📚 Trigonometric Ratios | 三角比
Trigonometric ratios provide the tools to relate angles and side lengths in triangles, forming a bridge between geometry and algebra. At A-Level, you will extend these ratios from acute angles in right triangles to any real number via the unit circle, explore their reciprocal forms, manipulate identities, and solve equations that model periodic behaviour. A thorough grasp of sine, cosine, tangent and their graphs is essential for calculus, mechanics and further pure mathematics.
三角比建立了三角形中角度与边长之间的关系，是连接几何与代数的桥梁。在 A-Level 阶段，你需要将这些比从直角三角形中的锐角推广到任意实数角（通过单位圆），掌握它们的倒数形式，灵活运用恒等式，并求解描述周期现象的方程。透彻理解正弦、余弦、正切及其图像，对微积分、力学和进阶纯数学都至关重要。
1. Understanding Trigonometric Ratios | 认识三角比
A trigonometric ratio is the quotient of two side lengths in a right-angled triangle for a given acute angle. The three primary ratios – sine (sin), cosine (cos) and tangent (tan) – are functions that map an angle to a real number. Beyond right triangles, these ratios are defined on the Cartesian plane using the coordinates of points on a circle, which allows angles of any magnitude and even negative angles to be handled consistently.
三角比是直角三角形中给定锐角的两条边长之比。三个基本比——正弦 (sin)、余弦 (cos) 和正切 (tan)——都是将角度映射为实数的函数。在直角三角形之外，这些比可以通过圆上点的坐标在笛卡尔平面上定义，从而能够统一处理任意大小的角度乃至负角。
2. Right-Triangle Definitions (SOH CAH TOA) | 直角三角形定义 (SOH CAH TOA)
For an acute angle θ in a right-angled triangle, label the side opposite θ as 'opposite', the side next to θ (not the hypotenuse) as 'adjacent', and the longest side as 'hypotenuse'. Then:
sinθ = opposite / hypotenuse
cosθ = adjacent / hypotenuse
tanθ = opposite / adjacent
The mnemonic 'SOH CAH TOA' helps recall these definitions. It is important to note that these ratios depend only on the angle and not on the size of the triangle, because all right triangles with the same acute angle are similar.
对于直角三角形中的锐角 θ，将 θ 的对边记作“对边”，与 θ 相邻的非斜边记作“邻边”，最长的边记作“斜边”。于是有：
sinθ = 对边 / 斜边
cosθ = 邻边 / 斜边
tanθ = 对边 / 邻边
记忆口诀“SOH CAH TOA”可帮助回想这些定义。需要注意的是，这些比值只与角度有关，而与三角形的大小无关，因为所有含相同锐角的直角三角形都是相似的。
3. The Unit Circle Definition | 单位圆定义
with anticlockwise taken as positive. The point where the terminal arm meets the circle has coordinates (cosθ, sinθ). Consequently, tanθ = sinθ / cosθ (provided cosθ ≠ 0). This definition handles angles greater than 90°, up to 360° and beyond, as well as negative angles.
为了将三角比扩展到 0° 到 90° 之外，考虑一个以原点为圆心、半径为 1 的圆。角度 θ 从正 x 轴开始度量，逆时针方向为正。终边与单位圆的交点坐标为 (cosθ, sinθ)。因此，tanθ = sinθ / cosθ（前提是 cosθ ≠ 0）。这个定义可以处理大于 90° 的角、直到 360° 乃至更大的角，以及负角。
Using the unit circle, we see that sinθ and cosθ are the y- and x-coordinates, so they oscillate between −1 and 1. The tangent ratio, being a quotient, can take any real value.
利用单位圆，我们可以看到 sinθ 和 cosθ 分别是 y 坐标和 x 坐标，因此它们的值在 −1 和 1 之间振荡。正切比作为商，可以取任意实数值。
4. Radian Measure | 弧度制
Angles in A-Level mathematics are frequently given in radians rather than degrees. One radian is the angle subtended at the centre of a circle by an arc length equal to the radius. The conversion is:
π radians = 180°
rees to radians, multiply by π/180. Key radian measures to memorise: π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°), π (180°), 2π (360°). Radian mode must be used in calculus involving trigonometric functions.
在 A-Level 数学中，角度常用弧度而非度来表示。1 弧度是指当弧长等于半径时，圆心角的大小。换算关系为：
π 弧度 = 180°
因此，1 弧度 ≈ 57.3°。将角度乘以 π/180 即可转换为弧度。需要熟记的关键弧度值：π/6 (30°)、π/4 (45°)、π/3 (60°)、π/2 (90°)、π (180°)、2π (360°)。在涉及三角函数的微积分运算中，必须使用弧度制。
5. Exact Trigonometric Values | 精确三角值
Certain angles produce simple, exact values that can be derived from isosceles right triangles (45°) and equilateral triangles (30° and 60°). These values must be known by heart, as they appear frequently without calculator use.
| Angle θ (degrees) | Angle θ (radians) | sinθ | cosθ | tanθ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | 1/√2 = √2/2 | 1/√2 = √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
rpin many algebraic simplifications and are essential for solving equations such as sinθ = √3/2 without a calculator. You should be able to reproduce the two special triangles (45°-45°-90° and 30°-60°-90°) that generate these ratios.
