Trigonometric Identities and Equations | 三角恒等式与方程求解

📚 Trigonometric Identities and Equations | 三角恒等式与方程求解

In Edexcel A Level Mathematics, trigonometric identities and equations are central to Pure Mathematics. This topic brings together algebraic manipulation, exact values, and the periodic behaviour of sine, cosine, and tangent. A strong command of identities allows you to simplify expressions, prove results, and solve equations over specified intervals.

在 Edexcel A Level 数学中，三角恒等式与三角方程是纯数学的核心内容。这个主题综合了代数变形、特殊角的精确值以及正弦、余弦和正切的周期性质。熟练掌握恒等式能帮助你化简表达式、证明结论，并在指定区间内解方程。

1. The Pythagorean Identity | 毕达哥拉斯恒等式

The foundation of nearly every trigonometric simplification is the Pythagorean identity sin²θ + cos²θ = 1. It arises directly from the unit circle definition of sine and cosine.

几乎所有三角化简的基础都是毕达哥拉斯恒等式 sin²θ + cos²θ = 1。它直接来源于单位圆上正弦和余弦的定义。

Dividing sin²θ + cos²θ = 1 by cos²θ gives 1 + tan²θ = sec²θ. Dividing by sin²θ gives 1 + cot²θ = cosec²θ.

将 sin²θ + cos²θ = 1 除以 cos²θ 可得 1 + tan²θ = sec²θ；除以 sin²θ 可得 1 + cot²θ = cosec²θ。

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

For example, if sinθ = 3/5 and θ is acute, then cosθ = √(1 − sin²θ) = √(1 − 9/25) = √(16/25) = 4/5, and tanθ = sinθ/cosθ = 3/4.

例如，如果 sinθ = 3/5 且 θ 为锐角，则 cosθ = √(1 − sin²θ) = √(1 − 9/25) = √(16/25) = 4/5，且 tanθ = sinθ/cosθ = 3/4。

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2. Addition Formulae | 和差角公式

The addition formulae are given in the Edexcel formula booklet, but you must know when and how to apply them. They include sin(A ± B) and cos(A ± B).

和差角公式在 Edexcel 公式手册中给出，但你必须知道何时以及如何应用它们，包括 sin(A ± B) 和 cos(A ± B)。

For example, sin(A + B) = sinA cosB + cosA sinB, while cos(A + B) = cosA cosB − sinA sinB.

例如，sin(A + B) = sinA cosB + cosA sinB，而 cos(A + B) = cosA cosB − sinA sinB。

sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)

These formulae are essential when finding exact values such as sin 75° by writing 75° = 45° + 30°.

这些公式在求精确值（例如将 75° 写成 45° + 30° 求 sin 75°）时至关重要。

So sin 75° = sin(45° + 30°) = sin45° cos30° + cos45° sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

因此 sin 75° = sin(45° + 30°) = sin45° cos30° + cos45° sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4。

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3. Double Angle Formulae | 倍角公式

The double angle formulae follow from the addition formulae by setting A = B = θ. The most important are sin 2θ, cos 2θ, and tan 2θ.

倍角公式由和差角公式令 A = B = θ 得到。最重要的是 sin 2θ、cos 2θ 和 tan 2θ。

sin 2θ = 2 sinθ cosθ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan 2θ = 2tanθ / (1 − tan²θ)

The two alternative forms of cos 2θ are especially useful for integrating sin²θ or cos²θ later in the course.

cos 2θ 的两种等价形式在后续积分 sin²θ 或 cos²θ 时特别有用。

When solving equations, you can choose the form that matches the rest of the expression. For example, use cos 2θ = 1 − 2sin²θ in an equation involving sinθ.

在解方程时，你可以选择与其余表达式匹配的形式。例如，在涉及 sinθ 的方程中使用 cos 2θ = 1 − 2sin²θ。

If cosθ = 1/3, then cos 2θ = 2cos²θ − 1 = 2(1/9) − 1 = 2/9 − 1 = −7/9.

如果 cosθ = 1/3，则 cos 2θ = 2cos²θ − 1 = 2(1/9) − 1 = 2/9 − 1 = −7/9。

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4. Factor Formulae | 和差化积公式

The factor formulae convert sums and differences of sines and cosines into products. They are useful in proving identities and in solving equations where a sum is set equal to zero.

