Double Angle Formulas and Their Applications | IB数学：二倍角公式及其应用

📚 Double Angle Formulas and Their Applications | IB数学：二倍角公式及其应用

The double angle formulas are among the most frequently used identities in IB Mathematics, appearing in trigonometry, calculus, and geometry. They express trigonometric functions of \(2\theta\) in terms of functions of \(\theta\), allowing us to simplify expressions, solve equations, and evaluate integrals.

二倍角公式是IB数学中最常用的恒等式之一，出现在三角学、微积分和几何中。它们将 \(2\theta\) 的三角函数用 \(\theta\) 的三角函数表示，帮助我们化简表达式、解方程和计算积分。

1. The Basic Double Angle Formulas | 二倍角公式的基本形式

The three fundamental double angle formulas are:

三个基本的二倍角公式是：

sin 2θ = 2 sin θ cos θ

cos 2θ = cos² θ − sin² θ

tan 2θ = 2 tan θ / (1 − tan² θ)

These identities hold for all values of θ for which both sides are defined. For tan 2θ, we require θ ≠ π/2 + kπ and tan θ ≠ ±1.

这些恒等式对所有使两边都有定义的 θ 成立。对于 tan 2θ，需要 θ ≠ π/2 + kπ 且 tan θ ≠ ±1。

2. The Sine Double Angle Formula | 正弦二倍角公式

The formula \(\sin 2\theta = 2\sin\theta\cos\theta\) is derived directly from the sine addition formula \(\sin(A+B) = \sin A\cos B + \cos A\sin B\) by setting \(A = B = \theta\).

公式 \(\sin 2\theta = 2\sin\theta\cos\theta\) 直接由正弦和角公式 \(\sin(A+B) = \sin A\cos B + \cos A\sin B\) 令 \(A = B = \theta\) 得到。

This identity is especially useful in integration when we need to convert a product of sine and cosine into a single sine function.

这个恒等式在积分中特别有用，因为我们可以将正弦和余弦的乘积转化为单一的正弦函数。

For example, \(\int \sin 2x \, dx = -\frac{1}{2}\cos 2x + C\).

例如，\(\int \sin 2x \, dx = -\frac{1}{2}\cos 2x + C\)。

3. The Three Forms of Cos 2θ | cos 2θ 的三种形式

The cosine double angle formula has three equivalent forms, each useful in different contexts:

余弦二倍角公式有三种等价形式，每种形式在不同场合下各有用途：

Form 1: \(\cos 2\theta = \cos^2\theta – \sin^2\theta\) — the original form, useful for simplifying expressions with both squares.
形式1： \(\cos 2\theta = \cos^2\theta – \sin^2\theta\) —— 原始形式，适用于含两个平方的表达式。
Form 2: \(\cos 2\theta = 2\cos^2\theta – 1\) — this gives the power-reduction identity \(\cos^2\theta = \frac{1+\cos 2\theta}{2}\).
形式2： \(\cos 2\theta = 2\cos^2\theta – 1\) —— 由此得到降幂公式 \(\cos^2\theta = \frac{1+\cos 2\theta}{2}\)。
Form 3: \(\cos 2\theta = 1 – 2\sin^2\theta\) — this gives \(\sin^2\theta = \frac{1-\cos 2\theta}{2}\).
形式3： \(\cos 2\theta = 1 – 2\sin^2\theta\) —— 由此得到 \(\sin^2\theta = \frac{1-\cos 2\theta}{2}\)。

These forms are essential for integration of even powers of sine and cosine.

这些形式对计算正弦和余弦偶数次幂的积分至关重要。

4. The Tangent Double Angle Formula | 正切二倍角公式

The tangent double angle formula is derived from the tangent addition formula:

正切二倍角公式由正切和角公式推导得到：

tan 2θ = 2 tan θ / (1 − tan² θ)

This formula is valid when \(\theta \neq \frac{\pi}{4} + \frac{k\pi}{2}\) and \(\theta \neq \frac{\pi}{2} + k\pi\). It is commonly used in solving trigonometric equations and in coordinate geometry to find slopes of lines.

