Graphs and Properties of Trigonometric Functions in Radians | 弧度制下三角函数的图像与性质

📚 Graphs and Properties of Trigonometric Functions in Radians | 弧度制下三角函数的图像与性质

When angles are measured in radians, trigonometric functions take on a natural mathematical elegance. The graph of y = sin x, for example, has x-coordinates that correspond directly to real numbers, making the study of periodicity, amplitude, and transformations far more intuitive. In this article, we explore the graphs and key properties of sine, cosine, and tangent functions in radians, and we examine how transformations affect their appearance.

当角度以弧度为单位时，三角函数呈现出一种自然的数学优雅。例如，y = sin x 的图像，其 x 坐标直接对应实数，这使得对周期性、振幅和变换的研究变得更加直观。在本文中，我们探讨正弦、余弦和正切函数在弧度制下的图像与关键性质，并研究变换如何影响它们的形态。

1. Why Radians Matter | 为什么弧度至关重要

In degree measure, the sine function has a period of 360°, which is an arbitrary number. In radians, the period of sin x and cos x is 2π, a constant that arises naturally from the geometry of the unit circle. This makes calculus formulas cleaner, such as the derivative of sin x being cos x without any scaling factor.

在度数制下，正弦函数的周期为360°，这是一个人为设定的数。在弧度制下，sin x 与 cos x 的周期为 2π，这一常数自然地源自单位圆的几何性质。这使得微积分公式更加简洁，例如 sin x 的导数直接为 cos x，无需任何比例因子。

For any angle x measured in radians, the arc length on a unit circle is exactly x. Thus, the x-axis of a trigonometric graph represents both the angle and the corresponding arc length. This geometric interpretation is central to understanding why radians are the default unit in higher mathematics.

对于任意以弧度为单位的角度 x，单位圆上的弧长恰好为 x。因此，三角函数图像的 x 轴同时表示角度和对应的弧长。这一几何解释是理解为何弧度制成为高等数学默认单位的核心。

2. The Graph of y = sin x | y = sin x 的图像

The sine function is defined for all real numbers. Its graph is a smooth, continuous wave that oscillates between -1 and 1. The function is odd, meaning sin(-x) = -sin x, and its graph is symmetric about the origin.

正弦函数对所有实数都有定义。其图像是一条平滑、连续的波浪线，在 -1 和 1 之间振荡。该函数是奇函数，即 sin(-x) = -sin x，其图像关于原点对称。

Key features of y = sin x:

y = sin x 的关键特征：

Period: 2π — the graph repeats every 2π units.
周期：2π — 图像每 2π 个单位重复一次。
Amplitude: 1 — the maximum distance from the x-axis.
振幅：1 — 距 x 轴的最大距离。
Zeros at x = nπ, where n is any integer.
零点位于 x = nπ，其中 n 为任意整数。
Maximum value 1 at x = π/2 + 2nπ; minimum value -1 at x = 3π/2 + 2nπ.
在 x = π/2 + 2nπ 处取得最大值 1；在 x = 3π/2 + 2nπ 处取得最小值 -1。

sin(x + 2π) = sin x for all real x

sin(x + 2π) = sin x 对所有实数 x 成立

3. The Graph of y = cos x | y = cos x 的图像

The cosine function is also defined for all real numbers and shares the same period and amplitude as sine. However, it is an even function, meaning cos(-x) = cos x, and its graph is symmetric about the y-axis.

余弦函数同样对所有实数有定义，并且与正弦函数具有相同的周期和振幅。然而，它是一个偶函数，即 cos(-x) = cos x，其图像关于 y 轴对称。

Key features of y = cos x:

y = cos x 的关键特征：

Period: 2π; Amplitude: 1.
周期：2π；振幅：1。
Zeros at x = π/2 + nπ.
零点位于 x = π/2 + nπ。
Maximum value 1 at x = 2nπ; minimum value -1 at x = π + 2nπ.
在 x = 2nπ 处取最大值 1；在 x = π + 2nπ 处取最小值 -1。

Notice that the graph of y = cos x is exactly the graph of y = sin x shifted left by π/2. This phase relationship is expressed as cos x = sin(x + π/2).

注意，y = cos x 的图像恰好是 y = sin x 的图像向左平移 π/2 得到的。这一相位关系表示为 cos x = sin(x + π/2)。

4. The Graph of y = tan x | y = tan x 的图像

The tangent function behaves very differently from sine and cosine. Since tan x = sin x / cos x, it is undefined wherever cos x = 0, i.e., at x = π/2 + nπ. These points create vertical asymptotes on the graph.

