Basic Trigonometry | 基础三角学

📚 Basic Trigonometry | 基础三角学

Trigonometry is a branch of mathematics that explores the relationships between the angles and side lengths of triangles. From architecture to engineering and even music, trigonometric principles form the backbone of many real-world applications. In the IB Mathematics curriculum, mastering basic trigonometry means understanding angle measurement, trigonometric ratios in right-angled triangles, the unit circle, exact values of special angles, fundamental identities, graphing trigonometric functions, and solving simple trigonometric equations. This article will guide you through these essential concepts in a clear, dual-language format to support your revision.

三角学是数学的一个分支，研究三角形中角与边长之间的关系。从建筑到工程，甚至音乐，三角原理是许多现实应用的基础。在 IB 数学课程中，掌握基础三角学意味着要理解角度测量、直角三角形中的三角比、单位圆、特殊角的精确值、基本恒等式、三角函数的图像以及简单三角方程的求解。本文将以清晰的中英双语形式带你梳理这些核心概念，助力你的复习备考。

1. Angle Measurement: Degrees and Radians | 角度测量：度与弧度

Angles can be measured in degrees, where a full circle is 360°, or in radians, where a full circle is 2π rad. Radians are the SI unit for angle measure and are defined such that an arc length equal to the radius subtends an angle of 1 radian at the centre of a circle. The conversion factor is 180° = π rad. This relationship is essential because many advanced trigonometric formulas and calculus rely on radian measure.

角度可以用度来衡量，一个整圆为 360°，也可以用弧度来衡量，一个整圆为 2π 弧度。弧度是国际单位制的角度单位，其定义是当弧长等于半径时，该弧所对的圆心角为 1 弧度。转换因子是 180° = π 弧度。这一关系至关重要，因为许多高级三角公式和微积分都依赖于弧度制。

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For instance, 30° equals 30 × π/180 = π/6 rad. Conversely, π/3 rad equals π/3 × 180/π = 60°. Familiarity with common conversions allows you to switch seamlessly between the two systems.

角度转弧度，乘以 π/180；弧度转角度，乘以 180/π。例如，30° 等于 30 × π/180 = π/6 弧度。反过来，π/3 弧度等于 π/3 × 180/π = 60°。熟悉常见的换算能让你在两个体系之间自如切换。

30° = π/6 rad
45° = π/4 rad
60° = π/3 rad
90° = π/2 rad
180° = π rad
360° = 2π rad

这些常用换算关系应熟记于心。

2. Right-Angled Triangle Trigonometry | 直角三角形中的三角比

In a right-angled triangle, the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) are defined relative to one of the acute angles. The mnemonic SOH CAH TOA is a helpful reminder: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. These definitions only apply to right-angled triangles and provide a direct way to find unknown sides or angles.

在直角三角形中，正弦 (sin)、余弦 (cos) 和正切 (tan) 是针对其中一个锐角来定义的。记忆口诀 SOH CAH TOA 有助于记忆：正弦 = 对边/斜边，余弦 = 邻边/斜边，正切 = 对边/邻边。这些定义仅适用于直角三角形，为求解未知的边长或角度提供了直接方法。

If you know one acute angle and one side length, you can determine any other side using the appropriate ratio. For example, in a triangle with a 30° angle, if the hypotenuse is 10, the opposite side is 10 × sin(30°) = 5. Similarly, you can find an unknown acute angle by using the inverse trigonometric functions: θ = sin⁻¹(opp/hyp), cos⁻¹(adj/hyp), tan⁻¹(opp/adj).

如果已知一个锐角和一条边长，就可以通过相应的三角比求出其他边。例如，在一个含 30° 角的三角形中，若斜边长为 10，则对边为 10 × sin(30°) = 5。同样，利用反三角函数可以求出未知锐角：θ = sin⁻¹(对边/斜边)、cos⁻¹(邻边/斜边)、tan⁻¹(对边/邻边)。

3. The Unit Circle and Trigonometric Functions | 单位圆与三角函数

The unit circle is a circle of radius 1 centred at the origin of the coordinate plane. It extends trigonometric definitions to all real numbers, not just acute angles. For any angle θ measured anticlockwise from the positive x-axis, the coordinates of the point where the terminal side intersects the unit circle are (cos θ, sin θ). The tangent is then sin θ / cos θ, provided cos θ ≠ 0.

单位圆是以坐标原点为圆心、半径为 1 的圆。它将三角函数的定义拓展到全体实数，而不仅仅是锐角。对于从正 x 轴逆时针测量的任意角 θ，其终边与单位圆交点的坐标为 (cos θ, sin θ)。正切则为 sin θ / cos θ，前提是 cos θ ≠ 0。

This representation reveals the periodicity and symmetry of trigonometric functions. As θ increases beyond 90°, the signs of sine and cosine change according to the quadrant. In the first quadrant, both are positive; in the second, sine is positive, cosine negative; in the third, both negative; in the fourth, cosine positive, sine negative. The mnemonic “All Students Take Calculus” helps remember the sign patterns (All, Sine, Tangent, Cosine positive in QI, QII, QIII, QIV respectively).

