Graphs of Sine, Cosine and Tangent | 正弦、余弦和正切函数图像

📚 Graphs of Sine, Cosine and Tangent | 正弦、余弦和正切函数图像

In this revision article, we will explore the graphs of the three fundamental trigonometric functions: sine, cosine and tangent. These graphs are essential for the IGCSE Edexcel Mathematics syllabus, and understanding their shapes, key features and transformations will help you tackle many exam questions with confidence.

在本篇复习文章中，我们将探讨三个基本三角函数（正弦、余弦和正切）的图像。这些图像是 IGCSE Edexcel 数学大纲的核心内容，理解它们的形状、关键特征和变换，将帮助你在考试中自信应对许多题目。

1. The Sine Graph | 正弦函数图像

The sine graph, y = sin x, is a smooth, continuous wave that repeats every 360°. It starts at the origin (0, 0), rises to a maximum value of 1 at 90°, returns to 0 at 180°, falls to a minimum value of −1 at 270°, and completes one full cycle at 360°.

正弦函数图像 y = sin x 是一条平滑连续的波浪线，每 360° 重复一次。它从原点 (0, 0) 出发，在 90° 时上升到最大值 1，在 180° 时回到 0，在 270° 时下降到最小值 −1，并在 360° 时完成一个完整周期。

For x-values between 0° and 360°, the sine graph is positive in the first and second quadrants (0° to 180°) and negative in the third and fourth quadrants (180° to 360°).

在 0° 到 360° 范围内，正弦图像在第一、第二象限（0° 到 180°）为正，在第三、第四象限（180° 到 360°）为负。

The graph of y = sin x is an odd function, meaning that sin(−x) = −sin x. This gives the graph rotational symmetry about the origin.

y = sin x 是一个奇函数，即 sin(−x) = −sin x。这使得图像关于原点具有旋转对称性。

2. The Cosine Graph | 余弦函数图像

The cosine graph, y = cos x, also repeats every 360°. It starts at its maximum value of 1 at x = 0°, falls to 0 at 90°, reaches its minimum value of −1 at 180°, returns to 0 at 270°, and rises back to 1 at 360°.

余弦函数图像 y = cos x 同样每 360° 重复一次。它在 x = 0° 时从最大值 1 出发，在 90° 时降到 0，在 180° 时达到最小值 −1，在 270° 时回到 0，并在 360° 时回升到 1。

Notice that the cosine graph has the same shape as the sine graph but is shifted 90° to the left. In fact, cos x = sin(x + 90°).

注意，余弦图像与正弦图像形状相同，但向左平移了 90°。事实上，cos x = sin(x + 90°)。

The graph of y = cos x is an even function, meaning that cos(−x) = cos x. This gives the graph symmetry about the y-axis.

y = cos x 是一个偶函数，即 cos(−x) = cos x。这使得图像关于 y 轴对称。

3. The Tangent Graph | 正切函数图像

The tangent graph, y = tan x, is fundamentally different from sine and cosine. It repeats every 180° rather than 360°, and its range is all real numbers, not just between −1 and 1.

正切函数图像 y = tan x 与正弦、余弦图像有本质区别。它每 180° 重复一次，而不是 360°，其值域为全体实数，而不仅仅是 −1 到 1 之间。

At x = 0°, tan x = 0. The graph rises steeply as x approaches 90°, becoming infinitely large. At exactly x = 90°, the function is undefined, and the graph has a vertical asymptote. Similarly, asymptotes occur at x = 90° + 180°k, where k is any integer, such as 90°, 270°, 450°, etc.

在 x = 0° 时，tan x = 0。当 x 接近 90° 时，图像急剧上升，趋于无穷大。在 x = 90° 处，函数无定义，图像有一条竖直渐近线。类似地，渐近线出现在 x = 90° + 180°k 处，其中 k 为任意整数，如 90°、270°、450° 等。

The tangent function has the identity tan x = sin x ⁄ cos x, which explains why it is undefined wherever cos x = 0.

正切函数满足恒等式 tan x = sin x ⁄ cos x，这解释了为什么在 cos x = 0 处函数无定义。

4. Key Features — Amplitude and Period | 关键特征——振幅与周期

For sine and cosine graphs, the amplitude is half the distance between the maximum and minimum values. For y = sin x and y = cos x, the amplitude is 1. The period is the length of one complete cycle; for these basic graphs, the period is 360°.

对于正弦和余弦图像，振幅是最大值和最小值之间距离的一半。对于 y = sin x 和 y = cos x，振幅为 1。周期是一个完整循环的长度；对于这些基本图像，周期为 360°。

For the tangent graph, there is no amplitude in the usual sense (the range is infinite), but the period is 180°.

对于正切图像，通常意义上的振幅不存在（值域无限），但周期为 180°。

In general, for y = a sin(bx) and y = a cos(bx):

一般来说，对于 y = a sin(bx)

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