📚 Trigonometric Identities and Equations | 三角恒等式与方程
Trigonometric identities and equations are central to Edexcel A-level Mathematics. This topic combines algebraic manipulation, exact values, radian measure, and graphical thinking. You are expected to solve equations such as sin 2x = cos x, to prove identities such as sin²θ + cos²θ ≡ 1, and to use the R-formula for expressions like 3 sin θ + 4 cos θ. A strong method, careful attention to the given interval, and fluency with exact values will secure most marks.
三角恒等式与方程是 Edexcel A-level 数学的核心内容。该专题融合了代数变形、精确值、弧度制以及图形思维。你需要会解诸如 sin 2x = cos x 这样的方程，会证明 sin²θ + cos²θ ≡ 1 等恒等式，也会用 R 形式处理 3 sin θ + 4 cos θ 这类表达式。掌握扎实的方法、认真关注给定区间、熟悉特殊角的精确值，就能拿到大部分分数。
1. Radian Measure and the Unit Circle | 弧度制与单位圆
In A-level trigonometry, angles are usually measured in radians. One full turn is 2π radians, so π rad = 180° and 1 rad ≈ 57.3°. Radian measure is preferred because it makes derivatives and integrals of trigonometric functions much simpler.
在 A-level 三角学中，角度通常使用弧度制。一整圈为 2π 弧度，因此 π 弧度 = 180°，1 弧度约等于 57.3°。使用弧度制的原因在于它能让三角函数的求导和积分变得更加简单。
π rad = 180°
For a circle of radius r, the arc length s and sector area A are given by s = rθ and A = ½r²θ, where θ must be in radians. These formulas do not work directly with degrees, so converting to radians before applying them is essential.
对于半径为 r 的圆，弧长 s 和扇形面积 A 分别为 s = rθ 与 A = ½r²θ，其中 θ 必须使用弧度。这些公式不能直接使用角度制，因此代入前必须先将角度转换为弧度。
s = rθ, A = ½r²θ
On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate of a point on the circle. This geometric view helps you remember signs in each quadrant and the periodic behaviour of sine and cosine.
在单位圆上，cos θ 是点的横坐标，sin θ 是点的纵坐标。这种几何图像有助于你记住各象限的符号以及正弦、余弦的周期性变化。
2. Pythagorean Identities | 勾股恒等式
The most important identity is sin²θ + cos²θ ≡ 1. It comes directly from the equation of the unit circle x² + y² = 1. The symbol ≡ means the identity is true for all permissible values of θ, not just for particular solutions.
最重要的恒等式是 sin²θ + cos²θ ≡ 1。它直接来源于单位圆方程 x² + y² = 1。符号 ≡ 表示对所有允许的 θ 值恒成立，而不仅仅对某个特定解成立。
sin²θ + cos²θ ≡ 1
Dividing this identity by cos²θ and by sin²θ gives two very useful derived identities: 1 + tan²θ ≡ sec²θ and 1 + cot²θ ≡ cosec²θ. These are especially helpful when an equation contains tan θ and sec θ, or cot θ and cosec θ.
把这个恒等式分别除以 cos²θ 和 sin²θ，可以得到两个非常有用的导出恒等式：1 + tan²θ ≡ sec²θ 以及 1 + cot²θ ≡ cosec²θ。当方程中含有 tan θ 与 sec θ，或者 cot θ 与 cosec θ 时，它们特别有用。
1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ
For example, if sin θ = 3/5 and θ is obtuse, you can find cos θ using cos²θ = 1 − sin²θ = 16/25. Since θ is obtuse, cos θ is negative, so cos θ = −4/5. This sign choice is a common source of errors.
例如，若 sin θ = 3/5 且 θ 为钝角，你可以利用 cos²θ = 1 − sin²θ = 16/25 求出 cos θ。因为 θ 是钝角，cos θ 为负，所以 cos θ = −4/5。这种符号选择经常容易出错。
3. Solving Basic Trigonometric Equations | 解基本三角方程
To solve sin θ = k, cos θ = k, or tan θ = k, first find the principal value using your calculator or exact values. Then use the symmetry of the trig graphs or the CAST diagram to find all values in the required interval.
解 sin θ = k、cos θ = k 或 tan θ = k 时，先用计算器或特殊角的精确值求出主值。然后利用三角函数图像的对称性或者 CAST 图，在给定区间内找出所有解。
For sin θ = 0.5 in the interval 0 ≤ θ < 2π, the principal value is π/6. Since sine is also positive in the second quadrant, the second solution is π − π/6 = 5π/6. No other solutions lie in the interval.
例如，在区间 0 ≤ θ < 2π 内解 sin θ = 0.5，主值为 π/6。由于正弦在第二象限也为正，因此另一个解是 π − π/6 = 5π/6。该区间内没有其他解。
sin θ = 0.5 ⇒ θ = π/6, 5π/6
For tan θ = −1, the principal value is −π/4. Tangent has period π, so all solutions are of the form θ = −π/4 + nπ, where n is an integer. Always then filter these into the given interval.
对于 tan θ = −1，主值为 −π/4。正切函数的周期为 π，因此所有解可写为 θ = −π/4 + nπ，其中 n 为整数。最后再把这些解限制到题目给定的区间内。
4. Using the CAST Diagram | 使用 CAST 图
The CAST diagram tells you which trigonometric ratios are positive in each quadrant. In the first quadrant, All are positive; in the second, Sin only; in the third, Tan only; in the fourth, Cos only.
