Trigonometric Functions: Definitions, Graphs, and Properties | 三角函数定义、图像与性质

📚 Trigonometric Functions: Definitions, Graphs, and Properties | 三角函数定义、图像与性质

Trigonometric functions are central to IB Mathematics, appearing in topics ranging from geometry and periodic phenomena to calculus and complex numbers. This article provides a comprehensive review of their definitions, graphs, and key properties, with a focus on exam-relevant details.

三角函数是IB数学的核心内容，广泛出现在几何、周期现象、微积分和复数等专题中。本文将系统梳理它们的定义、图像和关键性质，并重点突出与考试相关的细节。

1. Angle Measure: Degrees and Radians | 角度度量：度与弧度

Before defining trigonometric functions, we must understand angle measure. In IB Mathematics, radians are the preferred unit. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

在定义三角函数之前，我们必须理解角度的度量。在IB数学中，弧度是首选单位。一弧度是圆周上长度等于半径的弧所对的圆心角。

π radians = 180°

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Common exact values include π/6 = 30°, π/4 = 45°, π/3 = 60°, and π/2 = 90°.

将度转换为弧度，乘以π/180；将弧度转换为度，乘以180/π。常见精确值包括π/6 = 30°，π/4 = 45°，π/3 = 60°，π/2 = 90°。

2. Right-Triangle Definitions | 直角三角形定义

For an acute angle θ in a right-angled triangle, the six trigonometric ratios are defined using the opposite side (O), adjacent side (A), and hypotenuse (H).

对于直角三角形中的一个锐角θ，六个三角比使用对边(O)、邻边(A)和斜边(H)来定义。

sin θ = O/H

正弦 sin θ = 对边/斜边
cos θ = A/H

余弦 cos θ = 邻边/斜边
tan θ = O/A

正切 tan θ = 对边/邻边
csc θ = H/O

余割 csc θ = 斜边/对边
sec θ = H/A

正割 sec θ = 斜边/邻边
cot θ = A/O

余切 cot θ = 邻边/对边

These definitions are valid only for 0 < θ < π/2. To extend to all real angles, we use the unit circle.

这些定义仅对0 < θ < π/2有效。为了扩展到所有实数角，我们使用单位圆。

3. Unit Circle Definition | 单位圆定义

On a unit circle centred at the origin, a point P(x, y) corresponds to an angle θ measured from the positive x-axis. Then cos θ = x and sin θ = y. This gives a natural extension to all real values of θ.

在以原点为圆心的单位圆上，点P(x, y)对应从x轴正方向量起的角θ。于是cos θ = x，sin θ = y。这给出了对所有实数θ的自然扩展。

cos²θ + sin²θ = 1

Because the radius is 1, the coordinates satisfy the Pythagorean identity. The tangent is defined as tan θ = sin θ / cos θ, provided cos θ ≠ 0.

因为半径为1，坐标满足勾股恒等式。正切定义为tan θ = sin θ / cos θ，前提是cos θ ≠ 0。

Common unit circle values are grouped by quadrant. For example, at θ = π/3, the coordinates are (1/2, √3/2), so sin(π/3) = √3/2 and cos(π/3) = 1/2.

常见的单位圆值按象限分组。例如，在θ = π/3处，坐标为(1/2, √3/2)，因此sin(π/3) = √3/2，cos(π/3) = 1/2。

4. Fundamental Identities | 基本恒等式

Several identities are essential for simplifying expressions and solving equations in IB exams.

有若干恒等式对化简表达式和解方程至关重要，是IB考试中的必备工具。

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

cot²θ + 1 = csc²θ

Tangent and cotangent are defined by tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Reciprocal identities give csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.

正切和余切定义为tan θ = sin θ / cos θ，cot θ = cos θ / sin θ。倒数恒等式给出csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。

5. Periodicity | 周期性

A function f is periodic with period p if f(θ + p) = f(θ) for all θ. The smallest positive such p is called the fundamental period.

