A-Level Mathematics: Derivation and Applications of Double Angle Formulas | A-Level数学：二倍角公式的推导与应用

📚 A-Level Mathematics: Derivation and Applications of Double Angle Formulas | A-Level数学：二倍角公式的推导与应用

The double angle formulas are among the most important tools in A-Level trigonometry. They allow us to express trigonometric functions of twice an angle in terms of functions of the original angle, and they appear repeatedly in solving equations, integration, differentiation, and geometric problems. This article will show you exactly where these formulas come from, how to remember them, and how to apply them under exam conditions.

二倍角公式是A-Level三角学中最重要的工具之一。它们帮助我们用一个角的正弦、余弦、正切来表示这个角两倍的正弦、余弦、正切。在解方程、积分、微分和几何问题中，二倍角公式反复出现。本文将向您展示这些公式的推导来源、记忆方法以及如何在考试条件下灵活应用。

1. The Addition Formulas | 加法公式回顾

Before deriving the double angle formulas, we must recall the addition formulas for sine, cosine, and tangent. These are standard identities that you are expected to know with certainty in the A-Level syllabus.

在推导二倍角公式之前，我们必须回顾正弦、余弦和正切的加法公式。这些是A-Level考纲中要求必须准确掌握的恒等式。

For any angles A and B, the following identities hold:

对于任意角 A 和 B，以下恒等式成立：

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)

These formulas are valid for all real A and B. In the tangent formula, the expression is undefined when the denominator equals zero, which corresponds to cases where A + B = 90° + 180°n.

这些公式对所有实数 A 和 B 都成立。在正切加法公式中，当分母为零时表达式无定义，这对应 A + B = 90° + 180°n 的情形。

2. Deriving sin(2θ) | 推导sin(2θ)

To derive the double angle formula for sine, we simply substitute A = θ and B = θ into the sine addition formula.

要推导正弦的二倍角公式，我们只需将 A = θ 和 B = θ 代入正弦加法公式。

Let A = θ and B = θ in the addition formula for sine:

令 A = θ，B = θ，代入正弦的加法公式：

sin(θ + θ) = sin θ cos θ + cos θ sin θ

The two terms on the right-hand side are identical, so we obtain:

右边两项完全相同，因此我们得到：

sin 2θ = 2 sin θ cos θ

This is the first double angle identity. It tells us that the sine of a double angle is twice the product of the sine and cosine of the original angle. This identity is extremely useful for simplifying expressions and solving trigonometric equations.

这就是第一个二倍角恒等式。它告诉我们二倍角的正弦等于原角正弦和余弦乘积的两倍。这一恒等式在化简表达式和求解三角方程时极为有用。

3. Deriving cos(2θ) in Three Forms | 推导cos(2θ)的三种形式

The cosine double angle formula is slightly richer because it can be written in three equivalent forms. Each form is useful in different contexts, especially in integration.

余弦的二倍角公式更加丰富，因为它可以写成三种等价的形式。每种形式在不同场景下都有用，尤其在积分中。

Start with the addition formula for cosine:

从余弦的加法公式开始：

cos(θ + θ) = cos θ cos θ − sin θ sin θ

Hence we get the basic form:

因此得到基本形式：

cos 2θ = cos²θ − sin²θ

Now, using the Pythagorean identity sin²θ + cos²θ = 1, we can replace sin²θ with 1 − cos²θ:

接着，利用勾股恒等式 sin²θ + cos²θ = 1，我们可以用 1 − cos²θ 替换 sin²θ：

cos 2θ = cos²θ − (1 − cos²θ) = 2 cos²θ − 1

Similarly, replacing cos²θ with 1 − sin²θ gives:

同理，用 1 − sin²θ 替换 cos²θ 得到：

cos 2θ = (1 − sin²θ) − sin²θ = 1 − 2 sin²θ

These three forms are all equally valid. Which one you choose depends on the problem. For example, when integrating sin²x or cos²x, you will need the forms involving only cos 2θ on the right-hand side.

这三种形式都同样有效。选择哪一种取决于具体问题。例如，在积分 sin²x 或 cos²x 时，你需要用到右边只含 cos 2θ 的形式。

4. Deriving tan(2θ) | 推导tan(2θ)

The tangent double angle formula follows from the tangent addition formula in the same way.