正切函数的行为与正弦和余弦函数截然不同。由于 tan x = sin x / cos x，它在 cos x = 0 处无定义，即在 x = π/2 + nπ 处。这些点形成了图像上的垂直渐近线。

Key features of y = tan x:

y = tan x 的关键特征：

Period: π — much shorter than sine and cosine.
周期：π — 比正弦和余弦的周期短得多。
Range: all real numbers.
值域：全体实数。
Vertical asymptotes at x = π/2 + nπ.
垂直渐近线位于 x = π/2 + nπ。
Zeros at x = nπ.
零点位于 x = nπ。

The graph crosses the x-axis at the origin and at every integer multiple of π, rising from -∞ to +∞ between consecutive asymptotes.

该图像在原点以及每个整数倍的 π 处穿过 x 轴，在相邻渐近线之间从 -∞ 上升至 +∞。

5. Amplitude Transformations | 振幅变换

For a function of the form y = A sin x or y = A cos x, the constant A affects the amplitude. The amplitude becomes |A|, meaning the graph is stretched or compressed vertically.

对于形如 y = A sin x 或 y = A cos x 的函数，常数 A 影响振幅。振幅变为 |A|，意味着图像在垂直方向上被拉伸或压缩。

If A is negative, the graph is reflected across the x-axis. For example, y = -2 sin x has amplitude 2 and is flipped upside down relative to y = 2 sin x.

若 A 为负数，图像则关于 x 轴翻转。例如，y = -2 sin x 的振幅为 2，并且相对于 y = 2 sin x 上下颠倒。

Amplitude = |A| for y = A sin x and y = A cos x

对于 y = A sin x 和 y = A cos x，振幅 = |A|

6. Period Transformations | 周期变换

When we multiply the input by a constant B, as in y = sin(Bx), the period changes. The new period is 2π / |B|. For tangent, the new period is π / |B|.

当我们对自变量乘以常数 B 时，如 y = sin(Bx)，周期随之改变。新周期为 2π / |B|。对于正切函数，新周期为 π / |B|。

This means that larger values of B produce more waves in the same horizontal interval, while smaller values of B stretch the graph horizontally.

这意味着较大的 B 值会在相同的水平区间内产生更多个波，而较小的 B 值则会在水平方向上拉伸图像。

Example: y = sin(2x) completes one full cycle between x = 0 and x = π, because its period is 2π/2 = π.

例如：y = sin(2x) 在 x = 0 到 x = π 之间完成一个完整周期，因为其周期为 2π/2 = π。

Period = 2π / |B| for sine and cosine; Period = π / |B| for tangent

正弦和余弦的周期 = 2π / |B|；正切的周期 = π / |B|

7. Phase Shifts: Horizontal Translations | 相位移动：水平平移

A horizontal shift is introduced by adding a constant C inside the argument: y = sin(x – C). The graph shifts to the right by C units if C is positive, and to the left if C is negative.

通过在自变量中加入常数 C 可以引入水平位移：y = sin(x – C)。若 C 为正，图像向右平移 C 个单位；若 C 为负，则向左平移。

This shift is called a phase shift. For example, y = sin(x – π/2) is the sine wave shifted right by π/2, which makes it identical to y = -cos x. Indeed, sin(x – π/2) = -cos x is a useful identity.

这种位移称为相位移动。例如，y = sin(x – π/2) 是正弦波向右平移 π/2，这使得它与 y = -cos x 完全相同。事实上，sin(x – π/2) = -cos x 是一个有用的恒等式。

When combined with reflections, phase shifts allow us to model many physical phenomena, such as alternating current or simple harmonic motion.

当与反射结合时，相位移动使我们能够模拟许多物理现象，如交流电或简谐运动。

8. Vertical Translations | 垂直平移

The transformation y = sin x + D shifts the graph vertically. If D is positive, the graph moves upward; if D is negative, it moves downward. The midline of the graph, previously y = 0, becomes y = D.

变换 y = sin x + D 会使图像在垂直方向上移动。若 D 为正，图像上移；若 D 为负，图像下移。图像的中线，原本为 y = 0，现在变为 y = D。

This transformation does not affect the period, amplitude, or phase of the function. It only changes the vertical position. For instance, y = cos x + 3 oscillates between 2 and 4.

