because division by zero is not allowed, secθ is undefined when cosθ = 0, and cosecθ is undefined when sinθ = 0. These domain restrictions are often tested in questions about functions and equations.

因为不能除以零，所以当 cosθ = 0 时 secθ 无定义，当 sinθ = 0 时 cosecθ 无定义。这些定义域限制经常出现在函数与方程题中。

---

3. Ratio Identities | 3. 商数关系

The ratio identities express tangent and cotangent in terms of sine and cosine. They are the most basic tools for simplifying trigonometric expressions.

商数关系用正弦和余弦表示正切和余切，是化简三角函数表达式最基本的工具。

tanθ = sinθ/cosθ, cotθ = cosθ/sinθ

The first identity is valid when cosθ ≠ 0, and the second is valid when sinθ ≠ 0. In many exam questions, replacing tanθ with sinθ/cosθ reveals hidden simplifications.

第一个关系式要求 cosθ ≠ 0，第二个关系式要求 sinθ ≠ 0。在很多考试题中，把 tanθ 换成 sinθ/cosθ 能让隐含的化简路径立刻显现。

---

4. Pythagorean Identities | 4. 毕达哥拉斯恒等式

From the unit circle equation x² + y² = 1, we immediately get the most important trigonometric identity. Dividing this identity by cos²θ or sin²θ produces two additional forms.

由单位圆方程 x² + y² = 1 立即可得最重要的三角恒等式。将等式两边分别除以 cos²θ 或 sin²θ，可以得到另外两种常用形式。

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

These identities are used constantly in integration, differentiation and equation solving. Notice that the second and third forms are not independent; they are simply rearrangements of the first after division.

这些恒等式在积分、微分和方程求解中反复出现。注意后两个形式并不是独立的，它们只是第一个恒等式两边同除后得到的结果。

---

5. Negative-Angle and Periodicity Identities | 5. 负角与周期性质

Sine and tangent are odd functions, while cosine is even. This gives the negative-angle identities, which are often used when reflecting graphs over the axes.

正弦和正切是奇函数，余弦是偶函数，由此得到负角恒等式。这些性质常用于研究函数图像关于坐标轴的对称性。

sin(-θ) = -sinθ, cos(-θ) = cosθ, tan(-θ) = -tanθ

Trigonometric functions also repeat at fixed intervals. Sine, cosine, secant and cosecant have period 2π, while tangent and cotangent have period π.

三角函数以固定间隔重复。正弦、余弦、正割和余割的周期为 2π，正切和余切的周期为 π。

sin(θ + 2π) = sinθ, cos(θ + 2π) = cosθ, tan(θ + π) = tanθ

Complementary angle relationships are also useful: sin(π/2 – θ) = cosθ and cos(π/2 – θ) = sinθ. These are sometimes called co-function identities.

余角关系同样有用：sin(π/2 – θ) = cosθ，cos(π/2 – θ) = sinθ。它们有时被称为余函数恒等式。

---

6. Compound Angle Formulas | 6. 和角与差角公式

Compound angle formulas express sine, cosine and tangent of A ± B in terms of functions of A and B. These formulas are the foundation for many other identities, such as double and half-angle formulas.

和角与差角公式用 A 和 B 的三角函数表示 A ± B 的正弦、余弦和正切。它们也是倍角、半角等许多恒等式的基础。

sin(A + B) = sinA cosB + cosA sinB

sin(A – B) = sinA cosB – cosA sinB

cos(A + B) = cosA cosB – sinA sinB

cos(A – B) = cosA cosB + sinA sinB

tan(A + B) = (tanA + tanB)/(1 – tanA tanB)

tan(A – B) = (tanA – tanB)/(1 + tanA tanB)

Exam tip: the cosine formulas have the opposite sign pattern to the sine formulas. Many students lose marks by writing cos(A + B) as cosA cosB + sinA sinB, which is incorrect.

考试提醒：余弦公式中的符号与正弦公式相反。很多学生会把 cos(A + B) 误写成 cosA cosB + sinA sinB，这是常见错误。

---

7. Double Angle Formulas | 7. 倍角公式

Set B = A in the compound angle formulas to obtain the double-angle formulas. The double-angle formula for cosine has three equivalent forms, each useful in different contexts.

在和角公式中令 B = A，就得到倍角公式。余弦的倍角公式有三种等价写法，分别适用于不同场景。

sin2θ = 2sinθ cosθ

cos2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ

tan2θ = 2tanθ/(1 – tan²θ)

The three forms of cos2θ are especially important in IB calculus. For example, using 2cos²θ – 1 = cos2θ gives a clean way to integrate cos²θ, because it can be rewritten as (1 + cos2θ)/2.

cos2θ 的三种形式在 IB 微积分中尤其重要。例如，利用 2cos²θ – 1 = cos2θ 可以把 cos²θ 改写成 (1 + cos2θ)/2，从而方便地求积分。

---

8. Half-Angle Formulas | 8. 半角公式

Rearrange the double-angle formulas to obtain half-angle identities. These are needed when integrating expressions such as sin²x or cos²x, and when solving certain trigonometric equations.

重新整理倍角公式可以得到半角恒等式。它们常用于计算 sin²x、cos²x 这类表达式的积分，也用于求解某些三角方程。

sin²(θ/2) = (1 – cosθ)/2

cos²(θ/2) = (1 + cosθ)/2

tan(θ/2) = (1 – cosθ)/sinθ = sinθ/(1 + cosθ)

When taking square roots of the first two formulas, the sign depends on the quadrant in which θ/2 lies. The rational form for tan(θ/2) avoids that sign ambiguity and is often preferred.

对前两个公式开方时，正负号取决于 θ/2 所在的象限。tan(θ/2) 的有理式形式避免了符号判断问题，因此更常被使用。

---

9. Product-to-Sum Identities | 9. 积化和差公式

Product-to-sum identities convert a product of two sine or cosine functions into a sum or difference. They appear in IB exam questions about integration and in solving trigonometric equations.

积化和差公式把两个正弦或余弦函数的乘积转化为和或差。它们在 IB 积分题和三角方程题中会出现。

2sinA cosB = sin(A + B) + sin(A – B)

2cosA sinB = sin(A + B) – sin(A – B)

更多咨询请联系16621398022（同微信）