📚 Trigonometric Functions | 三角函数
Trigonometric functions are a cornerstone of Edexcel A-Level Mathematics, extending far beyond right-angled triangle ratios into the study of periodic behaviour, wave modelling, and advanced calculus. A deep understanding of radian measure, the unit circle, function transformations, and identities is essential for success in Pure Mathematics and applied topics.
三角函数是 Edexcel A-Level 数学的核心内容，它不仅限于直角三角形中的边比，还延伸至周期现象研究、波动建模和高等微积分。对弧度制、单位圆、函数变换及恒等式的透彻理解，是学好纯数学与应用专题的关键。
1. Radian Measure and the Unit Circle | 弧度制与单位圆
n arc equal in length to the radius. This leads to the fundamental conversion: π radians = 180°, so 2π radians correspond to a full turn. The use of radians is essential in calculus, where derivative formulas for sine and cosine require the argument to be in radians.
弧度制为弧长、角度和半径之间的联系提供了一种自然方式。1 弧度定义为半径长度所对应的圆心角。由此得出基本换算：π 弧度 = 180°，因此 2π 弧度对应一整圈。在微积分中应用弧度制至关重要，因为正弦和余弦的导数公式要求自变量以弧度为单位。
On the unit circle (radius 1), any angle θ measured anticlockwise from the positive x-axis determines a point with coordinates (cos θ, sin θ). As θ varies, cos θ and sin θ trace the horizontal and vertical displacements, giving immediate geometric meaning to these functions.
在单位圆（半径为 1）中，任意从正 x 轴逆时针测量的角 θ 可确定一点，其坐标为 (cos θ, sin θ)。随着 θ 变化，cos θ 和 sin θ 描绘出水平和竖直位移，为这两个函数赋予了直接的几何意义。
2. Graphs of Sine, Cosine and Tangent | 正弦、余弦和正切函数的图像
The sine curve y = sin θ has a smooth wave shape, with period 2π and range [– The cosine curve y = cos θ is a horizontal shift of the sine curve: cos θ = sin(θ + π/2).
正弦曲线 y = sin θ 呈平滑波浪形，周期为 2π，值域为 [–1, 1]。它在 θ 的整数倍 π 处穿过 θ 轴，并在 π/2 + 2kπ 和 3π/2 + 2kπ 处分别达到最大值与最小值。余弦曲线 y = cos θ 是正弦曲线的水平平移：cos θ = sin(θ + π/2)。
al numbers. The graph rises sharply near asymptotes and passes through the origin with gradient 1.
正切函数 y = tan θ 的周期为 π，在 cos θ = 0 处即 θ = π/2 + kπ 存在竖直渐近线。其值域为全体实数。图像在渐近线附近急剧上升，并以斜率为 1 穿过原点。
3. Periodicity and Symmetry Properties | 周期性与对称性
Trigonometric functions are inherently periodic: sin(θ + 2π) = sin θ, cos(θ + 2π) = cos θ, and tan(θ + π) = tan θ. These identities underpin the repetition of their graphs and are vital when solving equations over large intervals.
三角函数具有固有的周期性： sin(θ + 2π) = sin θ，cos(θ + 2π) = cos θ，tan(θ + π) = tan θ。 这些恒等式支撑了图像的重现性，在解较大区间内的方程时至关重要。
Even-odd properties simplify many problems: cos(–θ) = cos θ (even), while sin(–θ) = –sin θ and tan(–θ) = –tan θ (odd). Supplementary and complementary angle properties, such as sin(π – θ) = sin θ and cos(π – θ) = –cos θ, help relate angles in different quadrants.