这种表示法揭示了三角函数的周期性和对称性。当 θ 增大超过 90° 时，正弦和余弦的符号会根据象限而变化。第一象限两者均为正；第二象限正弦为正、余弦为负；第三象限两者均为负；第四象限余弦为正、正弦为负。记忆口诀 “All Students Take Calculus” 帮助记住各象限为正的函数（第一象限 All，第二象限 Sine，第三象限 Tangent，第四象限 Cosine）。

Using the unit circle, we can define sine and cosine for any angle, even negative ones by rotating clockwise. This forms the foundation for modelling periodic phenomena like waves and oscillations.

借助单位圆，我们可以对任意角（甚至通过顺时针旋转得到的负角）定义正弦和余弦。这为模拟波动和振动等周期性现象奠定了基础。

4. Exact Values of Special Angles | 特殊角的精确值

Certain angles appear so frequently that their exact sine, cosine, and tangent values should be memorised. These typically include 0°, 30°, 45°, 60°, and 90° (and their radian equivalents). The values can be derived from right-angled triangles: the 45° right triangle (isosceles) gives sin 45° = cos 45° = 1/√2 = √2/2; the 30°-60° right triangle gives sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, and so on.

一些角出现频率极高，它们的正弦、余弦和正切的精确值应当熟记。这些角通常包括 0°、30°、45°、60° 和 90°（及其弧度等价值）。这些值可以通过特殊直角三角形推导：45° 直角等腰三角形得出 sin 45° = cos 45° = 1/√2 = √2/2；30°-60° 直角三角形得出 sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 等。

Here is a summary table of exact values:

下表总结了这些精确值：

Angle (degrees)	Radians	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined

Knowing these values without a calculator is crucial for solving trigonometric equations and for acquiring a deeper number sense in trigonometry.

不用计算器而熟知这些值，对求解三角方程和培养三角学中的数感至关重要。

5. Fundamental Trigonometric Identities | 基本三角恒等式

Trigonometric identities are equations that hold true for all permissible values of the variable. The most fundamental is the Pythagorean identity: sin²θ + cos²θ = 1. This follows directly from the unit circle, as the x- and y-coordinates of any point on the circle satisfy x² + y² = 1. From this identity, dividing by cos²θ (when cos θ ≠ 0) gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ. Although secant, cosecant and cotangent are less emphasised in basic trigonometry, they appear in IB Mathematics.

三角恒等式是对变量所有允许取值都成立的等式。最基本的是毕达哥拉斯恒等式：sin²θ + cos²θ = 1。这直接源于单位圆，因为圆上任意点的坐标满足 x² + y² = 1。由该恒等式，除以 cos²θ（当 cos θ ≠ 0）可得 1 + tan²θ = sec²θ，除以 sin²θ 可得 1 + cot²θ = csc²θ。尽管正割、余割和余切在基础三角学中不那么强调，但它们会在 IB 数学中出现。

sin²θ + cos²θ = 1

上述恒等式是核心，务必牢记。

Another key identity is the quotient identity: tan θ = sin θ / cos θ. Also, the symmetry identities are helpful: sin(-θ) = -sin θ (odd function), cos(-θ) = cos θ (even function), and tan(-θ) = -tan θ. These identities simplify equations and aid in graphing.

另一个关键恒等式是商数关系：tan θ = sin θ / cos θ。此外，对称恒等式也很实用：sin(-θ) = -sin θ（奇函数），cos(-θ) = cos θ（偶函数），tan(-θ) = -tan θ。这些恒等式能简化方程并有助于绘图。

6. Graphs of Sine and Cosine Functions | 正弦与余弦函数的图像

The sine and cosine functions are periodic with period 2π (in radians) or 360°. Their graphs are smooth, wave-like curves called sinusoids. The standard form is f(x) = A sin(Bx + C) + D or f(x) = A cos(Bx + C) + D, where A represents amplitude, B affects period (Period = 2π/|B|), C causes phase shift (horizontal shift = -C/B), and D is the vertical shift. Understanding these transformations is critical for modelling and for interpreting real-world data.

