Edexcel IGCSE Mathematics: Graphs of Sine, Cosine and Tangent Functions | Edexcel IGCSE数学：正弦、余弦与正切函数图像

📚 Edexcel IGCSE Mathematics: Graphs of Sine, Cosine and Tangent Functions | Edexcel IGCSE数学：正弦、余弦与正切函数图像

Trigonometric functions are among the most important ideas in mathematics, linking angles to ratios in right-angled triangles. In the Edexcel IGCSE course, however, we go beyond triangles and examine the graphs of sine, cosine and tangent functions. These graphs reveal repeating patterns of periodic behaviour that appear throughout science, engineering and music. Mastering their shapes, key values and transformations is essential for solving equations and for achieving top marks in the exam.

三角函数是数学中最重要的概念之一，它将角度与直角三角形中的比值联系起来。然而在 Edexcel IGCSE 课程中，我们超越三角形本身，深入研究正弦、余弦和正切函数的图像。这些图像揭示了周期性行为中不断重复的规律，这些规律在科学、工程和音乐中无处不在。掌握它们的形状、关键数值以及函数变换，对于解方程和在考试中取得高分至关重要。

1. The Sine Graph y = sin x | 正弦函数图像 y = sin x

The sine graph is the most fundamental of the three trigonometric graphs. For an angle x measured in degrees, sin x gives the y-coordinate of a point on the unit circle. The graph is a smooth wave that starts at the origin. At x = 0°, sin x = 0; as x increases toward 90°, the value rises to a maximum of 1; it then falls through 0 at 180°, reaches a minimum of -1 at 270°, and returns to 0 at 360°.

正弦图像是三种三角函数图像中最基本的。对于以度为单位的角度 x，sin x 给出单位圆上一点的 y 坐标。图像是一条平滑的波浪曲线，从原点开始。当 x = 0° 时，sin x = 0；随着 x 增大到 90°，函数值上升到最大值 1；随后在 180° 处下降到 0，在 270° 处达到最小值 -1，并在 360° 处回到 0。

The pattern then repeats exactly every 360°. We call this repeating interval the period. The greatest distance from the centre line (the x-axis) is called the amplitude; for y = sin x, the amplitude is 1. The key points to memorise are listed below.

这一模式随后每 360° 精确重复一次，我们称这个重复间隔为周期。离开中心线（即 x 轴）的最大距离称为振幅；对于 y = sin x，振幅为 1。需要牢记的关键点如下。

• (0°, 0), (90°, 1), (180°, 0), (270°, -1), (360°, 0) — five key points in one cycle | 五个关键点为一个周期
• Zeros at 0°, 180°, 360° | 函数零点位于 0°、180°、360°
• Maximum value 1 at 90°; minimum value -1 at 270° | 最大值 1 在 90° 处；最小值 -1 在 270° 处
• The graph is symmetric about the line x = 90° within each half-cycle | 在每个半周期内图像关于直线 x = 90° 对称

2. The Cosine Graph y = cos x | 余弦函数图像 y = cos x

The cosine graph has exactly the same wave shape and the same period of 360° as the sine graph, but it is shifted 90° to the left. At x = 0°, cos x = 1, so the graph begins at its maximum value. It falls through 0 at 90°, reaches -1 at 180°, rises back to 0 at 270°, and completes one full cycle at 360° where the value is 1 again.

余弦图像与正弦图像具有完全相同的波浪形状和 360° 的周期，但整体向左平移了 90°。当 x = 0° 时，cos x = 1，因此图像从最大值开始。它在 90° 处下降到 0，在 180° 处达到最小值 -1，在 270° 处回升到 0，并在 360° 处完成一个完整周期，此时函数值再次为 1。

Two identities are especially useful. First, cos(-x) = cos x, so the cosine graph is symmetrical about the y-axis; we call this an even function. Second, cos x = sin(x + 90°), which explains why the cosine wave is simply the sine wave shifted left by 90°. The key points are (0°, 1), (90°, 0), (180°, -1), (270°, 0), (360°, 1).