该公式在 \(\theta \neq \frac{\pi}{4} + \frac{k\pi}{2}\) 且 \(\theta \neq \frac{\pi}{2} + k\pi\) 时成立。它常用于解三角方程以及解析几何中求直线的斜率。

For example, if \(\tan \theta = \frac{1}{2}\), then \(\tan 2\theta = \frac{2 \times \frac{1}{2}}{1 – (\frac{1}{2})^2} = \frac{1}{3/4} = \frac{4}{3}\).

例如，若 \(\tan \theta = \frac{1}{2}\)，则 \(\tan 2\theta = \frac{2 \times \frac{1}{2}}{1 – (\frac{1}{2})^2} = \frac{1}{3/4} = \frac{4}{3}\)。

5. Derivation from Angle Addition Formulas | 由和角公式推导

Understanding the derivation helps memorisation and reveals the structure of these identities.

理解推导过程有助于记忆，并能揭示这些恒等式的结构。

Start with the sine addition formula:

从正弦和角公式出发：

sin(A+B) = sin A cos B + cos A sin B

Set \(A = B = \theta\) to obtain \(\sin 2\theta = 2\sin\theta\cos\theta\).

令 \(A = B = \theta\)，得到 \(\sin 2\theta = 2\sin\theta\cos\theta\)。

Similarly, \(\cos(A+B) = \cos A\cos B – \sin A\sin B\) gives \(\cos 2\theta = \cos^2\theta – \sin^2\theta\).

同理，\(\cos(A+B) = \cos A\cos B – \sin A\sin B\) 给出 \(\cos 2\theta = \cos^2\theta – \sin^2\theta\)。

Finally, using \(\tan(A+B) = \frac{\tan A + \tan B}{1-\tan A\tan B}\) with \(A=B=\theta\) yields the tangent double angle formula.

最后，令 \(\tan(A+B) = \frac{\tan A + \tan B}{1-\tan A\tan B}\) 中的 \(A=B=\theta\)，即得正切二倍角公式。

6. Simplifying Trigonometric Expressions | 化简三角表达式

Double angle formulas are powerful tools for rewriting complicated trigonometric expressions into simpler forms.

二倍角公式是将复杂三角表达式改写为简单形式的有力工具。

Example: Simplify \(\frac{\sin 2x}{1+\cos 2x}\).

例：化简 \(\frac{\sin 2x}{1+\cos 2x}\)。

Using \(\sin 2x = 2\sin x\cos x\) and \(1+\cos 2x = 1 + (2\cos^2 x – 1) = 2\cos^2 x\), we get:

使用 \(\sin 2x = 2\sin x\cos x\) 和 \(1+\cos 2x = 1 + (2\cos^2 x – 1) = 2\cos^2 x\)，得到：

\(\frac{2\sin x\cos x}{2\cos^2 x} = \frac{\sin x}{\cos x} = \tan x\)

Thus the whole expression reduces to \(\tan x\), which is much easier to analyse.

因此整个表达式化简为 \(\tan x\)，更便于分析。

7. Solving Trigonometric Equations | 解三角方程

Double angle formulas often transform equations involving \(2x\) into equations involving \(x\), making them solvable by factoring or using basic trigonometric values.

二倍角公式常把含 \(2x\) 的方程转化为含 \(x\) 的方程，从而可以通过因式分解或基本三角函数值求解。

Example: Solve \(\cos 2x = \cos x\) for \(0 \le x < 2\pi\).

例：解方程 \(\cos 2x = \cos x\)，其中 \(0 \le x < 2\pi\)。

Using \(\cos 2x = 2\cos^2 x – 1\), the equation becomes:

利用 \(\cos 2x = 2\cos^2 x – 1\)，方程变为：

2 cos² x − 1 = cos x → 2 cos² x − cos x − 1 = 0

Factor: \((2\cos x + 1)(\cos x – 1) = 0\). Thus \(\cos x = 1\) or \(\cos x = -\frac{1}{2}\).