正切的二倍角公式同样可以由正切加法公式推导得出。

Substitute A = θ and B = θ into the tangent addition formula:

令 A = θ，B = θ，代入正切加法公式：

tan(θ + θ) = (tan θ + tan θ) / (1 − tan θ · tan θ)

Simplify the numerator and denominator:

化简分子和分母：

tan 2θ = 2 tan θ / (1 − tan²θ)

This formula is valid only when tan θ is defined and the denominator 1 − tan²θ is not zero. In particular, tan 2θ is undefined when θ = 45° + 90°n, because then tan²θ = 1 and the denominator becomes zero.

该公式仅在 tan θ 有定义且分母 1 − tan²θ 不为零时成立。特别地，当 θ = 45° + 90°n 时，tan²θ = 1，分母为零，tan 2θ 无定义。

5. Key Identities Summary | 关键恒等式总结

For quick revision, here is the complete list of double angle formulas you must remember.

为方便快速复习，下面列出你必须记住的全部二倍角公式。

Function	Identity
sin 2θ	2 sin θ cos θ
cos 2θ	cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ
tan 2θ	2 tan θ / (1 − tan²θ)

You should also remember the reverse half-angle identities, which are derived by rearranging the double angle forms:

你还应该记住由二倍角公式变形得到的半角恒等式：

sin²θ = (1 − cos 2θ) / 2

cos²θ = (1 + cos 2θ) / 2

These are often called the power-reduction formulas because they reduce a squared trigonometric function to a linear expression involving cos 2θ.

这些通常被称为“降幂公式”，因为它们将平方的三角函数化为涉及 cos 2θ 的一次表达式。

6. Solving Trigonometric Equations | 解三角方程

One of the most common exam applications of double angle formulas is solving trigonometric equations that involve both 2θ and θ. The strategy is to express everything in terms of a single trigonometric function.

二倍角公式最常见的考试应用之一是求解同时包含 2θ 和 θ 的三角方程。解题策略是将所有项表示为关于同一个三角函数的表达式。

Example: Solve the equation cos 2θ + sin θ = 0 for 0° ≤ θ < 360°.

范例：解方程 cos 2θ + sin θ = 0，其中 0° ≤ θ < 360°。

Using the form cos 2θ = 1 − 2 sin²θ, we substitute:

使用形式 cos 2θ = 1 − 2 sin²θ，代入方程：

1 − 2 sin²θ + sin θ = 0

Rearrange into a quadratic in sin θ:

将其整理为关于 sin θ 的二次方程：

2 sin²θ − sin θ − 1 = 0

Factorise:

因式分解：

(2 sin θ + 1)(sin θ − 1) = 0

Thus sin θ = −1/2 or sin θ = 1. For 0° ≤ θ < 360°, the solutions are θ = 90°, θ = 210°, θ = 330°. Always check the required domain when giving final answers.

因此 sin θ = −1/2 或 sin θ = 1。在 0° ≤ θ < 360° 范围内，解为 θ = 90°，θ = 210°，θ = 330°。写出最终答案时务必检查题目要求的定义域。

7. Integration Applications | 积分应用

Double angle formulas are indispensable when integrating squares of sine and cosine. Without them, integrals like ∫ sin²x dx cannot be evaluated directly using basic rules.

二倍角公式在积分正弦和余弦的平方时不可或缺。没有它们，像 ∫ sin²x dx 这样的积分无法直接用基本规则求值。

Use the power-reduction identity sin²x = (1 − cos 2x)/2 to rewrite the integrand:

使用降幂恒等式 sin²x = (1 − cos 2x)/2 改写被积函数：

∫ sin²x dx = ∫ (1 − cos 2x)/2 dx

= ½ ∫ (1 − cos 2x) dx

Now integrate term by term:

现在逐项积分：

∫ sin²x dx = ½ (x − ½ sin 2x) + C = x/2 − (sin 2x)/4 + C

Similarly, ∫ cos²x dx = x/2 + (sin 2x)/4 + C. These results appear frequently in AS and A2 pure mathematics papers, especially when calculating areas under curves or volumes of revolution.

类似地，∫ cos²x dx = x/2 + (sin 2x)/4 + C。这些结果在AS和A2纯数学考试中经常出现，尤其是在计算曲线下面积或旋转体体积时。

8. Differentiation Applications | 微分应用

Double angle identities can also simplify differentiation. For instance, the derivative of sin²x can be found by the chain rule, but the double angle form gives a neater answer.

