IB Mathematics: Core Concepts of Basic Trigonometry | IB数学：基础三角学核心概念

📚 IB Mathematics: Core Concepts of Basic Trigonometry | IB数学：基础三角学核心概念

Trigonometry is a fundamental branch of mathematics that studies relationships between angles and sides of triangles. In the IB Mathematics curriculum, a solid grasp of basic trigonometry is essential for solving problems in geometry, calculus, and even physics. This article covers the core concepts you need to master, from angle measurement to trigonometric identities and applications.

三角学是数学中研究角度与三角形边之间关系的基础分支。在IB数学课程中，扎实掌握基础三角学对于解决几何、微积分乃至物理中的问题至关重要。本文将涵盖你需要掌握的核心概念，从角度测量到三角恒等式及应用。

1. Degrees and Radians | 角度制与弧度制

Angles can be measured in degrees or radians. One full revolution equals 360 degrees, which is equivalent to 2π radians. The radian is the standard unit in advanced mathematics because it simplifies calculus formulas. To convert, use: 180° = π rad.

角度可以用度或弧度来度量。一整圈等于360度，相当于2π弧度。弧度是高等数学中的标准单位，因为它简化了微积分公式。换算关系为：180° = π 弧度。

角度转弧度：弧度 = 角度 × π / 180

弧度转角度：角度 = 弧度 × 180 / π

Common angles: 0°, 30°, 45°, 60°, 90° correspond to 0, π/6, π/4, π/3, π/2 radians. You should memorize these conversions instantly.

常见角度：0°、30°、45°、60°、90°分别对应0、π/6、π/4、π/3、π/2弧度。你应该能立即记住这些换算。

2. Right-Triangle Definitions | 直角三角形中的定义

For a right triangle with an acute angle θ, the six trigonometric ratios are defined using the opposite side (opp), adjacent side (adj), and hypotenuse (hyp):

对于含锐角θ的直角三角形，六个三角比使用对边、邻边和斜边定义：

sin θ = opp / hyp
cos θ = adj / hyp
tan θ = opp / adj = sin θ / cos θ

The reciprocal functions are: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. These definitions are valid for acute angles in a right triangle.

倒数函数为：csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。这些定义在直角三角形中对锐角成立。

3. The Unit Circle | 单位圆

The unit circle is a circle of radius 1 centered at the origin. For any angle θ, the point P(cos θ, sin θ) lies on the circle. This extends the definitions of trigonometric functions to all real angles, not just acute ones.

单位圆是半径为1、圆心在原点的圆。对于任意角θ，点P(cos θ, sin θ)位于圆上。这将三角函数的定义扩展到所有实数角度，而不仅仅是锐角。

Using the unit circle, we can determine the signs of functions in each quadrant. Quadrant I: all positive; Quadrant II: sin positive; Quadrant III: tan positive; Quadrant IV: cos positive. A common mnemonic is “ASTC” or “All Students Take Calculus.”

利用单位圆，我们可以确定各象限中函数的符号。第一象限：全正；第二象限：sin为正；第三象限：tan为正；第四象限：cos为正。常用口诀”ASTC”或”All Students Take Calculus”来记忆。

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

You must know the exact values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90°. These values appear frequently in IB exams and are derived from equilateral and isosceles right triangles.

你必须掌握0°、30°、45°、60°和90°的正弦、余弦和正切精确值。这些值在IB考试中频繁出现，可由等边三角形和等腰直角三角形推导。

θ	0°	30°	45°	60°	90°
sin θ	0	½	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	½	0
tan θ	0	√3/3	1	√3	undefined

Notice that sin θ increases from 0 to 1, cos θ decreases from 1 to 0, and tan θ is undefined at 90° because cos 90° = 0.

注意sin θ从0增加到1，cos θ从1减小到0，tan θ在90°处无定义，因为cos 90° = 0。

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

Trigonometric identities are equations that hold for all values of the variable. The most important one is the Pythagorean identity:

三角恒等式是对变量所有取值都成立的等式。最重要的是毕达哥拉斯恒等式：

sin² θ + cos² θ = 1

Dividing by cos² θ gives: 1 + tan² θ = sec² θ. Dividing by sin² θ gives: cot² θ + 1 = csc² θ. These identities are used to simplify expressions and prove other identities.