此变换不影响函数的周期、振幅或相位，仅改变其垂直位置。例如，y = cos x + 3 在 2 和 4 之间振荡。

In real-world applications, vertical shifts represent a baseline or average value, such as the mean temperature over a year.

在现实应用中，垂直平移代表基线或平均值，例如一年中的平均气温。

9. Combining Transformations | 综合变换

In the general form y = A sin(B(x – C)) + D, each parameter plays a distinct role:

在一般形式 y = A sin(B(x – C)) + D 中，每个参数扮演着不同的角色：

A controls the amplitude (vertical stretch/compression and reflection if negative).
A 控制振幅（垂直拉伸/压缩，若为负数则反射）。
B controls the period (horizontal compression/stretch).
B 控制周期（水平压缩/拉伸）。
C controls the phase shift (horizontal translation).
C 控制相位移动（水平平移）。
D controls the vertical translation (midline shift).
D 控制垂直平移（中线移动）。

When graphing such a function, it is often easiest to follow this order: first identify the amplitude and period, then sketch the unshifted graph, and finally apply the horizontal and vertical translations.

在绘制此类函数图像时，最容易的顺序通常是：首先确定振幅和周期，然后画出未平移的图像，最后进行水平与垂直平移。

10. Solving Equations Graphically | 用图像法解方程

The graphs of trigonometric functions can be used to solve equations. For example, to solve sin x = 0.5 for 0 ≤ x < 2π, locate the horizontal line y = 0.5 on the graph of y = sin x. It intersects the curve at x = π/6 and x = 5π/6.

三角函数的图像可以用来解方程。例如，在 0 ≤ x < 2π 范围内解 sin x = 0.5，可在 y = sin x 的图像上找到水平直线 y = 0.5。它与曲线相交于 x = π/6 和 x = 5π/6。

Because of periodicity, the full solution set is x = π/6 + 2nπ and x = 5π/6 + 2nπ, for any integer n. The graph reminds us that there are always two solutions within each period unless the equation involves a tangent function, where there is exactly one solution per period.

由于周期性，完整解集为 x = π/6 + 2nπ 和 x = 5π/6 + 2nπ，其中 n 为任意整数。图像提醒我们，在每个周期内通常有两个解，除非方程涉及正切函数——正切在每个周期内只有一个解。

11. Applications in Modelling | 在建模中的应用

Trigonometric functions in radians are widely used to model periodic phenomena. For instance, the tide height at a harbour can be modelled as h(t) = A sin(Bt – C) + D, where t is time in hours.

弧度制下的三角函数被广泛用于模拟周期现象。例如，港口的水位高度可以建模为 h(t) = A sin(Bt – C) + D，其中 t 是以小时为单位的时间。

In such models, the amplitude reflects the maximum deviation from the average level, the period corresponds to the natural cycle such as 12 hours for tides, and the phase shift accounts for the time of the first high tide.

在此类模型中，振幅反映距平均水位的最大偏差，周期对应自然周期（如潮汐的12小时），而相位移动则解释首次高潮的时间。

Another example is simple harmonic motion, where the displacement of a spring is given by x(t) = A cos(ωt + φ). Understanding the graph helps physicists predict the position at any time.

另一个例子是简谐运动，其中弹簧的位移为 x(t) = A cos(ωt + φ)。理解图像有助于物理学家预测任意时刻的位置。

12. Summary Table of Key Properties | 关键性质总结表

The following table summarises the essential properties of the three basic trigonometric functions in radians.

下表总结了弧度制下三个基本三角函数的基本性质。

Function \| 函数	Period \| 周期	Amplitude \| 振幅	Range \| 值域	Zeros \| 零点
sin x	2π	1	[-1, 1]	nπ
cos x	2π	1	[-1, 1]	π/2 + nπ
tan x	π	Not defined \| 无定义	All real numbers \| 全体实数	nπ

Mastery of these graphs and properties is essential for solving trigonometric equations, performing transformations, and succeeding in calculus. Always sketch the base graph first, then apply transformations step by step.

掌握这些图像和性质对于解三角方程、进行变换以及在微积分中取得成功至关重要。务必先画出基础图像，然后逐步应用变换。

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Graphs and Properties of Trigonometric Functions in Radians | 弧度制下三角函数的图像与性质