有两个恒等式尤其重要。第一，cos(-x) = cos x，因此余弦图像关于 y 轴对称，我们称之为偶函数。第二，cos x = sin(x + 90°)，这说明余弦波实际上就是正弦波向左平移 90°。关键点为 (0°, 1)、(90°, 0)、(180°, -1)、(270°, 0)、(360°, 1)。

3. The Tangent Graph y = tan x | 正切函数图像 y = tan x

The tangent graph looks completely different from sine and cosine. At x = 0°, tan x = 0. As x approaches 90° from the left, tan x increases without bound, heading toward positive infinity. At exactly 90° the tangent is undefined, so the graph has a vertical asymptote. Just to the right of 90°, tan x starts from very large negative values and rises to 0 at 180°.

正切图像与正弦、余弦截然不同。当 x = 0° 时，tan x = 0。当 x 从左侧趋近 90° 时，tan x 无限增大，趋向正无穷。在恰好 90° 处正切无定义，因此图像拥有一条垂直渐近线。在 90° 的右侧，tan x 从极大的负值开始，并在 180° 处上升到 0。

The pattern repeats every 180°, so the period of y = tan x is 180°, not 360°. The tangent graph has no amplitude because its y-values are unbounded; it rises from negative infinity to positive infinity between consecutive asymptotes. Its zeros occur at 0°, 180°, 360° and all other multiples of 180°. The asymptotes occur at 90° + 180°n, where n is any integer.

这一模式每 180° 重复一次，因此 y = tan x 的周期是 180°，而不是 360°。正切图像没有振幅，因为其 y 值无界；它在相邻两条渐近线之间从负无穷上升到正无穷。它的零点出现在 0°、180°、360° 以及其他所有 180° 的倍数处。渐近线出现在 x = 90° + 180°n，其中 n 为任意整数。

4. Comparing the Three Graphs | 三种图像对比

It is vital to compare the three graphs side by side, because the exam often asks you to identify a graph from its equation or to state which function a sketch represents. The table below summarises the most important differences.

将三种图像并排对比至关重要，因为考试经常要求你根据方程识别图像，或判断一张草图代表哪个函数。下表总结了最重要的区别。

Remember: sine starts at 0 and rises; cosine starts at 1 and falls; tangent passes through the origin and shoots up to an asymptote. These three opening behaviours alone are enough to identify any graph in a multiple-choice question.

请记住：正弦从 0 开始并上升；余弦从 1 开始并下降；正切穿过原点并迅速趋近渐近线。仅凭这三种起始形态，就足以在选择题中识别出任何一张图像。

5. Amplitude y = a sin x | 振幅 y = a sin x

When a function is written in the form y = a sin x, the constant a changes the amplitude. The graph is stretched or compressed vertically by a factor of |a|. For example, y = 2 sin x has a maximum of 2 and a minimum of -2, so its range is -2 ≤ y ≤ 2, while y = ½ sin x only rises to ½ and falls to -½.

当函数写成 y = a sin x 的形式时，常数 a 会改变振幅。图像在垂直方向上被拉伸或压缩 |a| 倍。例如，y = 2 sin x 的最大值为 2，最小值为 -2，因此其值域为 -2 ≤ y ≤ 2；而 y = ½ sin x 只上升到 ½，下降到 -½。

If a is negative, the graph is reflected in the x-axis; for example, y = -sin x is the sine wave flipped upside down. The amplitude is always recorded as a positive value, so for y = -3 cos x the amplitude is 3. In every case, the period remains unchanged at 360°.

如果 a 为负数，图像会关于 x 轴反射；例如 y = -sin x 就是正弦波上下翻转。振幅始终记为正值，因此 y = -3 cos x 的振幅为 3。在每种情况下，周期保持不变，仍为 360°。

6. Period y = sin bx | 周期 y = sin bx

The coefficient b inside the sine function changes the period. For y = sin bx, where b > 0, the new period is found using the formula below.

正弦函数内部的系数 b 会改变周期。对于 y = sin bx（其中 b > 0），新周期用下面的公式求出。

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