这些精确值支撑着大量代数化简，也是不借助计算器求解诸如 sinθ = √3/2 等方程的基础。你应当能够画出产生这些比值的两个特殊三角形（45°-45°-90° 和 30°-60°-90°）。
6. Signs of Trigonometric Ratios and ASTC | 三角比的符号与 ASTC
The sign of a trigonometric ratio depends on the quadrant in which the terminal arm of angle θ lies. The mnemonic 'All Stations To Central' or 'All Silver Tea Cups' (ASTC) tells us which ratios are positive in each quadrant:
- Quadrant I (0° to 90°): All (sin, cos, tan) positive.
- Quadrant II (90° to 180°): Only Sine positive.
- Quadrant III (180° to 270°): Only Tangent positive.
- Quadrant IV (270° to 360°): Only Cosine positive.
The signs can be remembered by counting anti-clockwise: A (all), S (sin), T (tan), C (cos). For negative angles, move clockwise. This rule is crucial when solving trigonometric equations with multiple solutions.
三角比的正负号取决于角 θ 终边所在的象限。助记口诀“ASTC”或“All Silver Tea Cups”告诉我们各象限中哪些比为正：
- 第一象限 (0° 至 90°): 所有 (sin, cos, tan) 都为正。
- 第二象限 (90° 至 180°): 只有 正弦 为正。
- 第三象限 (180° 至 270°): 只有 正切 为正。
- 第四象限 (270° 至 360°): 只有 余弦 为正。
按照逆时针方向可记为 A (全部)、S (正弦)、T (正切)、C (余弦)。对于负角，沿顺时针方向判断。这一规则在求解具有多个解的三角方程时至关重要。
7. Reciprocal Ratios: sec θ, cosec θ, cot θ | 倒数比：正割、余割、余切
The three reciprocal trigonometric functions are defined as:
sec θ ≡ 1 / cos θ
cosec θ ≡ 1 / sin θ
cot θ ≡ 1 / tan θ ≡ cos θ / sin θ
Note that 'cosec' is sometimes abbreviated as 'csc' 90°, 270°, etc. cot θ is undefined when sin θ = 0, i.e. at 0°, 180°, etc. Their ranges are |sec θ| ≥ 1, |cosec θ| ≥ 1, while cot θ can take any real value.
三个倒数三角函数定义如下：
sec θ ≡ 1 / cos θ
cosec θ ≡ 1 / sin θ
cot θ ≡ 1 / tan θ ≡ cos θ / sin θ
注意 'cosec' 有时也缩写为 'csc'。这些函数的定义域受到分母不能为零的限制。例如，当 cos θ = 0 时 sec θ 无定义，即在 θ = 90°、270° 等处；当 sin θ = 0 时 cot θ 无定义，即在 0°、180° 等处。它们的值域为 |sec θ| ≥ 1，|cosec θ| ≥ 1，而 cot θ 可以取任意实数。
8. Pythagorean Identities | 毕达哥拉斯恒等式
x² + y² = 1 and the definitions x = cos θ, y = sin θ, we obtain the fundamental identity:
sin²θ + cos²θ ≡ 1
Dividing this identity by cos²θ (cos θ ≠ 0) yields:
1 + tan²θ ≡ sec²θ
Dividing by sin²θ (sin θ ≠ 0) gives:
1 + cot²θ ≡ cosec²θ
These identities are used to simplify expressions, prove other identities, and solve trigonometric equations where substitution reduces an expression to a single ratio. When using them, remember to check for restrictions on θ that would make a denominator zero.
由单位圆方程 x² + y² = 1 以及定义 x = cos θ，y = sin θ，可得到基本恒等式：
sin²θ + cos²θ ≡ 1
将该式除以 cos²θ（cos θ ≠ 0）得到：
1 + tan²θ ≡ sec²θ
除以 sin²θ（sin θ ≠ 0）得到：
1 + cot²θ ≡ cosec²θ
这些恒等式用于化简表达式、证明其他恒等式，以及在解三角方程时通过代换将表达式化为单一比值的方程。使用时，注意检查那些会导致分母为零的 θ 取值。
9. Solving Basic Trigonometric Equations | 解基本三角方程
A typical equation like sin θ = k (where |k| exam, you are given an interval, for example 0 ≤ θ < 360° or 0 ≤ θ < 2π. The strategy is:
- Find the principal solution using inverse functions or exact values.
- Use the ASTC diagram to determine the quadrants where the ratio has the required sign.
