Clever Applications of Angle Addition Formulas in Problem Solving | 和角公式在解题中的巧妙运用

📚 Clever Applications of Angle Addition Formulas in Problem Solving | 和角公式在解题中的巧妙运用

The angle addition formulas are among the most powerful tools in trigonometry. They appear in almost every A-level mathematics exam, either directly or through derived identities such as double-angle and auxiliary-angle forms. Mastering their clever application can greatly simplify complex problems.

和角公式是三角学中最强大的工具之一。在A-level数学考试中，它们几乎无处不在，有时直接使用，有时通过倍角公式、辅助角公式等衍生形式出现。熟练掌握它们的巧妙运用，可以大大化简复杂问题。

1. The Basic Forms | 和角公式的基本形式

For any angles A and B, the standard addition and subtraction formulas are:

对于任意角 A 和 B，标准的和角与差角公式如下：

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

And for cosine:

余弦公式为：

cos(A+B) = cosA cosB − sinA sinB
cos(A−B) = cosA cosB + sinA sinB

For tangent:

正切公式为：

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

A common memory aid is to say “sine keeps the sign, cosine changes the sign” – meaning sin(A+B) has a plus in the formula while sin(A−B) has a minus, whereas cos(A+B) has a minus while cos(A−B) has a plus.

一个常用的记忆口诀是“正弦符号不变，余弦符号相反”——即 sin(A+B) 展开后中间为加号，sin(A−B) 为减号；而 cos(A+B) 中间为减号，cos(A−B) 为加号。

2. Exact Evaluation of Non-Special Angles | 非特殊角的精确求值

The formulas allow us to compute exact values for angles that are not usually memorised, such as 15°, 75°, 105°.

利用和角公式，我们可以精确求出一些通常不需要背记的角度值，例如 15°、75°、105°。

Example: find the exact value of sin75°.

例如：求 sin75° 的精确值。

sin75° = sin(45° + 30°) = sin45° cos30° + cos45° sin30°

Substituting the known values:

代入已知值：

= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

This exact form is much more useful than a rounded decimal in algebra and calculus problems.

这个精确形式在代数与微积分问题中远比四舍五入的小数更有用。

3. Deriving Double-Angle Formulas | 从和角公式推导倍角公式

Setting B = A in the addition formulas immediately gives the double-angle identities.

在和角公式中令 B = A，即可直接得到倍角公式。

sin2A = 2 sinA cosA

For cosine:

余弦的倍角公式为：

cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A

And for tangent:

正切的倍角公式为：

tan2A = 2tanA / (1 − tan²A)

These forms are extremely useful when solving equations involving sin2A or cos2A, especially when factorising is needed.

这些形式在解含 sin2A 或 cos2A 的方程时非常有用，尤其是在需要因式分解的情况下。

4. Combining asinx + bcosx into a Single Trigonometric Function | 将 asinx + bcosx 合并为单个三角函数

The auxiliary-angle formula is a direct consequence of the angle addition identity. It expresses expressions like Rsin(x+α) in terms of sinx and cosx.

辅助角公式是和角公式的直接推论，它把 Rsin(x+α) 这样的表达式展开成含 sinx 与 cosx 的形式。

Suppose we have asinx + bcosx. Set:

假设有 asinx + bcosx，令：

R = √(a² + b²), and α = arctan(b/a) (for a > 0)

Then:

则：

asinx + bcosx = R sin(x + α)

Example: rewrite 3sinx + 4cosx in the form Rsin(x+α). Here R = √(3² + 4²) = 5, and α = arctan(4/3). Thus the expression becomes 5sin(x + arctan(4/3)).

例：将 3sinx + 4cosx 化为 Rsin(x+α) 的形式。这里 R = √(3² + 4²) = 5，α = arctan(4/3)。因此原式化为 5sin(x + arctan(4/3))。

This technique is essential for finding maximum and minimum values, solving equations, and sketching graphs.

这一技巧在求最大值、最小值、解方程及画图时都至关重要。

5. Applications in Triangles | 在三角形中的应用

In any triangle ABC, we have A + B + C = π. This leads to useful relations involving the sum or difference of angles.

在任意三角形 ABC 中，有 A + B + C = π。由此可导出很多涉及角度和差的常用关系。

For example, since C = π − (A+B), we have:

例如，因为 C = π − (A+B)，所以有：

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

Similarly:

同样地：

cosC = −cos(A+B) = sinA sinB − cosA cosB

In many triangle-solving questions, you may need to prove statements such as sin(A+B) = sinC or use tan(A+B) when given tanA and tanB.

在许多解三角形问题中，你可能需要证明 sin(A+B) = sinC 这样的等式，或者在已知 tanA、tanB 时使用 tan(A+B)。

6. Proving Trigonometric Identities | 证明三角恒等式

The angle addition formulas are the backbone of countless identity proofs. A typical exam question asks you to show that sin(A+B)sin(A−B) = sin²A − sin²B.

和角公式是无数恒等式证明的基石。一个典型的考试题是证明 sin(A+B)sin(A−B) = sin²A − sin²B。

Using the formulas, we expand:

利用公式展开：

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

This is a difference of squares:

这是平方差：

= sin²A cos²B − cos²A sin²B

Then replace cos²B by 1 − sin²B and cos²A by 1 − sin²A:

然后用 cos²B = 1 − sin²B 和 cos²A = 1 − sin²A 代换：

= sin²A(1 − sin²B) − (1 − sin²A)sin²B = sin²A − sin²B

Thus the identity holds. Notice how the formulas turn a seemingly complicated product into a simple algebraic expression.