正弦和余弦函数是周期函数，周期为 2π（弧度制）或 360°。它们的图像是光滑的波状曲线，称为正弦曲线。标准形式为 f(x) = A sin(Bx + C) + D 或 f(x) = A cos(Bx + C) + D，其中 A 代表振幅，B 影响周期（周期 = 2π/|B|），C 引起相位移（水平位移 = -C/B），D 为垂直位移。理解这些变换对于建模和解读实际数据至关重要。

For the basic y = sin x graph, the curve starts at (0,0), rises to a maximum of 1 at π/2, returns to 0 at π, reaches a minimum of -1 at 3π/2, and completes a full cycle at 2π. The y = cos x graph starts at (0,1), drops to 0 at π/2, reaches -1 at π, and returns to 1 at 2π. Both graphs oscillate between -1 and 1 and exhibit wave-like properties.

就基本的 y = sin x 图像而言，曲线始于 (0,0)，在 π/2 处升至最大值 1，在 π 处回到 0，在 3π/2 处达到最小值 -1，并在 2π 完成一个完整周期。y = cos x 的图像始于 (0,1)，在 π/2 处降为 0，在 π 处达到 -1，并在 2π 处回到 1。两幅图像均在 -1 与 1 之间振荡，并展现出波状特性。

7. Graph of the Tangent Function | 正切函数的图像

The tangent function, y = tan x, has a completely different appearance. Defined as sin x / cos x, it is periodic with period π (or 180°), not 2π. The graph has vertical asymptotes wherever cos x = 0, i.e., at x = π/2 + kπ for any integer k. This is because tan x becomes undefined at these points. The curve passes through the origin and increases from -∞ to +∞ between each pair of asymptotes, crossing the x-axis at integer multiples of π.

正切函数 y = tan x 的图像截然不同。由于定义为 sin x / cos x，它的周期是 π（或 180°），而非 2π。图像在 cos x = 0 处存在垂直渐近线，即 x = π/2 + kπ（k 为整数）。这是因为在这些点上 tan x 无定义。曲线通过原点，并在每一对渐近线之间从 -∞ 递增至 +∞，在 π 的整数倍处穿过 x 轴。

In the standard transformation y = A tan(Bx + C) + D, the period is π/|B|, and the vertical asymptotes are shifted accordingly. Tangent graphs are useful for modelling unbounded periodic behaviour, although they appear less frequently in elementary real-world scenarios than sine and cosine curves.

在标准变换 y = A tan(Bx + C) + D 中，周期为 π/|B|，垂直渐近线也相应平移。正切图像可用于模拟无界的周期性行为，尽管在初等实际场景中不如正余弦曲线常见。

8. Solving Simple Trigonometric Equations | 解简单三角方程

Trigonometric equations ask for the values of the variable that make a given trigonometric statement true. A simple equation like sin x = 0.5 has infinitely many solutions because sine is periodic. First, find the principal value, typically using inverse functions and a calculator or exact values. For sin x = 0.5, the principal values in the range 0 ≤ x < 2π are x = π/6 and x = 5π/6, because sine is positive in the first and second quadrants.

三角方程是求解使给定三角表达式成立的变量值。像 sin x = 0.5 这样的简单方程有无穷多个解，因为正弦是周期函数。首先，求出主值，通常使用反函数以及计算器或精确值。对于 sin x = 0.5，在 0 ≤ x < 2π 范围内的主值为 x = π/6 和 x = 5π/6，因为正弦在第一和第二象限为正。

To find the general solution, add integer multiples of the period. Thus for sin x = 0.5, the general solutions are x = π/6 + 2πk or x = 5π/6 + 2πk, k ∈ Z. For cosine equations, if cos x = a, principal solutions usually come in symmetric pairs around 0 or π, and general solution adds 2πk. For tangent equations, the period is π, so if tan x = b, then x = arctan(b) + πk. Always check the given domain to select the appropriate solutions.

为得出通解，需加上周期的整数倍。因此 sin x = 0.5 的通解为 x = π/6 + 2πk 或 x = 5π/6 + 2πk，k ∈ Z。对于余弦方程，若 cos x = a，主值通常以 0 或 π 为中心成对出现，通解则加 2πk。对于正切方程，周期为 π，因此若 tan x = b，则 x = arctan(b) + πk。务必根据给定定义域选取合适的解。

9. Angles of Elevation and Depression | 仰角与俯角

Basic trigonometry is frequently applied to problems involving angles of elevation and depression, often in right-angled triangle contexts. The angle of elevation is the angle above the horizontal when an observer looks up at an object. The angle of depression is the angle below the horizontal when an observer looks down. These angles are equal when the object and observer are on parallel lines of sight, due to alternate interior angles.

