Cosine Function: Graph and Properties | IB数学：余弦函数图像与性质

📚 Cosine Function: Graph and Properties | IB数学：余弦函数图像与性质

The cosine function is one of the fundamental trigonometric functions studied in IB Mathematics. Its wave-like graph appears in physics, engineering, and many other fields. Understanding its shape and properties is essential for solving problems involving periodic behaviour.

余弦函数是IB数学中学习的基本三角函数之一。它的波浪形图像出现在物理、工程以及许多其他领域。理解其图像形状和性质，对于解决涉及周期行为的问题至关重要。

1. Definition from the Unit Circle | 单位圆定义

On the unit circle, an angle θ is measured from the positive x-axis. The x-coordinate of the point where the terminal side intersects the circle is defined as cos θ. As θ increases from 0 to 2π, the x-coordinate traces out the cosine values.

在单位圆上，角θ从x轴正方向开始度量。终边与圆交点的x坐标被定义为cos θ。当θ从0增加到2π时，x坐标的变化就描绘出了余弦值。

The cosine function can be extended to all real numbers by allowing the angle to rotate multiple times around the circle, both clockwise and counterclockwise.

通过允许角度在圆上顺时针或逆时针旋转多圈，余弦函数可以扩展到所有实数。

2. The Basic Cosine Graph | 基本余弦图像

The graph of y = cos x is a smooth curve that starts at its maximum value when x = 0. At x = 0, cos 0 = 1. The curve then decreases, crosses zero at x = π/2, reaches its minimum at x = π, and returns to 1 at x = 2π.

y = cos x的图像是一条平滑曲线，在x = 0处从最大值开始。当x = 0时，cos 0 = 1。然后曲线下降，在x = π/2处穿过零点，在x = π处达到最小值，并在x = 2π处回到1。

y = cos x, for 0 ≤ x ≤ 2π

One complete wave from 0 to 2π is called one cycle. The shape repeats every 2π radians.

从0到2π的一个完整波形称为一个周期。这个形状每2π弧度重复一次。

3. Domain and Range | 定义域与值域

For the function f(x) = cos x, the domain is all real numbers: x ∈ ℝ. Because the unit circle x-coordinate never exceeds 1 in absolute value, the range is [-1, 1].

对于函数f(x) = cos x，定义域是所有实数：x ∈ ℝ。因为单位圆上的x坐标绝对值始终不超过1，所以值域是[-1, 1]。

Domain: (-∞, ∞) Range: [-1, 1]

定义域：(-∞, ∞) 值域：[-1, 1]

These are the first two properties checked when sketching or analysing a cosine function.

这是绘制或分析余弦函数时首先要检查的两个性质。

4. Periodicity | 周期性

A function is periodic if f(x + p) = f(x) for some positive constant p. For the basic cosine function, the smallest positive period is 2π.

如果存在正常数p使得f(x + p) = f(x)，则称该函数是周期函数。对于基本余弦函数，最小正周期是2π。

cos(x + 2π) = cos x

In general, if the function is written as y = a cos(bx), the period is given by Period = 2π / |b|. This is a key formula in IB examinations.

一般地，如果函数写成y = a cos(bx)，则周期为周期 = 2π / |b|。这是IB考试中的一个关键公式。

5. Even Function Symmetry | 偶函数对称性

The cosine function is an even function, meaning its graph is symmetric about the y-axis. Algebraically, this is expressed as:

余弦函数是偶函数，这意味着其图像关于y轴对称。代数上表示为：

cos(-x) = cos x

When you reflect the right half of the cosine graph across the y-axis, you obtain the left half. This property is often tested together with transformations.

将余弦图像的右半部分关于y轴反射，就得到左半部分。这个性质通常与图像变换一起考查。

6. Amplitude and Vertical Shift | 振幅与垂直平移

For a function of the form y = a cos x, the amplitude is |a|. It represents half the distance between the maximum and minimum values.

对于形如y = a cos x的函数，振幅为|a|。它表示最大值与最小值之间距离的一半。

If a > 0, the graph opens upward normally; if a < 0, the graph is reflected across the x-axis. The maximum and minimum values become |a| and -|a| respectively.

如果a > 0，图像正常开口向上；如果a < 0，图像关于x轴翻转。最大值和最小值分别变为|a|和-|a|。

Adding a constant d gives y = a cos x + d, which shifts the entire graph vertically by d units. The new range becomes [d – |a|, d + |a|].

加上常数d得到y = a cos x + d，这会将整个图像垂直平移d个单位。新的值域变为[d – |a|, d + |a|]。

Transform	Effect
y = a cos x	Vertical stretch/compression by \|a\|, reflection if a < 0
y = cos x + d	Vertical shift by d units

在形如y = a cos x的函数中，振幅为|a|。它表示最大值与最小值之间距离的一半。

如果a > 0，图像正常开口向上；如果a < 0，图像关于x轴翻转。最大值和最小值分别变为|a|和-|a|。

加上常数d得到y = a cos x + d，这会将整个图像垂直平移d个单位。新的值域变为[d – |a|, d + |a|]。

变换	效果
y = a cos x	关于x轴垂直伸缩/压缩，a < 0时翻转
y = cos x + d	垂直平移d个单位

7. Phase Shift (Horizontal Translation) | 相位移动（水平平移）

The general cosine function is often written as y = a cos(bx – c) + d. The horizontal shift, also called the phase shift, is determined by solving bx – c = 0.

