Graphs of sine, cosine and tangent | 正弦、余弦与正切函数图像

📚 Graphs of sine, cosine and tangent | 正弦、余弦与正切函数图像

re studied with x measured in radians unless a question states degrees. They are periodic functions, so their graphs repeat at regular intervals and can be used to solve equations and model oscillations.

在 Edexcel A-Level 数学中，y = sin x、y = cos x 和 y = tan x 的图像通常以弧度制表示 x，除非题目明确使用角度制。它们都是周期函数，图像按固定间隔重复，因此可用来解方程和描述振动现象。

ns are defined for all real x, while the tangent function is undefined wherever cos x = 0.

正弦函数和余弦函数的定义域为全体实数，而正切函数在 cos x = 0 处无定义。

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1. The core trigonometric ratios and their graph domains | 核心三角比及其图像定义域

The three trigonometric functions are built from the ratios of sides in a right-angled triangle, but their graphs extend these ratios to all real angles using the unit circle or periodic extension.

三个三角函数来自直角三角形中边的比值，但它们的图像通过单位圆或周期延拓，将这些比值推广到全体实数角。

tion is undefined at x = π/2 + nπ, where n is an integer, because division by zero occurs.

对 y = sin x 和 y = cos x 来说，每个实数 x 都有唯一的 y 值。对 y = tan x 来说，函数在 x = π/2 + nπ（n 为整数）处无定义，因为此时出现了除以零的情况。

You must be able to sketch all three graphs for at least one full period and to label key points exactly.

你必须能够至少画出一个完整周期内这三个函数的图像，并准确标出关键点。

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2. Graph of y = sin x: wave from the origin | y = sin x 的图像：从原点出发的波

, returns to 0 at x = π, falls to a minimum of −1 at x = 3π/2, and returns to 0 at x = 2π.

y = sin x 的图像从原点出发，函数值为 0；在 x = π/2 处升至最大值 1；在 x = π 处回到 0；在 x = 3π/2 处降至最小值 −1；在 x = 2π 处再次回到 0。

is rotationally symmetric about the origin, making sine an odd function: sin(−x) = −sin x.

它的基本周期为 2π，因此对任意 x 都有 sin(x + 2π) = sin x。图像关于原点中心对称，所以正弦是奇函数：sin(−x) = −sin x。

t x = nπ, where n is any integer. These are the points where the graph crosses the x-axis.

sin x 的零点位于 x = nπ（n 为任意整数），这些点是图像与 x 轴的交点。

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3. Graph of y = cos x: the same wave shifted | y = cos x 的图像：平移后的同一波形

to 0 at x = π/2, reaches −1 at x = π, rises to 0 at x = 3π/2, and returns to 1 at x = 2π.

y = cos x 的图像与正弦图像形状相同，只是向左平移了 π/2。它从点 (0,1) 开始，在 x = π/2 处降到 0，在 x = π 处达到 −1，在 x = 3π/2 处升到 0，在 x = 2π 处回到 1。

etric about the y-axis, so cos(−x) = cos x; it is an even function. Its period is also 2π.

余弦图像关于 y 轴对称，因此 cos(−x) = cos x；它是偶函数。周期同样为 2π。

hows that the cosine graph is simply the sine graph shifted left by π/2. This identity is useful when transforming graphs.

恒等式 cos x = sin(x + π/2) 表明余弦图像就是正弦图像向左平移 π/2 得到的结果。这一关系在图像变换中非常有用。

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4. Graph of y = tan x: period, zeros and asymptotes | y = tan x 的图像：周期、零点与渐近线

The tangent graph repeats every π, not every 2π. It passes through the origin and is increasing on every interval between consecutive vertical asymptotes.

正切图像每 π 重复一次，而不是每 2π。它经过原点，并在相邻两条竖直渐近线之间的每个区间上递增。

x and cos x = 0 at these points. The graph approaches these lines but never touches them.

竖直渐近线出现在 x = π/2 + nπ（n 为整数），因为 tan x = sin x / cos x，而在这些点 cos x = 0。图像无限接近这些直线但永不接触。

xactly where sin x = 0. The graph has no maximum or minimum value, so it has no amplitude.

正切函数的零点位于 x = nπ，也就是 sin x = 0 的位置。图像没有最大值或最小值，因此没有振幅。

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5. Amplitude, period, and key features | 振幅、周期与关键特征

For y = a sin(bx + c) + d, the amplitude is |a| he centre line to a maximum or minimum. The basic sine and cosine graphs have amplitude 1.

对 y = a sin(bx + c) + d 来说，振幅是 |a|，即中心线到最大值或最小值的竖直距离。基本正弦和余弦图像的振幅都是 1。

The period of y = sin(bx) or y = cos(bx) is 2π/|b|. The tangent graph y = tan(bx) has period π/|b|.

y = sin(bx) 或 y = cos(bx) 的周期为 2π/|b|；正切图像 y = tan(bx) 的周期为 π/|b|。

Function	Amplitude	Period	Zeros	Symmetry
sin x	1	2π	nπ	Odd
cos x	1	2π	π/2 + nπ	Even
tan x	None	π	nπ	Odd

andard angles are essential for accurate sketches. For example, sin(π/6) = 1/2, cos(π/4) = √2/2 and tan(π/3) = √3.

标准角的精确值对于准确画图至关重要。例如 sin(π/6) = 1/2，cos(π/4) = √2/2，tan(π/3) = √3。

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6. Using radians and degrees | 弧度制与角度制的使用

Edexcel exam questions usually use radians for trigonometric graphs and calculus, but they may include degrees in geometric contexts. You must label the x-axis correctly: 0, π/2, π, 3π/2, 2π when working in radians.

Edexcel 考试题中，三角图像和微积分通常使用弧度制，但几何情境下也可能使用角度制。必须正确标注 x 轴：使用弧度时标为 0、π/2、π、3π/2、2π。

y π/180. To convert radians to degrees, multiply by 180/π. Exact values such as sin(π/3) = √3/2 must be known without a calculator.

将角度转换为弧度，乘以 π/180；将弧度转换为角度，乘以 180/π。必须不借助计算器掌握精确值，如 sin(π/3) = √3/2。

In a sketch, the horizontal scale changes completely if you switch from radians to degrees. A period of 2π in radians corresponds to 360° in degrees.

在草图中，如果从弧度制切换为角度制，水平刻度会完全不同。弧度制下的周期 2π 对应角度制下的 360°。

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7. Transformations of trigonometric graphs | 三角函数的图像变换

graph is y = a sin(bx + c) + d. The value a stretches the graph vertically by a factor of |a| and reflects it in the x-axis if a is negative.

一般变换后的正弦或余弦图像为 y =

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