基础三角学常应用于涉及仰角和俯角的问题，通常出现在直角三角形背景下。仰角是指观察者抬头看物体时，视线与水平线之间的夹角。俯角则是观察者低头看物体时，视线与水平线之间的夹角。当物体和观察者处于平行视线上时，根据内错角相等原理，这两个角相等。

A typical problem: from the top of a cliff 50 m high, the angle of depression to a boat is 20°. To find the horizontal distance d from the base of the cliff, recognise that the angle of depression equals the angle of elevation from the boat, forming a right triangle where tan(20°) = 50/d. Solving gives d = 50 / tan(20°). Such exercises reinforce the practical use of trigonometric ratios.

一个典型问题：从 50 米高的悬崖顶部测得对一艘船的俯角为 20°。为求悬崖底部到船的水平距离 d，应意识到俯角等于从船处看悬崖的仰角，这构成了一个直角三角形，其中 tan(20°) = 50/d。解得 d = 50 / tan(20°)。这类练习强化了三角比的实际应用。

10. The Sine Rule and Cosine Rule (Basic Introduction) | 正弦定理与余弦定理（基础介绍）

While right-angled triangle trigonometry is the starting point, IB Mathematics also introduces the sine and cosine rules to handle non-right-angled triangles. The sine rule states: a/sin A = b/sin B = c/sin C, where sides a, b, c are opposite angles A, B, C respectively. It is used when we know either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA), though the latter requires careful checking for the ambiguous case.

尽管直角三角形是三角学的起点，IB 数学也会介绍正弦定理和余弦定理以处理非直角三角形。正弦定理指出：a/sin A = b/sin B = c/sin C，其中边 a、b、c 分别对应角 A、B、C。当已知两角一边（AAS 或 ASA）或两边及其一边的对角（SSA）时使用该定理，但后者需要小心验证是否存在多解情况。

The cosine rule connects sides and an included angle: c² = a² + b² – 2ab cos C. This is essentially an extension of the Pythagorean theorem and is useful when given two sides and the included angle (SAS) or when all three sides are known (SSS). Both rules expand the solver’s toolkit beyond right triangles and appear in many IB exam questions.

余弦定理将边长与夹角联系起来：c² = a² + b² – 2ab cos C。它本质上是毕达哥拉斯定理的延伸，在已知两边及其夹角（SAS）或已知三边（SSS）时非常有用。这两个定理将解题工具从直角三角形拓展到一般三角形，并在许多 IB 考题中出现。

11. Strategies and Common Mistakes | 解题策略与常见错误

When solving trigonometry problems, always sketch a diagram when possible. Label known sides and angles, and decide whether the problem involves right-angled definitions, the unit circle, or the sine/cosine rules. Check that your calculator is in the correct mode (degrees or radians) — a tiny oversight that often leads to completely wrong answers. Pay attention to the domain of solutions; a general solution set might need to be restricted to a specific interval.

在解三角问题时，尽可能画出图形。标出已知的边和角，并判断问题是涉及直角定义、单位圆还是正余弦定理。务必检查计算器是否处于正确的模式（角度制或弧度制）——这一小小的疏忽常常导致完全错误的答案。注意解的取值范围；通解可能需要限制在特定区间内。

Common pitfalls include: forgetting that sin⁻¹(x) returns only a principal value and missing other solutions; misapplying SOH CAH TOA when the triangle is not right-angled; and confusing amplitude with vertical shift in graph transformations. Regular practice with past paper questions will build confidence and accuracy.

常见误区包括：忘记 sin⁻¹(x) 只返回主值而遗漏其他解；在非直角三角形中误用 SOH CAH TOA；以及在图像变换中混淆振幅与垂直位移。定期练习历年真题将建立信心并提升准确性。

12. Summary and Practice Tips | 总结与练习建议

Basic trigonometry builds a framework that supports more advanced topics such as vectors, complex numbers, and calculus. Mastery requires moving comfortably between degrees and radians, memorising exact values, understanding the unit circle, and being fluent with identities and graph transformations. Start by reviewing the key definitions daily, then gradually increase difficulty by tackling equation-solving and application problems.

基础三角学构建了一个支撑更高级主题（如向量、复数和微积分）的框架。要精通它，需要自如地在角度与弧度之间转换，记住精确值，理解单位圆，并熟练运用恒等式和图像变换。建议先每天复习关键定义，然后逐步增加难度，解决方程求解和应用题。

Use active recall: draw the sine and cosine graphs from memory, derive the Pythagorean identity from the unit circle, and test yourself on special angle values. Work through IB-style questions that combine multiple concepts, as the exam often links trigonometry with algebra or geometry. With consistent effort, you will develop strong trigonometric intuition.

运用主动回忆法：凭记忆画出正弦和余弦图像，从单位圆推导毕达哥拉斯恒等式，并自测特殊角的值。练习将多个概念结合起来的 IB 风格题目，因为考试常将三角学与代数或几何结合。持之以恒，你将培养出牢固的三角直觉。

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Basic Trigonometry | 基础三角学