因式分解：\((2\cos x + 1)(\cos x – 1) = 0\)。因此 \(\cos x = 1\) 或 \(\cos x = -\frac{1}{2}\)。

The solutions are \(x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}\) in the given interval.

在给定区间内，解为 \(x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}\)。

8. Applications in Integration | 在积分中的应用

One of the most common uses of double angle formulas is in evaluating integrals of squared trigonometric functions.

二倍角公式最常见的用途之一是计算三角函数的平方积分。

For example, \(\int \sin^2 x \, dx\) can be evaluated using \(\sin^2 x = \frac{1-\cos 2x}{2}\):

例如，\(\int \sin^2 x \, dx\) 可用 \(\sin^2 x = \frac{1-\cos 2x}{2}\) 计算：

∫ sin² x dx = ∫ (1 − cos 2x)/2 dx = x/2 − sin 2x/4 + C

Similarly, \(\int \cos^2 x \, dx = \int (1+\cos 2x)/2 \, dx = x/2 + \sin 2x/4 + C\).

同理，\(\int \cos^2 x \, dx = \int (1+\cos 2x)/2 \, dx = x/2 + \sin 2x/4 + C\)。

These integrals appear frequently in IB HL calculus questions and in finding areas under curves.

这些积分在IB高级水平微积分题目以及求曲线面积时频繁出现。

9. Geometric and Physical Applications | 几何与物理应用

Double angle formulas are not just abstract identities; they have concrete applications in geometry and physics.

二倍角公式不是抽象的恒等式，它们在几何和物理中有具体应用。

In projectile motion, the range \(R\) of a projectile launched with speed \(u\) at angle \(\theta\) is given by \(R = \frac{u^2 \sin 2\theta}{g}\). Using \(\sin 2\theta = 2\sin\theta\cos\theta\) or recognizing that maximum range occurs when \(\sin 2\theta = 1\) (i.e., \(\theta = 45^\circ\)) is a direct application.

在抛体运动中，以速度 \(u\)、角度 \(\theta\) 发射的物体射程为 \(R = \frac{u^2 \sin 2\theta}{g}\)。利用 \(\sin 2\theta = 2\sin\theta\cos\theta\) 或识别最大射程出现在 \(\sin 2\theta = 1\) 即 \(\theta = 45^\circ\) 时，就是直接应用。

In geometry, the area of an isosceles triangle with equal sides \(r\) and included angle \(2\theta\) is \(\frac{1}{2}r^2\sin 2\theta = r^2\sin\theta\cos\theta\).

在几何中，腰长为 \(r\)、顶角为 \(2\theta\) 的等腰三角形面积为 \(\frac{1}{2}r^2\sin 2\theta = r^2\sin\theta\cos\theta\)。

These examples show how the double angle formula simplifies real-world calculations.

这些例子展示了二倍角公式如何简化现实中的计算。

10. Common Mistakes and Tips | 常见错误与技巧

Students often make mistakes when applying double angle formulas. Here are some common pitfalls and how to avoid them:

学生在应用二倍角公式时常犯错。以下是一些常见陷阱及避免方法：

Mistake: Writing \(\sin 2\theta = 2\sin\theta\) without the \(\cos\theta\) factor. Remember the factor of \(\cos\theta\) is essential.
错误：将 \(\sin 2\theta\) 写成 \(2\sin\theta\) 而漏掉 \(\cos\theta\)。记住 \(\cos\theta\) 因子必不可少。
Mistake: Confusing \(\cos 2\theta = \cos^2\theta – \sin^2\theta\) with \(\cos^2\theta + \sin^2\theta = 1\). The plus sign is only for Pythagorean identity.
错误：将 \(\cos 2\theta = \cos^2\theta – \sin^2\theta\) 与 \(\cos^2\theta + \sin^2\theta = 1\) 混淆。加号只属于勾股恒等式。
Tip: When solving equations, always check the domain of θ and whether tan 2θ is undefined.
技巧：解方程时，始终检查 θ 的定义域以及 tan 2θ 是否无定义。
Tip: For integration of \(\sin^2 x\) or \(\cos^2 x\), always use power-reduction forms derived from cos 2θ.
技巧：计算 \(\sin^2 x\) 或 \(\cos^2 x\) 的积分时，务必使用由 cos 2θ 导出的降幂公式。

