Simplifying a cos x ± b sin x: The Harmonic Form | 辅助角形式：化简 a cos x ± b sin x

📚 Simplifying a cos x ± b sin x: The Harmonic Form | 辅助角形式：化简 a cos x ± b sin x

This revision guide covers the Edexcel A-Level Pure Mathematics topic of rewriting expressions of the form a cos x ± b sin x as a single sine or cosine function. This technique, often called the harmonic form or R-form, is essential for solving trigonometric equations, finding maximum and minimum values, and modelling periodic behaviour.

本复习指南涵盖 Edexcel A-Level 纯数学中把 a cos x ± b sin x 形式改写为单一正弦或余弦函数的内容。这种技巧通常称为辅助角形式或 R 形式，对于解三角方程、求最大值与最小值以及建立周期模型至关重要。

1. What Is the Harmonic Form? | 什么是辅助角形式？

An expression such as 3 cos x + 4 sin x contains both sine and cosine terms with the same angle x. It is not immediately clear what its maximum value is or how to solve an equation like 3 cos x + 4 sin x = 2. The harmonic form rewrites the expression as a single trigonometric function, usually R cos(x − α) or R sin(x + α).

像 3 cos x + 4 sin x 这样的表达式同时含有同角 x 的正弦项和余弦项。我们很难直接看出它的最大值，也不容易解出 3 cos x + 4 sin x = 2 这样的方程。辅助角形式把它改写为单一三角函数，通常写成 R cos(x − α) 或 R sin(x + α)。

The letters R and α are chosen so that the original two-term expression becomes one term with an amplitude R and a phase shift α. This makes many problems much easier to handle.

我们选取 R 和 α，使原来的两项表达式变成一项，振幅为 R，相移为 α。这样很多问题都更容易处理。

2. Why Do We Need It? | 为什么需要它？

In Edexcel exam questions, you may be asked to express a cos x + b sin x in the form R cos(x − α). Once this is done, you can read off the maximum and minimum values instantly, solve equations that would otherwise require complicated identities, and sketch transformations of basic trigonometric curves.

在 Edexcel 考试题中，你可能会被要求把 a cos x + b sin x 表示成 R cos(x − α) 的形式。完成后，你可以立刻读出最大值和最小值，解出原本需要复杂恒等式的方程，并画出基本三角曲线的变换图形。

Without the harmonic form, solving an equation like 5 cos x + 12 sin x = 6 would be very difficult. With the harmonic form, it reduces to a simple equation such as 13 cos(x − 67.4°) = 6.

如果没有辅助角形式，解 5 cos x + 12 sin x = 6 这样的方程会非常困难。使用辅助角形式后，它就化简为 13 cos(x − 67.4°) = 6 这样的简单方程。

3. The Standard Forms | 标准形式

There are four standard ways to write the harmonic form. The two most common in Edexcel exams are given below, where R > 0 and α is an acute angle or an angle adjusted to the correct quadrant.

辅助角形式有四种标准写法。Edexcel 考试中最常见的两种如下，其中 R > 0，α 是锐角或按正确象限调整后的角。

a cos x + b sin x = R cos(x − α)

a cos x + b sin x = R sin(x + α)

You can also use the equivalent forms a cos x − b sin x = R cos(x + α) or a sin x ± b cos x = R sin(x ± β). The key is to match the coefficients of cos x and sin x correctly.

你也可以使用等价形式 a cos x − b sin x = R cos(x + α) 或 a sin x ± b cos x = R sin(x ± β)。关键是要正确匹配 cos x 和 sin x 的系数。

4. Deriving R and α | R 与 α 的推导

Consider the identity for cos(x − α):

考虑 cos(x − α) 的恒等式：

R cos(x − α) = R cos x cos α + R sin x sin α

If this must equal a cos x + b sin x, then comparing coefficients gives

如果它必须等于 a cos x + b sin x，那么比较系数可得

R cos α = a, R sin α = b

Squaring and adding these two equations eliminates α and gives R:

将两式平方后相加可消去 α，得到 R：

R = √(a² + b²)

Dividing the second equation by the first gives α:

用第二式除以第一式可得到 α：

tan α = b/a

Therefore α is found by taking the inverse tangent, but you must always check the quadrant using the signs of a and b.