奇偶性质能简化许多问题：cos(–θ) = cos θ（偶函数），而 sin(–θ) = –sin θ 和 tan(–θ) = –tan θ（奇函数）。补角和余角性质，例如 sin(π – θ) = sin θ 和 cos(π – θ) = –cos θ，有助于联系不同象限的角。
4. Transformations of Trigonometric Graphs | 三角函数的图像变换
nction of the form y = a sin(bθ + c) + d or y = a cos(bθ + c) + d, the parameters control:
- Amplitude: |a| (vertical stretch)
- Period: 2π/|b| for sine and cosine; π/|b| for tangent
- Phase shift: –c/b (horizontal translation)
- Vertical shift: d
These transformations can be combined, and the order of operations (e.g., stretch then translate) matters when interpreting the shift from the standard position.
理解图像变换对建模与草图绘制至关重要。对于 y = a sin(bθ + c) + d 或 y = a cos(bθ + c) + d 形式的函数，各参数控制：
- 振幅：|a|（纵向拉伸）
- 周期：正弦与余弦为 2π/|b|；正切为 π/|b|
- 相位移：–c/b（水平平移）
- 纵向位移：d
这些变换可以组合，且在解读从标准位置开始的位移时，运算顺序（例如先拉伸再平移）至关重要。
For y = a tan(bθ + c) + d, the period becomes π/|b| ift accordingly. When sketching, find the new central point and asymptote positions first.
对于 y = a tan(bθ + c) + d，周期变为 π/|b|，渐近线也相应平移。绘制草图时，应先确定新的中心点与渐近线位置。
5. Inverse Trigonometric Functions | 反三角函数
To define inverse functions, trigonometric functions must be restricted to principal domains where they are one-to-one. The standard restrictions are:
- y = arcsin x: domain [–1, 1], range [–π/2, π/2]
- y = arccos x: domain [–1, 1], range [0, π]
- y = arctan x: domain ℝ, range (–π/2, π/2)
in⁻¹ x, cos⁻¹ x, tan⁻¹ x is also commonly used, but must not be confused with reciprocals.
要定义反函数，必须将三角函数限制在其为一一对应的主值区间内。标准的限制为：
- y = arcsin x：定义域 [–1, 1]，值域 [–π/2, π/2]
- y = arccos x：定义域 [–1, 1]，值域 [0, π]
- y = arctan x：定义域 ℝ，值域 (–π/2, π/2)
它们分别是相应限定图像关于直线 y = x 的反射。符号 sin⁻¹ x、cos⁻¹ x、tan⁻¹ x 也常被使用，但切勿与倒数混淆。
6. Reciprocal Trigonometric Functions | 倒数三角函数
n θ. Their graphs feature vertical asymptotes where the original sine, cosine, or tangent are zero, and they are useful in identities and calculus.
另有三个函数被定义为倒数：sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ = cos θ/sin θ。它们的图像在原正弦、余弦或正切为零处出现竖直渐近线，这些函数在恒等式变换与微积分中十分有用。
s of y = cosec θ and y = cot θ similarly show periodic behaviour with vertical asymptotes.
例如，y = sec θ 在 cos θ = 1 处最小值为 1，并在 cos θ 的零点附近趋于 ±∞。y = cosec θ 和 y = cot θ 的图形同样表现出具有竖直渐近线的周期行为。
7. Trigonometric Identities | 三角恒等式
The Pythagorean identities stem from the unit circle:
sin²θ + cos²θ ≡ 1
1 + tan²θ ≡ sec²θ
1 + cot²θ ≡ cosec²θ
Dividing the base identity by cos²θ or sin²θ yields the others. These are used to simplify expressions, prove equivalences, and solve equations.
源自单位圆的毕达哥拉斯恒等式有：sin²θ + cos²θ ≡ 1，以及由此除以 cos²θ 或 sin²θ 得到的 1 + tan²θ ≡ sec²θ 和 1 + cot²θ ≡ cosec²θ。这些恒等式用于化简表达式、证明等价关系和解方程。
Compound angle formulas are key Pure Mathematics tools:
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)
From these, double-angle formulas emerge: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ, and tan 2θ = 2 tan θ / (1 – tan²θ).