如果对一切θ都有f(θ + p) = f(θ)，则函数f是周期函数，最小正周期p称为基本周期。

sin θ and cos θ have period 2π.

sin θ 和 cos θ 的周期为2π。
tan θ and cot θ have period π.

tan θ 和 cot θ 的周期为π。
csc θ and sec θ have period 2π.

csc θ 和 sec θ 的周期为2π。

Periodicity is the reason trigonometric functions are ideal for modelling repeating phenomena such as waves, tides, and seasons.

周期性是三角函数适合模拟波、潮汐和季节等重复现象的原因。

6. Symmetry and Parity | 对称性与奇偶性

Understanding whether a function is even or odd helps predict its graph and simplify integrals and series.

理解函数是偶函数还是奇函数有助于预测其图像，并简化积分和级数运算。

cos θ is even: cos(−θ) = cos θ.

cos θ 是偶函数：cos(−θ) = cos θ。
sin θ is odd: sin(−θ) = −sin θ.

sin θ 是奇函数：sin(−θ) = −sin θ。
tan θ is odd: tan(−θ) = −tan θ.

tan θ 是奇函数：tan(−θ) = −tan θ。

The graph of y = cos θ is symmetric about the y-axis, while y = sin θ and y = tan θ are symmetric about the origin.

y = cos θ的图像关于y轴对称，而y = sin θ和y = tan θ的图像关于原点对称。

7. Graphs of Sine and Cosine | 正弦与余弦图像

The graph of y = sin θ starts at (0, 0), rises to a maximum of 1 at π/2, returns to 0 at π, falls to −1 at 3π/2, and completes one cycle at 2π. The graph of y = cos θ is a horizontal shift of the sine graph by π/2 to the left.

y = sin θ的图像从(0,0)出发，在π/2处上升到最大值1，在π处回到0，在3π/2处下降到−1，并在2π处完成一个周期。y = cos θ的图像是正弦图像向左平移π/2的结果。

Both curves have amplitude 1, range [−1, 1], and period 2π. Their intercepts and turning points are listed below.

两条曲线的振幅均为1，值域为[−1, 1]，周期为2π。它们的零点和极值点如下表所示。

Function	Zeros	Maxima	Minima
y = sin θ	θ = nπ	θ = π/2 + 2nπ, y = 1	θ = 3π/2 + 2nπ, y = −1
y = cos θ	θ = π/2 + nπ	θ = 2nπ, y = 1	θ = π + 2nπ, y = −1

n ∈ ℤ

8. Graph of Tangent and Asymptotes | 正切图像与渐近线

The graph of y = tan θ has vertical asymptotes where cos θ = 0, i.e., at θ = π/2 + nπ. Between consecutive asymptotes, the curve increases from −∞ to +∞, passing through 0 at θ = nπ.

y = tan θ的图像在cos θ = 0处，即θ = π/2 + nπ处，有垂直渐近线。在相邻渐近线之间，曲线从−∞增加到+∞，并在θ = nπ处通过0。

The period of tan θ is π, not 2π. There is no maximum or minimum value, so the range is all real numbers. The graph has rotational symmetry about the origin.

tan θ的周期是π，不是2π。它没有最大值或最小值，因此值域为全体实数。图像关于原点具有旋转对称性。

Similarly, y = cot θ has asymptotes at θ = nπ and zeros at θ = π/2 + nπ.

类似地，y = cot θ在θ = nπ处有渐近线，在θ = π/2 + nπ处有零点。

9. Amplitude, Period and Phase Shift | 振幅、周期与相位偏移

IB problems frequently involve functions of the form y = a sin(bθ − c) + d or y = a cos(bθ − c) + d. Each parameter transforms the graph in a specific way.

IB题目经常涉及 y = a sin(bθ − c) + d 或 y = a cos(bθ − c) + d 形式的函数。每个参数都以特定方式变换图像。

|a| is the amplitude: it stretches or compresses the graph vertically.

|a| 是振幅：它使图像在竖直方向上拉伸或压缩。
2π/|b| is the period: the horizontal length of one complete cycle.

