Circular Functions: The Unit Circle and Trigonometric Graphs | 圆函数：单位圆与三角图像

📚 Circular Functions: The Unit Circle and Trigonometric Graphs | 圆函数：单位圆与三角图像

In IB Mathematics, circular functions extend the right‑triangle definitions of sine, cosine and tangent to any real‑number input. By placing an angle in standard position on the unit circle, the coordinates of the terminal point give cos θ and sin θ directly, and the ratio y/x yields tan θ. This unified approach reveals periodicity, symmetry and a rich set of identities, making it the backbone of trigonometric analysis. Understanding circular functions is essential for solving equations, modelling periodic phenomena and succeeding in the Analysis and Approaches or Applications and Interpretation courses.

在IB数学中，圆函数将正弦、余弦和正切的直角三角形定义推广到任意实数输入。将角置于单位圆的标准位置，其终边与单位圆的交点坐标直接给出 cos θ 和 sin θ，比值 y/x 则给出 tan θ。这种统一的观点揭示了周期性、对称性以及丰富的恒等式，构成了三角分析的主干。理解圆函数对于解方程、建立周期现象模型以及在分析与方法或应用与解释课程中取得成功都至关重要。

1. Radian Measure and Arc Length | 弧度制与弧长

Circular functions are built on radian measure, where one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. The conversion between degrees and radians uses the fact that 180° = π rad. Radians simplify calculus results and make arc length (ℓ = rθ) and sector area (A = ½ r²θ) formula‑friendly. The IB expects you to work fluently in both measures, so memorising common values like 30° = π/6, 45° = π/4 and 60° = π/3 is essential.

圆函数建立在弧度制之上：1弧度是弧长等于半径的圆弧所对的圆心角。度与弧度的换算基于 180° = π 弧度。弧度使微积分结果更为简洁，并让弧长公式 ℓ = rθ 与扇形面积公式 A = ½ r²θ 使用起来十分自然。IB要求你在两种制式间熟练转换，因此熟记诸如 30° = π/6、45° = π/4 和 60° = π/3 等常用值是必不可少的。

Angles can exceed 2π or be negative; the directed angle is measured from the positive x‑axis. This generality allows circular functions to be defined for all real numbers, not just acute angles.

角可以大于 2π 或为负值；有向角从正 x 轴开始度量。这种一般性使得圆函数能够对全体实数定义，而不仅仅是锐角。

2. The Unit Circle Definition | 单位圆定义

The unit circle is centred at (0,0) with radius 1. For any angle θ in standard position, the terminal ray intersects the circle at a point P(x,y). By definition, cos θ = x, sin θ = y. This simple geometric model immediately shows that −1 ≤ sin θ ≤ 1 and −1 ≤ cos θ ≤ 1, because P lies on the circle x² + y² = 1. The fundamental identity sin²θ + cos²θ = 1 is just a restatement of the circle equation.

单位圆是以 (0,0) 为圆心、半径为 1 的圆。对于任意处于标准位置的角 θ，其终边与圆相交于点 P(x,y)。根据定义，cos θ = x，sin θ = y。这一简洁的几何模型立刻表明 −1 ≤ sin θ ≤ 1 且 −1 ≤ cos θ ≤ 1，因为 P 始终在圆 x² + y² = 1 上。基本恒等式 sin²θ + cos²θ = 1 实际上就是圆的方程的重述。

Tangent is defined as tan θ = sin θ / cos θ, geometrically corresponding to the slope of the terminal ray or the y‑coordinate of the point where the terminal ray meets the vertical line x = 1. This definition explains why tan θ is undefined when cos θ = 0.

正切定义为 tan θ = sin θ / cos θ，几何上对应于终边的斜率或终边与直线 x = 1 交点的 y 坐标。这一定义解释了为何 cos θ = 0 时 tan θ 无定义。

3. Sine, Cosine and Tangent as Circular Functions | 正弦、余弦与正切作为圆函数

The word “circular” highlights that these functions arise naturally from the circle. As θ increases, the point P travels around the unit circle, generating wave‑like changes in x and y. Plotting (θ, sin θ) gives the sine wave; plotting (θ, cos θ) gives the cosine wave, shifted by π/2. Tangent, by contrast, repeats every π and has vertical asymptotes where cos θ = 0. Recognising this motion helps you visualise graphs and symmetries without memorisation.

“圆函数”一词强调这些函数自然地从圆中产生。随着 θ 的增大，点 P 沿单位圆周运动，使 x 和 y 产生波浪式的变化。画出 (θ, sin θ) 得到正弦波；画出 (θ, cos θ) 得到余弦波，两者相差 π/2 的相位。正切则每 π 重复一次，并在 cos θ = 0 处有垂直渐近线。认识到这种运动有助于你直观地想象图像和对称性，而无需死记硬背。

For any real θ, the definitions remain valid: cos(θ + 2π) = cos θ, sin(θ + 2π) = sin θ. This periodicity is a direct consequence of the circular path returning to its starting point after a full revolution.