和差化积公式将正弦、余弦的和差转化为乘积形式。它们在证明恒等式以及解和差为零的方程时非常有用。

sin P + sin Q = 2 sin((P+Q)/2) cos((P−Q)/2)
sin P − sin Q = 2 cos((P+Q)/2) sin((P−Q)/2)
cos P + cos Q = 2 cos((P+Q)/2) cos((P−Q)/2)
cos P − cos Q = −2 sin((P+Q)/2) sin((P−Q)/2)

These are less frequently the first step, but they can make an otherwise complicated proof much shorter.

这些公式通常不是解题的第一步，但它们可以让原本复杂的证明大大缩短。

For example, sin 5x + sin 3x can be written as 2 sin 4x cos x, which is much easier to integrate or solve if it equals zero.

例如，sin 5x + sin 3x 可以写成 2 sin 4x cos x，如果它等于零，这样处理起来会容易得多。

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5. Expressing a sin θ + b cos θ | 辅助角公式（R 形式）

Any expression of the form a sinθ + b cosθ can be written as R sin(θ + α) or R cos(θ − α), where R = √(a² + b²).

任何形如 a sinθ + b cosθ 的表达式都可以写成 R sin(θ + α) 或 R cos(θ − α)，其中 R = √(a² + b²)。

The angle α is chosen so that a = R cosα and b = R sinα for R sin(θ + α). This is sometimes called the R-alpha method.

对于 R sin(θ + α)，角度 α 的选择应满足 a = R cosα 且 b = R sinα。这种方法有时称为 R-α 方法。

a sinθ + b cosθ = R sin(θ + α), R = √(a² + b²), tan α = b/a

This transformation is essential for finding maximum and minimum values of an expression, or for solving equations of the form a sinθ + b cosθ = c.

这一变换对于求表达式的最大值和最小值，或求解形如 a sinθ + b cosθ = c 的方程至关重要。

For example, 3 sinθ + 4 cosθ can be written as 5 sin(θ + 53.1°) because R = √(3² + 4²) = 5 and tan α = 4/3.

例如，3 sinθ + 4 cosθ 可以写成 5 sin(θ + 53.1°)，因为 R = √(3² + 4²) = 5，且 tan α = 4/3。

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6. Solving Basic Trigonometric Equations | 解基本三角方程

When solving sinθ = k, cosθ = k, or tanθ = k, you must give all solutions in the required interval, usually 0° ≤ θ < 360° or 0 ≤ θ < 2π.

在解 sinθ = k、cosθ = k 或 tanθ = k 时，你必须给出指定区间内的所有解，通常是 0° ≤ θ < 360° 或 0 ≤ θ < 2π。

Start by finding the principal value from your calculator, then use the symmetry of the trigonometric graphs to locate the remaining solutions.

首先利用计算器求出主值，然后利用三角函数图像的对称性找出其余解。

For sine, if θ = α is a solution, then θ = 180° − α is also a solution. For cosine, θ = 360° − α is another solution.

对于正弦，如果 θ = α 是一个解，那么 θ = 180° − α 也是解；对于余弦，θ = 360° − α 是另一个解。

Always check the interval and the mode of your calculator, as mixing degrees and radians is a common source of error.

务必检查区间和计算器的角度单位，混淆角度制和弧度制是常见的错误来源。

For example, solve cosθ = 1/2 for 0° ≤ θ < 360°. The principal value is 60°, and the second solution is 360° − 60° = 300°.

例如，在 0° ≤ θ < 360° 内解 cosθ = 1/2。主值为 60°，第二个解为 360° − 60° = 300°。

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7. General Solutions and Periodicity | 通解与周期性

Trigonometric functions are periodic, so equations often have infinitely many solutions unless a finite interval is specified.

三角函数具有周期性，因此如果不限定有限区间，方程通常有无穷多个解。

For sinθ and cosθ, the period is 360° or 2π, so adding 360°n (or 2πn) to any solution produces another solution. For tanθ, the period is 180° or π.

对于 sinθ 和 cosθ，周期为

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