2π/|b| 是周期：一个完整周期的水平长度。
c/b is the phase shift: the graph shifts right if c/b > 0, left if c/b < 0.

c/b 是相位偏移：当c/b > 0时图像右移，当c/b < 0时图像左移。
d is the vertical translation: the midline of the graph is y = d.

d 是竖直平移：图像的中轴线为y = d。

Period = 2π / |b|

For example, y = 3 sin(2θ − π) + 1 has amplitude 3, period π, phase shift π/2 to the right, and vertical shift 1 upward.

例如，y = 3 sin(2θ − π) + 1 的振幅为3，周期为π，相位向右偏移π/2，竖直方向向上平移1。

10. Transformations of Graphs | 图像变换

Given a base graph such as y = sin θ, you should be able to sketch its transformations quickly. Horizontal stretch/compression is caused by changing b, while vertical stretch/compression is caused by changing a.

给定基础图像如y = sin θ，你应该能够快速画出其变换后的图像。改变b会导致水平拉伸/压缩，改变a会导致竖直拉伸/压缩。

Reflection in the x-axis: y = −f(θ).

x轴反射：y = −f(θ)。
Reflection in the y-axis: y = f(−θ).

y轴反射：y = f(−θ)。
Vertical translation: y = f(θ) + d.

竖直平移：y = f(θ) + d。
Horizontal translation: y = f(θ − c).

水平平移：y = f(θ − c)。

When sketching, always label the midline, amplitude, key intercepts, and one full period. This demonstrates clear communication in IB assessments.

画图时，务必标出中轴线、振幅、关键零点和至少一个完整周期。这能体现IB评估中要求的清晰表达。

11. Inverse Trigonometric Functions | 反三角函数

In IB Mathematics AA, inverse trigonometric functions are introduced with restricted domains to ensure they are one-to-one.

在IB数学AA中，反三角函数通过限制定义域来保证其一一对应。

y = arcsin x ⇔ x = sin y, −π/2 ≤ y ≤ π/2

y = arccos x ⇔ x = cos y, 0 ≤ y ≤ π

y = arctan x ⇔ x = tan y, −π/2 < y < π/2

These restricted ranges are important for evaluating compositions such as sin(arcsin x) and for solving trigonometric equations with principal values.

这些受限值域对于计算sin(arcsin x)等复合表达式以及使用主值求解三角方程非常重要。

12. Exam Tips and Common Pitfalls | 考试技巧与常见错误

Finally, here are practical tips to avoid losing marks in IB trigonometry questions.

最后，这里有一些实用技巧，帮助你在IB三角题中避免失分。

Always work in radians unless the question explicitly uses degrees.

除非题目明确使用度，否则始终使用弧度。
Do not confuse degrees with radians: 1° ≠ 1 rad.

不要混淆度与弧度：1° ≠ 1 rad。
Check the domain before solving equations; discard extraneous solutions.

求解方程前先检查定义域，舍弃增根。
For transformed graphs, write the parameters a, b, c, d clearly and verify the period and amplitude on the graph.

对于变换后的图像，清楚写出参数a、b、c、d，并在图像上验证周期和振幅。
Memorise exact values for angles 0, π/6, π/4, π/3, π/2.

熟记角度0、π/6、π/4、π/3、π/2的精确值。
When using the sine rule or cosine rule in non-right triangles, label sides and angles consistently.

在非直角三角形中使用正弦定理或余弦定理时，保持边和角的标记一致。

Mastering these definitions, graphs, and properties will give you a solid foundation for both Paper 1 and Paper 2 in IB Mathematics.

掌握这些定义、图像和性质，将为你在IB数学Paper 1和Paper 2中取得好成绩打下坚实基础。

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Trigonometric Functions: Definitions, Graphs, and Properties | 三角函数定义、图像与性质