二倍角恒等式也可以简化微分运算。例如，sin²x 的导数可以用链式法则求，但利用二倍角公式可以得到更简洁的结果。

Consider y = sin²x. Differentiating with respect to x:

考虑 y = sin²x。对 x 求导：

dy/dx = 2 sin x cos x

But 2 sin x cos x = sin 2x, so we can write:

而 2 sin x cos x = sin 2x，因此我们可以写：

dy/dx = sin 2x

This result is easy to remember and often quicker to use than expanding a chain-rule expression. Similarly, d/dx (cos²x) = −2 sin x cos x = −sin 2x. Being able to move comfortably between these forms is a valuable exam skill.

这个结果容易记忆，并且通常比展开链式法则表达式更快。类似地，d/dx (cos²x) = −2 sin x cos x = −sin 2x。能够熟练地在这些形式之间转换是一项宝贵的应试技能。

9. R-Formula Connections | 与R公式的联系

The double angle formulas are also connected to the R-formula (harmonic form) used to rewrite expressions such as a sin x + b cos x as R sin(x + α).

二倍角公式还与R公式（和谐形式）有关，后者用于将 a sin x + b cos x 这样的表达式改写为 R sin(x + α)。

For example, the expression sin 2x + cos 2x can be rewritten in the form √2 sin(2x + 45°). The double angle identity allows us to first express sin 2x and cos 2x in terms of sin x and cos x, although in practice we often apply the R-formula directly to the whole expression with argument 2x.

例如，表达式 sin 2x + cos 2x 可以改写为 √2 sin(2x + 45°)。二倍角恒等式允许我们先将 sin 2x 和 cos 2x 用 sin x 和 cos x 表示，虽然在实践中我们通常直接将R公式应用于整个含 2x 的表达式。

Another connection is in solving equations of the form a cos 2θ + b sin θ + c = 0. Using a double angle identity reduces such an equation to a quadratic in either sin θ or cos θ, after which the R-formula or general solution method can be applied.

另一个联系体现在解形如 a cos 2θ + b sin θ + c = 0 的方程时。使用二倍角恒等式可将此类方程化为关于 sin θ 或 cos θ 的二次方程，之后可以用R公式或通解方法求解。

10. Common Exam Pitfalls | 常见考试误区

Many students lose marks on double angle questions due to small sign or domain errors. Here are the most common pitfalls and how to avoid them.

许多学生在二倍角题目中因为符号或定义域的小错误而失分。以下是最常见的误区和避免方法。

Pitfall: Confusing the three forms of cos 2θ. Remember that 2 cos²θ − 1 is positive, while 1 − 2 sin²θ is negative when sin²θ is large. Check by testing θ = 45°: cos 90° = 0, so both forms give zero.

误区：混淆 cos 2θ 的三种形式。记住 2 cos²θ − 1 为正，而 1 − 2 sin²θ 在 sin²θ 较大时为负。可代入 θ = 45° 验证：cos 90° = 0，所以两种形式都为零。
Pitfall: Forgetting the domain when solving equations. Always find all solutions within the given interval and be careful with quadrants.

误区：解方程时遗忘定义域。始终求出给定区间内的所有解，并注意象限的判断。
Pitfall: Using tan 2θ when tan θ is undefined. Check whether θ = 90° + 180°n is included in the domain; if so, tan 2θ may need separate treatment.

误区：在 tan θ 无定义时使用 tan 2θ。检查定义域中是否包含 θ = 90° + 180°n；如果包含，则 tan 2θ 可能需要单独处理。
Pitfall: Misapplying integration limits after substitution. When using a double angle identity to integrate, you must adjust limits if you use a u-substitution, or keep the variable x and integrate directly.

误区：代入积分限时出错。当使用二倍角恒等式积分时，如果使用换元法，必须调整积分限；或者保持变量 x 直接积分。

By practising these types of problems carefully, you can avoid these common errors and secure full marks in double angle questions.

通过认真练习这些类型的问题，你可以避免这些常见错误，并在二倍角题目中获得满分。

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A-Level Mathematics: Derivation and Applications of Double Angle Formulas | A-Level数学：二倍角公式的推导与应用