CAST 图告诉你每个象限中哪些三角函数为正。第一象限 A：全部为正；第二象限 S：只有 sin 为正；第三象限 T：只有 tan 为正；第四象限 C：只有 cos 为正。
| Quadrant | Positive functions | English keyword |
|---|---|---|
| 1 | sin, cos, tan | All |
| 2 | sin only | Sin |
| 3 | tan only | Tan |
| 4 | cos only | Cos |
To use CAST, mark the angle from the positive x-axis, identify the quadrants where the function has the required sign, and then find all corresponding angles. For example, cos θ = −√3/2 gives θ = 5π/6 and θ = 7π/6 in 0 ≤ θ < 2π.
使用 CAST 图时，从 x 轴正方向标注角度，确定三角函数具有所需符号的象限，然后找出所有对应角。例如，cos θ = −√3/2 在 0 ≤ θ < 2π 内的解为 θ = 5π/6 和 θ = 7π/6。
cos θ = −√3/2 ⇒ θ = 5π/6, 7π/6
Many candidates lose marks by stopping after the calculator principal value. The CAST diagram should be used as a check that every solution in the requested range has been found.
很多考生会因只写出计算器给出的主值而失分。应将 CAST 图作为一种检查手段，确保在题目要求的范围内已经找出所有解。
5. Double-Angle Identities | 二倍角公式
The double-angle formulas are essential for solving equations and simplifying expressions. For sine, sin 2θ = 2 sin θ cos θ. For cosine, there are three equivalent forms, and choosing the right one can make a problem much easier.
二倍角公式是解方程和化简表达式的重要工具。正弦二倍角公式为 sin 2θ = 2 sin θ cos θ。余弦二倍角公式有三种等价形式，选择合适的形式可以大大简化问题。
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
For tangent, the formula is tan 2θ = 2 tan θ / (1 − tan²θ). This is useful when an equation already involves tan θ, or when proving identities with tangent terms.
正切的二倍角公式为 tan 2θ = 2 tan θ / (1 − tan²θ)。当方程中已经含有 tan θ，或者在证明带有正切项的恒等式时，它非常有用。
tan 2θ = 2 tan θ / (1 − tan²θ)
For example, to solve cos 2θ = sin θ, replace cos 2θ with 1 − 2 sin²θ. This gives 1 − 2 sin²θ = sin θ, which is a quadratic in sin θ. Factorising gives (2 sin θ − 1)(sin θ + 1) = 0, so sin θ = 1/2 or sin θ = −1.
例如，解方程 cos 2θ = sin θ 时，可将 cos 2θ 替换为 1 − 2 sin²θ，得到 1 − 2 sin²θ = sin θ，这是关于 sin θ 的二次方程。因式分解得到 (2 sin θ − 1)(sin θ + 1) = 0，所以 sin θ = 1/2 或 sin θ = −1。
cos 2θ = sin θ ⇒ sin θ = 1/2 or sin θ = −1
6. Compound-Angle Formulas | 复合角公式
The compound-angle formulas expand sin(A ± B), cos(A ± B), and tan(A ± B). They are given in the Edexcel formula booklet, but you must know how to apply them confidently and accurately.
复合角公式用于展开 sin(A ± B)、cos(A ± B) 和 tan(A ± B)。它们虽然列在 Edexcel 公式表中，但你必须能够自信、准确地运用这些公式。
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
For example, sin 75° can be written as sin(45° + 30°). Substituting exact values gives sin 75° = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.
例如，sin 75° 可以写成 sin(45° + 30°)。代入特殊角的精确值后得到 sin 75° = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4。
sin 75° = (√6 + √2)/4
These formulas also help when solving equations such as sin(θ + π/3) = cos(θ − π/6). Expanding both sides turns the equation into a linear relationship between sin θ and cos θ, which can then be simplified using tan θ.
这些公式在解诸如 sin(θ + π/3) = cos(θ − π/6) 这样的方程时也很有帮助。展开两边后，方程变成 sin θ 和 cos θ 之间的线性关系，随后可以利用 tan θ 进行化简。
7. The R-Formula | R 形式化简
The R-formula expresses a sin θ + b cos θ as R sin(θ + α) or R cos(θ − α), where R = √(a² + b²). This is particularly useful for finding maximum and minimum values or solving equations of the form 3 sin θ + 4 cos θ = 2.
R 形式可将 a sin θ + b cos θ 表达为 R sin(θ + α) 或 R cos(θ − α)，其中 R = √(a² + b²)。这在求最大值和最小值，或者解 3 sin θ + 4 cos θ = 2 这类方程时特别有用。
a sin θ + b cos θ = R sin(θ + α), R = √(a² + b²)
To find α, compare coefficients: R cos α = a and R sin α = b, so tan α = b/a. Make sure the quadrant of α is consistent with the signs of a and b, not just the calculator value of arctan(b/a).
求 α 时，可比较系数：R cos α = a，R sin α = b，因此 tan α = b/a。要注意 α 所在象限应与 a 和 b 的符号一致，而不能只依赖计算器给出的 arctan(b/a) 值。
For 3 sin θ + 4 cos θ, R = √(3² + 4²) = 5. The expression can be written as 5 sin(θ + 53.13°) or, in radians, 5 sin(θ + 0.927). The maximum value is therefore 5 and the minimum is −5.
对于 3 sin θ + 4 cos θ，R = √(3² + 4²) = 5。该表达式可以写成 5 sin(θ + 53.13°)，或用弧度表示 5 sin(θ + 0.927)。因此它的最大值为 5，最小值为 −5。
3 sin θ + 4 cos θ = 5 sin(θ + 0.927)
8. Proving Trigonometric Identities | 证明三角恒等式
In proof questions, you are usually asked to show that one side of an identity is equal to the other. Always start from the more complicated side and transform it step by step. Common strategies include changing everything to sine and cosine, using Pythagorean identities, and factoring
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导