11. Practice Problems | 练习与答案

Test your understanding with these selected problems. Solutions are provided below.

请用以下精选题目测试你的理解。答案附在下方。

Problem 1	If \(\sin \theta = \frac{3}{5}\) and \(\theta\) is in quadrant II, find \(\sin 2\theta\).
问题1	已知 \(\sin \theta = \frac{3}{5}\) 且 \(\theta\) 在第二象限，求 \(\sin 2\theta\)。
Problem 2	Solve \(\tan 2x = 1\) for \(0 \le x < \pi\).
问题2	解方程 \(\tan 2x = 1\)，其中 \(0 \le x < \pi\)。
Problem 3	Evaluate \(\int_0^{\pi/2} \cos^2 x \, dx\).
问题3	计算 \(\int_0^{\pi/2} \cos^2 x \, dx\)。

Solutions:

答案：

1. Since \(\theta\) is in quadrant II, \(\cos \theta = -\sqrt{1-\sin^2\theta} = -\frac{4}{5}\). Thus \(\sin 2\theta = 2 \cdot \frac{3}{5} \cdot (-\frac{4}{5}) = -\frac{24}{25}\).

1. 因为 \(\theta\) 在第二象限，\(\cos \theta = -\sqrt{1-\sin^2\theta} = -\frac{4}{5}\)。所以 \(\sin 2\theta = 2 \cdot \frac{3}{5} \cdot (-\frac{4}{5}) = -\frac{24}{25}\)。

2. \(\tan 2x = 1\) implies \(2x = \frac{\pi}{4} + k\pi\), so \(x = \frac{\pi}{8} + \frac{k\pi}{2}\). Within \(0 \le x < \pi\), the solutions are \(x = \frac{\pi}{8}, \frac{5\pi}{8}\).

2. \(\tan 2x = 1\) 得 \(2x = \frac{\pi}{4} + k\pi\)，所以 \(x = \frac{\pi}{8} + \frac{k\pi}{2}\)。在 \(0 \le x < \pi\) 内，解为 \(x = \frac{\pi}{8}, \frac{5\pi}{8}\)。

3. \(\int_0^{\pi/2} \cos^2 x \, dx = \int_0^{\pi/2} \frac{1+\cos 2x}{2} dx = \left[\frac{x}{2} + \frac{\sin 2x}{4}\right]_0^{\pi/2} = \frac{\pi}{4}\).

3. \(\int_0^{\pi/2} \cos^2 x \, dx = \int_0^{\pi/2} \frac{1+\cos 2x}{2} dx = \left[\frac{x}{2} + \frac{\sin 2x}{4}\right]_0^{\pi/2} = \frac{\pi}{4}\)。

12. Summary | 总结

The double angle formulas are indispensable in IB Mathematics. They connect trigonometric functions of \(2\theta\) to those of \(\theta\), enabling simplification, equation solving, and integration. Mastery of these identities and their derivations will save time in exams and build a strong foundation for higher-level mathematics.

二倍角公式在IB数学中不可或缺。它们将 \(2\theta\) 的三角函数与 \(\theta\) 的三角函数联系起来，帮助化简、解方程和积分。熟练掌握这些恒等式及其推导过程，将在考试中节省时间，并为更高阶的数学打下坚实基础。

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Double Angle Formulas and Their Applications | IB数学：二倍角公式及其应用