两边除以cos² θ得：1 + tan² θ = sec² θ。两边除以sin² θ得：cot² θ + 1 = csc² θ。这些恒等式用于化简表达式和证明其他恒等式。

Also remember the negative angle identities: sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ. These reflect the odd/even symmetry of the functions.

还要记住负角恒等式：sin(−θ) = −sin θ，cos(−θ) = cos θ，tan(−θ) = −tan θ。这些反映了函数的奇偶对称性。

6. Graphs of Trigonometric Functions | 三角函数图像

The graph of y = sin x is a wave that oscillates between −1 and 1 with period 2π. The graph of y = cos x has the same shape but is shifted left by π/2. The graph of y = tan x has vertical asymptotes at x = π/2 + nπ and period π.

y = sin x的图像是在−1和1之间振荡的波形，周期为2π。y = cos x的图像形状相同，但向左平移π/2。y = tan x的图像在x = π/2 + nπ处有垂直渐近线，周期为π。

Amplitude (for sin and cos): half the distance between maximum and minimum values.
Period: the smallest positive interval after which the graph repeats.
Phase shift: horizontal translation of the graph.

振幅（对sin和cos）：最大值与最小值之间距离的一半。
周期：图像重复的最小正间隔。
相位移动：图像的水平平移。

For y = a sin(bx + c) + d, the amplitude is |a|, the period is 2π/|b|, the phase shift is −c/b, and the vertical shift is d.

对于y = a sin(bx + c) + d，振幅为|a|，周期为2π/|b|，相位移动为−c/b，垂直移动为d。

7. Solving Trigonometric Equations | 解三角方程

Solving trigonometric equations involves finding all angles that satisfy the equation within a given domain. A basic method is to isolate the function and then use reference angles and symmetry.

解三角方程涉及在给定定义域内找到所有满足方程的角。基本方法是分离出函数，然后利用参考角和对称性。

For example, solve sin θ = ½ for 0° ≤ θ < 360°. The reference angle is 30°. Since sin is positive in quadrants I and II, the solutions are θ = 30° and θ = 150°.

例如，在0° ≤ θ < 360°范围内解sin θ = ½。参考角为30°。由于sin在第一、二象限为正，解为θ = 30°和θ = 150°。

For more complex equations like sin² θ = ½, use the square root property and consider both positive and negative roots: sin θ = ±√2/2. Then list all solutions in the domain.

对于更复杂的方程如sin² θ = ½，使用平方根性质并同时考虑正负根：sin θ = ±√2/2。然后列出定义域内的所有解。

8. Sine and Cosine Rules | 正弦定理与余弦定理

For any triangle (not necessarily right-angled), the sine rule and cosine rule relate side lengths to angles. These are essential in solving non-right triangles.

对于任意三角形（不一定是直角三角形），正弦定理和余弦定理将边长与角联系起来。它们在解非直角三角形时至关重要。

正弦定理：a / sin A = b / sin B = c / sin C

余弦定理：c² = a² + b² − 2ab cos C

The sine rule is useful when you know two angles and one side (AAS) or two sides and a non-included angle (SSA). The cosine rule is used when you know three sides (SSS) or two sides and the included angle (SAS).

正弦定理适用于已知两角一边（AAS）或两边及一对角（SSA）的情况。余弦定理适用于已知三边（SSS）或两边及其夹角（SAS）的情况。

When using the sine rule with SSA, watch for the ambiguous case: there may be two possible triangles, one triangle, or no triangle.

使用正弦定理处理SSA时，注意歧义情况：可能有两个三角形、一个三角形或没有三角形。

9. Area of a Triangle | 三角形的面积

The standard area formula ½ × base × height works for right triangles, but for other triangles we can use the formula involving two sides and the included angle:

标准面积公式½ × 底 × 高适用于直角三角形，但对于其他三角形，我们可以使用包含两边及其夹角的公式：

面积 = ½ ab sin C

Here, a and b are two sides of the triangle, and C is the angle between them. This formula is especially useful when the height is not easily determined.