- tan θ = k, add 180° (π rad) to the principal value for the second solution in the interval.
- Generate further solutions by adding or subtracting multiples of the period (360° or 2π for sin and cos, 180° or π for tan).
Always express your final answers in the required unit (degrees or radians) and check they lie within the given interval.
诸如 sin θ = k（其中 |k| ≤ 1）这样的方程，由于正弦具有周期性，会存在无穷多个解。在考试中，题目会给定一个区间，例如 0 ≤ θ < 360° 或 0 ≤ θ < 2π。解题策略为：
- 利用反函数或精确值求出主值。
- 使用 ASTC 图确定该比值为所需正负号的象限。
- 利用对称性：对于 sin θ = k，第二个解为 180° − θ（度）或 π − θ（弧度）；对于 cos θ = k，第二个解为 360° − θ（或 2π − θ）；对于 tan θ = k，在主值上加 180°（π 弧度）得到区间内的第二个解。
- 通过加减周期的整数倍（sin、cos 周期为 360° 或 2π，tan 周期为 180° 或 π）生成更多解。
答案务必以题目要求的单位（度或弧度）表示，并检查它们是否落在给定区间内。
10. Graphs of sin θ, cos θ and tan θ | sin θ、cos θ 和 tan θ 的图像
The graphs of the trigonometric functions reveal their periodic nature, symmetries and key features:
- tarts at 0, rises to 1 at π/2, goes back to 0 at π, to −1 at 3π/2, and returns to 0 at 2π.
- rts at 1 at θ=0, falls to 0 at π/2, to −1 at π, returns to 0 at 3π/2, and back to 1 at 2π.
- ptotes at θ = π/2 + nπ, where cos θ = 0. Passes through the origin and is an odd function.
Understanding these graphs helps in visualising solutions to equations and in sketching transformations.
三角函数的图像揭示了它们的周期性、对称性及主要特征：
- y = sin θ: 周期 2π (360°)，振幅 1，奇函数 [sin(−θ) = −sin θ]。起点为 0，在 π/2 处升至 1，π 处回到 0，3π/2 处降至 −1，2π 处回到 0。
- y = cos θ: 周期 2π，振幅 1，偶函数 [cos(−θ) = cos θ]。θ=0 时从 1 开始，π/2 处降至 0，π 处降至 −1，3π/2 处回到 0，2π 处回到 1。
- y = tan θ: 周期 π (180°)，无振幅（无界）。在 θ = π/2 + nπ 处有垂直渐近线（cos θ = 0）。过原点，是奇函数。
理解这些图像有助于直观地看出方程的解以及绘制图像变换。
11. Transformations of Trigonometric Graphs | 三角图像变换
y = a sin(bx + c) + d or y = a cos(bx + c) + d. The parameters have the following effects:
- a (amplitude): vertical stretch factor. The graph oscillates between −|a| and +|a|. Peak-to-peak height is 2|a|.
- b (frequency): horizontal stretch. The period becomes 2π / |b| (or 360° / |b|).
- c izontal translation by −c/b. For sin(bx + c), set bx + c = 0 to find the start of a cycle.
- d (vertical shift): the whole graph moves up by d. The centre line is y = d.
For tan(bx + c) + d, the period becomes π / |b|, and asymptotes shift accordingly. Transformations are often tested by asking you to sketch a graph, state its range, or find possible equations from a given graph.
一般的正弦或余弦函数可写作 y = a sin(bx + c) + d 或 y = a cos(bx + c) + d。各参数的影响如下：
- a（振幅）：垂直拉伸因子。图像在 −|a| 和 +|a| 之间振荡，峰峰高度为 2|a|。
- b（频率）：水平拉伸。周期变为 2π / |b|（或 360° / |b|）。
- c（相位移动）：水平平移 −c/b。对于 sin(bx + c)，令 bx + c = 0 可找到周期的起点。
- d（垂直移动）：整个图像上移 d。中心线为 y = d。
对于 tan(bx + c) + d，周期变为 π / |b|，渐近线也相应平移。图像变换的考察常要求你画出草图、说明值域，或从给定图像写出可能的函数表达式。
12. Trigonometric Ratios in Any Triangle | 任意三角形中的三角比
While the basic ratios are defined for right-angled triangles, trigonometry also extends to non-right triangles via the sine rule and cosine rule. These utilise the same sine and cosine ratios. For any triangle with sides a, b, c opposite angles A, B, C:
Sine Rule: a / sin A = b / sin B = c / sin C (or sin A / a = sin B / b = sin C / c)
Cosine Rule: a² = b² + c² − 2bc cos A
s case: if given two sides and a non-included angle, there may be two possible triangles (one acute and one obtuse). In such cases, use the fact that sin(180° − θ) = sin θ to check for a second valid solution.
虽然基本的三角比是针对直角三角形定义的，但三角学也通过正弦定理和余弦定理推广到了非直角三角形，它们同样使用
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