因此 α 可以通过取反正切求得，但你必须根据 a 和 b 的符号检查象限。

5. Choosing the Correct Quadrant | 选择正确的象限

The equation tan α = b/a alone does not uniquely determine α because the tangent function has period 180°. You must also look at the signs of a = R cos α and b = R sin α to decide which quadrant α lies in.

仅凭 tan α = b/a 无法唯一确定 α，因为正切函数的周期是 180°。你还必须根据 a = R cos α 和 b = R sin α 的符号判断 α 所在的象限。

The table below summarises the quadrant choice for α when writing a cos x + b sin x = R cos(x − α).

下表总结了在写 a cos x + b sin x = R cos(x − α) 时 α 的象限选择。

Sign of a	Sign of b	Quadrant of α	α range
positive	positive	First	0° to 90°
negative	positive	Second	90° to 180°
negative	negative	Third	180° to 270°
positive	negative	Fourth	270° to 360°

In many basic exam questions, a and b are both positive, so α is simply an acute angle. When one sign is negative, you must add or subtract 180° or use a negative angle such as −60° instead of 300°.

在许多基础考试题中，a 和 b 都是正数，所以 α 就是一个锐角。当有一个符号为负时，你必须加或减 180°，或者使用 −60° 这样的负角而不是 300°。

6. Worked Example 1: 3 cos x + 4 sin x | 例题1：3 cos x + 4 sin x

Express 3 cos x + 4 sin x in the form R cos(x − α), where R > 0 and 0° < α < 90°.

把 3 cos x + 4 sin x 表示成 R cos(x − α) 的形式，其中 R > 0，0° < α < 90°。

First find R using R = √(a² + b²):

首先用 R = √(a² + b²) 求 R：

R = √(3² + 4²) = √(9 + 16) = √25 = 5

Then find α using tan α = b/a:

然后用 tan α = b/a 求 α：

tan α = 4/3, α = arctan(4/3) ≈ 53.13°

Since both a = 3 and b = 4 are positive, α is in the first quadrant. Therefore

因为 a = 3 和 b = 4 都是正数，所以 α 在第一象限。因此

3 cos x + 4 sin x = 5 cos(x − 53.13°)

In radians, this is approximately 5 cos(x − 0.927 rad). This single-term form immediately shows that the maximum value is 5 and the minimum value is −5.

用弧度表示，这大约是 5 cos(x − 0.927 rad)。这个单项形式立刻表明最大值是 5，最小值是 −5。

7. Worked Example 2: cos x − √3 sin x | 例题2：cos x − √3 sin x

Express cos x − √3 sin x in the form R cos(x + α), where R > 0 and 0° < α < 90°.

把 cos x − √3 sin x 表示成 R cos(x + α) 的形式，其中 R > 0，0° < α < 90°。

First compare with R cos(x + α) = R cos x cos α − R sin x sin α. Here the coefficient of cos x is 1 and the coefficient of sin x is −√3. We set R cos α = 1 and R sin α = √3, which makes the expression R cos(x + α) because the sine term is negative.

首先与 R cos(x + α) = R cos x cos α − R sin x sin α 比较。这里 cos x 的系数是 1，sin x 的系数是 −√3。我们令 R cos α = 1，R sin α = √3，这样该表达式就是 R cos(x + α)，因为正弦项为负。

Find R:

求 R：

R = √(1² + (√3)²) = √(1 + 3) = √4 = 2

Find α:

求 α：

tan α = √3/1 = √3, α = 60°

Therefore

因此

cos x − √3 sin x = 2 cos(x + 60°)

This is an important check: a negative sine coefficient produces a plus sign inside the cosine bracket.

这是一个重要的检查：正弦项系数为负时，余弦括号内会出现加号。

8. Solving Equations with the Harmonic Form | 用辅助角形式解方程

Once an expression is in harmonic form, solving equations becomes straightforward. For example, solve 3 cos x + 4 sin x = 2 for 0° ≤ x ≤ 360°.

一旦表达式化为辅助角形式，解方程就变得简单。例如，解 3 cos x + 4 sin x = 2，其中 0° ≤ x ≤ 360°。

From Worked Example 1, we have 5 cos(x − 53.13°) = 2, so

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Simplifying a cos x ± b sin x: The Harmonic Form | 辅助角形式：化简 a cos x ± b sin x