这里a和b是三角形的两条边，C是它们之间的夹角。当高不易确定时，这个公式特别有用。

Another powerful formula is Heron’s formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a + b + c)/2 is the semi-perimeter. It works for any triangle when all three sides are known.

另一个强大的公式是海伦公式：面积 = √[s(s−a)(s−b)(s−c)]，其中s = (a + b + c)/2是半周长。当已知三边时适用于任何三角形。

10. Inverse Trigonometric Functions | 反三角函数

The inverse functions of sin, cos, and tan are denoted sin⁻¹, cos⁻¹, and tan⁻¹ (or arcsin, arccos, arctan). They return an angle for a given ratio. For example, sin⁻¹(0.5) = 30° (in the principal range).

sin、cos和tan的反函数分别记作sin⁻¹、cos⁻¹和tan⁻¹（或arcsin、arccos、arctan）。它们对给定的比值返回角度。例如，sin⁻¹(0.5) = 30°（在主值范围内）。

IB usually restricts the outputs of inverse functions to specific intervals to make them single-valued:

IB通常将反函数的输出限制在特定区间内以保证单值性：

arcsin x: −90° ≤ θ ≤ 90° (or −π/2 ≤ θ ≤ π/2)
arccos x: 0° ≤ θ ≤ 180° (or 0 ≤ θ ≤ π)
arctan x: −90° < θ < 90° (or −π/2 < θ < π/2)

Remember that arcsin(arccos) and arccos(arcsin) compositions require careful consideration of domains.

记住，arcsin和arccos等复合函数需要考虑定义域。

11. Applications: Bearings and Elevation | 应用：方位角与仰角

Trigonometry is applied in real-world contexts such as bearings, angles of elevation and depression, and periodic phenomena. A bearing is an angle measured clockwise from north, usually written as three digits (e.g., 045°).

三角学应用于现实场景，如方位角、仰角和俯角以及周期现象。方位角是从正北方向顺时针测量的角度，通常用三位数表示（如045°）。

For an observer looking at an object above, the angle of elevation is the angle between the horizontal line and the line of sight. If the object is below horizontal, it is the angle of depression.

当观察者看向上方的物体时，仰角是水平线与视线之间的夹角。如果物体在水平线下方，则为俯角。

Example: A tower casts a shadow of 40 m when the sun’s elevation angle is 35°. The height h of the tower satisfies tan 35° = h/40, so h = 40 × tan 35° ≈ 28 m.

示例：当太阳仰角为35°时，一座塔投下40米的影子。塔的高度h满足tan 35° = h/40，因此h = 40 × tan 35° ≈ 28米。

12. Common Pitfalls and Exam Tips | 常见易错点与考试技巧

Students often make mistakes by mixing degrees and radians on their calculator. Always check the calculator mode before solving a problem with a specific unit.

学生常因计算器上混用角度制和弧度制而出错。在解决特定单位的问题前，务必检查计算器的模式。

Another common error is forgetting the periodic nature of trigonometric equations. For example, sin θ = 0 has solutions θ = 0°, 180°, 360°, … not just θ = 0°.

另一个常见错误是忘记三角方程的周期性。例如，sin θ = 0的解为θ = 0°, 180°, 360°, …，而不仅仅是θ = 0°。

Always draw a unit circle or graph to visualize solutions.
When using the sine rule, check if the ambiguous case applies.
Use exact values for special angles instead of decimals when required.

始终画单位圆或图像来可视化解。
使用正弦定理时，检查是否适用歧义情况。
当需要精确值时，使用特殊角的精确值而不是小数。

Finally, practice converting between the different forms of identities and apply them flexibly in proofs and problem-solving. Mastery of these core concepts will build a strong foundation for more advanced topics such as calculus and complex numbers.

最后，练习在不同形式的恒等式之间转换，并在证明和解题中灵活应用。掌握这些核心概念将为微积分和复数等更高级主题打下坚实基础。

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IB Mathematics: Core Concepts of Basic Trigonometry | IB数学：基础三角学核心概念