因此命题得证。可以看到，公式把看似复杂的乘积转化为简单的代数式。

7. Finding Maximum and Minimum Values | 求最大值与最小值

When a trigonometric expression is transformed into a single sine or cosine function, its range becomes immediate.

当三角函数表达式被合并成单一的正弦或余弦函数后，其取值范围就一目了然。

Example: find the maximum and minimum values of y = sinx + cosx.

例：求 y = sinx + cosx 的最大值与最小值。

We write:

我们写作：

sinx + cosx = √2 sin(x + π/4)

Since the sine function always lies between −1 and 1, the maximum is √2 and the minimum is −√2.

由于正弦函数总在 −1 和 1 之间，因此最大值为 √2，最小值为 −√2。

This method also helps determine the amplitude, period, and phase shift of the graph.

这种方法还能帮助确定图像的振幅、周期和相位移动。

8. Solving Trigonometric Equations | 解三角方程

Many equations that initially look difficult can be solved by rewriting them with an auxiliary angle.

许多表面上困难的三角方程，通过辅助角改写后即可轻松求解。

Example: solve sinθ + √3 cosθ = 1 for 0 ≤ θ < 2π.

例：在 0 ≤ θ < 2π 内解方程 sinθ + √3 cosθ = 1。

Here R = √(1² + (√3)²) = 2, and tanα = √3, so α = π/3 (since a > 0, b > 0). Hence:

这里 R = √(1² + (√3)²) = 2，tanα = √3，所以 α = π/3（因为 a > 0，b > 0）。因此：

2 sin(θ + π/3) = 1

So sin(θ + π/3) = 1/2. Then θ + π/3 = π/6 or 5π/6 (within the required range). Subtracting π/3 gives:

于是 sin(θ + π/3) = 1/2。在给定范围内，θ + π/3 = π/6 或 5π/6。减去 π/3 得：

θ = −π/6 (discard) or θ = π/2

Also consider the periodicity of sine – the other solution in [0,2π) is θ = π/2 only, because the full solution set includes 7π/6 which comes from the second quadrant angle? Let’s be precise: sin(θ+π/3)=1/2 has general solutions θ+π/3 = π/6 + 2kπ or θ+π/3 = 5π/6 + 2kπ. Thus θ = −π/6 + 2kπ or θ = π/2 + 2kπ. For 0 ≤ θ < 2π, the valid solutions are θ = 11π/6 from the first family (with k=1) and θ = π/2 from the second family.

同时要注意正弦的周期性——刚才需更精确：sin(θ+π/3)=1/2 的通解为 θ+π/3 = π/6 + 2kπ 或 θ+π/3 = 5π/6 + 2kπ。因此 θ = −π/6 + 2kπ 或 θ = π/2 + 2kπ。在 0 ≤ θ < 2π 内，第一族给出 θ = 11π/6（k=1），第二族给出 θ = π/2。

So the solutions are θ = π/2 and 11π/6.

所以解为 θ = π/2 和 11π/6。

9. Applications in Calculus | 在微积分中的应用

Angle addition formulas are also essential when integrating products of sine and cosine. The product-to-sum identities are derived directly from the addition formulas.

和角公式在积分正弦与余弦的乘积时同样不可或缺。积化和差公式正是由和角公式直接推导而来。

Recall that:

回忆：

sinA cosB = ½[sin(A+B) + sin(A−B)]

Example: find ∫ sin3x cos2x dx.

例：求 ∫ sin3x cos2x dx。

Using the identity:

利用上述恒等式：

sin3x cos2x = ½[sin5x + sinx]

Integrating term by term:

逐项积分：

∫ sin3x cos2x dx = ½[−cos5x/5 − cosx] + C = −(1/10)cos5x − (1/2)cosx + C

Without converting the product to a sum, such an integral would be very tedious to compute.

如果不把乘积转化为和差形式，计算这样的积分将非常繁琐。

10. Common Pitfalls and Key Reminders | 常见易错点与关键提醒

When using the angle addition formulas, students often make sign mistakes or forget the domain restrictions in the tangent formula.

在使用和角公式时，学生常在符号上出错，或者忽略正切公式的定义域条件。

For cosine, remember that cos(A+B) has a minus sign, and cos(A−B) has a plus sign. Many incorrect signs originate here.

对于余弦，注意 cos(A+B) 中间为减号，cos(A−B) 中间为加号。很多错误符号都源于此。
The tangent formula is undefined when tanA tanB = 1 (for the plus form) or tanA tanB = −1 (for the minus form). Always check such cases separately.

正切公式在 tanA tanB = 1（对于加的形式）或 tanA tanB = −1（对于减的形式）时无定义。遇到这些情况要单独讨论。
When solving equations, do not forget the periodic nature of trigonometric functions. Write the general solution first, then restrict to the required interval.

解方程时不要忘记三角函数的周期性。先写出通解，再限制到指定区间。
When applying the auxiliary-angle formula, the quadrant of α must match the signs of a and b. For a < 0, add π to the principal value of arctan(b/a).

使用辅助角公式时，α 所在的象限必须与 a、b 的符号一致。当 a < 0 时，要在 arctan(b/a) 的主值上加上 π。

Mastering these details will help you avoid silent mistakes and use the formulas with confidence in exams.

掌握这些细节，可以帮助你避免隐性错误，并在考试中自信地运用这些公式。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Clever Applications of Angle Addition Formulas in Problem Solving | 和角公式在解题中的巧妙运用