一般的余弦函数常写成y = a cos(bx – c) + d。水平平移，也称为相位移动，通过解方程bx – c = 0来确定。

Phase shift = c / b

If c / b > 0, the graph shifts to the right by c / b units. If c / b < 0, the graph shifts to the left.

如果c / b > 0，图像向右平移c / b个单位。如果c / b < 0，图像向左平移。

A common mistake is to directly read the shift as c. Remember to divide by b first.

一个常见错误是直接将平移量读作c。记住先除以b。

8. Sketching Transformations | 绘制变换图像

When sketching y = a cos(bx – c) + d, follow these steps:

绘制y = a cos(bx – c) + d时，请遵循以下步骤：

Identify amplitude |a|, period 2π/|b|, phase shift c/b, and vertical shift d.

确定振幅|a|、周期2π/|b|、相位移动c/b和垂直平移d。
Draw the horizontal line y = d as the central axis.

绘制水平线y = d作为中心轴。
Mark the maximum y = d + |a| and minimum y = d – |a|.

标记最大值y = d + |a|和最小值y = d – |a|。
Start the first cycle at x = c/b. The period ends at x = c/b + 2π/|b|.

第一个周期从x = c/b开始。周期结束于x = c/b + 2π/|b|。
Divide the period into four equal parts to locate key points: max, zero, min, zero, max.

将周期分成四等份以定位关键点：最大值、零点、最小值、零点、最大值。

Always label the axes and at least one complete cycle.

始终标注坐标轴以及至少一个完整周期。

9. Zeros and Critical Points | 零点和极值点

For y = cos x, the zeros occur at x = π/2 + kπ, where k is any integer. The maximum points occur at x = 2kπ with value 1, and minimum points at x = π + 2kπ with value -1.

对于y = cos x，零点出现在x = π/2 + kπ，其中k是任意整数。最大值点出现在x = 2kπ处，值为1；最小值点出现在x = π + 2kπ处，值为-1。

For a transformed function y = a cos(bx – c) + d, solve the corresponding equations to find these points.

对于变换后的函数y = a cos(bx – c) + d，通过解相应方程来找到这些点。

These features are frequently used in examination questions to reconstruct the equation of a cosine function from its graph.

这些特征经常在考试题中用于根据图像反求余弦函数的方程。

10. Relationship with Sine Function | 与正弦函数的关系

The cosine and sine functions are closely related. The cosine graph is exactly the sine graph shifted left by π/2 radians:

余弦函数与正弦函数密切相关。余弦图像正是正弦图像向左平移π/2弧度：

cos x = sin(x + π/2)

Equivalently, sin x = cos(x – π/2). This relationship is useful when converting between the two functions.

等价地，sin x = cos(x – π/2)。这一关系在两种函数之间转换时很有用。

Notice that the cosine graph is a sine graph with the same amplitude and period, only shifted.

注意余弦图像就是具有相同振幅和周期的正弦图像，只是发生了平移。

11. Real-World Applications | 实际应用

The cosine function models many periodic phenomena, such as alternating current, sound waves, tidal heights, and pendulum motion.

余弦函数可以模拟许多周期现象，例如交流电、声波、潮汐高度和单摆运动。

For example, the displacement of a vibrating spring can be written as x(t) = A cos(ωt + φ), where A is amplitude, ω is angular frequency, and φ is the phase constant.

例如，振动弹簧的位移可以写为x(t) = A cos(ωt + φ)，其中A是振幅，ω是角频率，φ是相位常数。

In IB exam problems, you may be given data that fits a cosine model and asked to find the parameters a, b, c, and d.

在IB考试问题中，可能会给出符合余弦模型的数据，并要求你求出参数a、b、c和d。

12. Summary and Exam Tips | 总结与考试要点

To master the cosine function, remember the key properties: domain ℝ, range [-1, 1], period 2π, even symmetry, and specific zeros and extrema.

要掌握余弦函数，请记住关键性质：定义域ℝ，值域[-1, 1]，周期2π，偶函数对称性，以及特定的零点和极值点。

When transforming y = a cos(bx – c) + d, always compute amplitude, period, phase shift, and vertical shift before sketching.

在研究y = a cos(bx – c) + d的变换时，在绘制图像之前，务必先计算振幅、周期、相位移动和垂直平移。

Use the unit circle to remember exact values at key angles.

使用单位圆记住关键角度的精确值。
Check whether the graph starts at a maximum (cosine) or at zero (sine).

检查图像是从最大值（余弦）开始还是从零（正弦）开始。
When solving equations involving cos, use the general solution formulas.

解涉及cos的方程时，使用通解公式。

Practice sketching a variety of cosine functions to build visual intuition. This will help you quickly interpret graphs in the exam.

通过练习绘制各种余弦函数来建立直观认识。这将帮助你在考试中快速解读图像。

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Cosine Function: Graph and Properties | IB数学：余弦函数图像与性质