和差角公式是纯数学的关键工具：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)。由此可导出倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ，以及 tan 2θ = 2 tan θ / (1 – tan²θ)。
8. Solving Trigonometric Equations | 解三角方程
The key to solving trigonometric equations over a given interval is to use periodicity and symmetry to find all solutions. After obtaining a principal solution (e.g., using inverse functions or known exact values), additional solutions are generated by considering the CAST diagram or the graph of the function.
求解给定区间内的三角方程的关键，是利用周期性和对称性找出所有解。在得到主值解（例如通过反函数或已知的精确值）之后，借助 CAST 图或函数图像可产生其余解。
ple, to solve 2 sin θ = 1 for 0 ≤ θ < 2π: sin θ = 1/2 gives principal θ = π/6. Since sin(π – θ) = sin θ, θ = π – π/6 = 5π/6 is also a solution. Quadratic forms like 2cos²θ – cos θ – 1 = 0 are treated by substituting x = cos θ, solving for x, then finding θ.
例如，解 2 sin θ = 1（0 ≤ θ < 2π）：sin θ = 1/2 得主值 θ = π/6；由于 sin(π – θ) = sin θ，θ = π – π/6 = 5π/6 也是解。处理二次型如 2cos²θ – cos θ – 1 = 0 时，可令 x = cos θ，解关于 x 的方程，再求 θ。
When solving equations involving sec, cosec, or cot, it is often best to rewrite in terms of sine and cosine first. Always check the domain and ensure that solutions lie within the specified interval; answers are expected in radians unless degrees are specified.
当解含有 sec、cosec 或 cot 的方程时，通常最好先转化为正弦和余弦。务必检验定义域并确保解落在指定区间内；除非题目指定使用度数，否则答案应以弧度给出。
9. Small Angle Approximations | 小角度近似
When an angle is measured in radians and is small, the following approximations hold:
sin θ ≈ θ
cos θ ≈ 1 – θ²/2
tan θ ≈ θ
These are derived from the Maclaurin series expansions, neglecting terms of order θ³ and higher. They are particularly useful in mechanics and when evaluating limits of rational expressions involving trigonometric functions.
当角度以弧度为单位且很小时，有以下近似公式：sin θ ≈ θ，cos θ ≈ 1 – θ²/2，tan θ ≈ θ。这些近似源自麦克劳林级数展开，并忽略了 θ³ 及更高阶项。它们在力学和处理含有三角函数的极限时特别有用。
For instance, limθ→0 ll θ, or by rewriting as 3·(sin 3θ)/(3θ). The approximation allows quick estimation and error bounds when the angle is known to be sufficiently small.
例如，利用当 θ 很小时 sin 3θ ≈ 3θ，可得 limθ→0 (sin 3θ)/θ = 3，或通过变形为 3·(sin 3θ)/(3θ) 求解。当已知角度足够小时，近似公式可实现快速估算并给出误差范围。
10. Applications to Modelling | 建模应用
phenomena can be modelled using sine and cosine functions. The general model y = a sin(b(t – tides, temperature variations, pendulum motion, and AC voltages. Here, t represents time, |a| is the amplitude, 2π/|b| is the period, (c, d) is a point of the mean level crossing.
许多现实世界中的周期现象都可以用正弦或余弦函数建模。通用模型 y = a sin(b(t – c)) + d（或余弦等价形式）可描述潮汐、温度变化、单摆运动和交变电压。其中 t 表示时间，|a| 为振幅，2π/|b| 为周期，(c, d) 为平均水位穿越点。
To fit a model to data, identify the amplitude from the vertical range, the period from the time interval of one complete cycle, the vertical shift as the mean of maximum and minimum values, and the phase shift by substituting a known coordinate. Solving trigonometric equations within the model answers practical questions such as “when does the tide reach a certain height?”.
在数据拟合模型时，可从纵向范围确定振幅，从一个完整周期的时间间隔确定周期，纵向位移为最大值与最小值的平均值，相位移则通过代入已知坐标求得。解模型中的三角方程可以回答诸如“潮汐何时达到某一高度？”的实际问题。
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