对于任意实数 θ，该定义始终有效：cos(θ + 2π) = cos θ，sin(θ + 2π) = sin θ。这种周期性是绕圆一周后回到起点的直接结果。

4. Exact Values for Special Angles | 特殊角的精确值

IB examinations frequently require exact trigonometric values without a calculator. Using the unit circle or the standard 30‑60‑90 and 45‑45‑90 triangles, you can derive coordinates for multiples of π/6 and π/4. The table below summarises the essential first‑quadrant values; symmetry then gives values in other quadrants.

IB考试经常要求不借助计算器给出精确的三角值。利用单位圆或标准的 30‑60‑90 和 45‑45‑90 三角形，你可以推导出 π/6 和 π/4 整数倍角的坐标。下表总结了第一象限的关键值；再利用对称性便可得到其他象限的值。

θ (degrees)	θ (radians)	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined

Always simplify radicals and rationalise denominators unless instructed otherwise. Knowing these exact values allows you to solve equations like sin x = √2/2 quickly and to evaluate compound angles.

除非另有说明，始终化简根式并有理化分母。掌握这些精确值可以让你快速求解诸如 sin x = √2/2 的方程，并计算复合角。

5. Periodic Nature and Symmetry | 周期性及对称性

Sine and cosine have period 2π, while tangent has period π. The unit circle also reveals key symmetries: cos(−θ) = cos θ (even function), sin(−θ) = −sin θ (odd function) and tan(−θ) = −tan θ. Reflection properties, such as sin(π − θ) = sin θ and cos(π + θ) = −cos θ, come from the circle’s symmetry about the axes and origin. These relationships are the foundation for solving cast‑diagram problems and simplifying complicated expressions.

正弦与余弦的周期为 2π，正切的周期为 π。单位圆还揭示了关键的对称性：cos(−θ) = cos θ（偶函数），sin(−θ) = −sin θ（奇函数），tan(−θ) = −tan θ。诸如 sin(π − θ) = sin θ 和 cos(π + θ) = −cos θ 等反射性质，来源于圆周关于坐标轴和原点的对称性。这些关系是解决象限图问题与化简复杂表达式的基础。

When evaluating a trigonometric expression, always note the quadrant of the angle to determine the correct sign. Using the ASTC mnemonic (All Students Take Calculus) or a unit‑circle sketch helps avoid sign errors.

在计算三角表达式时，务必注意角所处的象限以确定正确的符号。使用 ASTC（全正、正弦正、正切正、余弦正）记忆口诀或画单位圆草图，有助于避免符号错误。

6. Trigonometric Identities from the Unit Circle | 从单位圆导出的三角恒等式

The unit circle provides a visual proof of the Pythagorean identity: sin²θ + cos²θ = 1. Dividing this identity by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ yields 1 + cot²θ = csc²θ. These three Pythagorean forms are indispensable for simplifying expressions and solving equations. Further identities, such as the compound‑angle formulas, can be derived geometrically and are widely tested in IB.

单位圆为毕达哥拉斯恒等式 sin²θ + cos²θ = 1 提供了直观的证明。将该恒等式两边同除以 cos²θ 得到 1 + tan²θ = sec²θ；同除以 sin²θ 则得到 1 + cot²θ = csc²θ。这三种毕达哥拉斯形式在化简表达式和解方程时不可或缺。此外，诸如复合角公式等其他恒等式可以通过几何方法推导得出，也是IB考试中常考的内容。

Key identities to master:

需要掌握的关键恒等式：

sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B

These identities are most powerful when combined with exact values, enabling you to find, for example, cos 75° by writing it as cos(45°+30°).

这些恒等式与精确值结合时威力最大，例如，你可以将 cos 75° 写成 cos(45°+30°) 来求值。

7. Graphs of Circular Functions | 圆函数的图像

The graph of y = sin x is a wave oscillating between −1 and 1 with x‑intercepts at integer multiples of π. The cosine graph is the sine graph shifted left by π/2. Both have a smooth, continuous shape with a period of 2π. The tangent graph consists of repeated branches with vertical asymptotes at x = π/2 + kπ, for k ∈ ℤ. Its period is π, and it is strictly increasing on each open interval between asymptotes.

y = sin x 的图像是在 −1 与 1 之间振荡的波形，x 截距为 π 的整数倍。余弦图像是正弦图像向左平移 π/2 的结果。两者都是光滑连续的，周期为 2π。正切图像由重复的分支构成，在 x = π/2 + kπ（k ∈ ℤ）处有垂直渐近线。其周期为 π，并且在渐近线之间的每个开区间上严格递增。

Labelling key points — maxima, minima, intercepts — on one period helps you sketch accurate graphs. Always mark the axis scales and indicate the period length when drawing these functions in IB exams.

标注一个周期内的关键点——最大值、最小值、截距——有助于绘出准确的图像。在IB考试中绘制这些函数时，务必标出坐标轴尺度并注明周期长度。

8. Transformations of Trigonometric Graphs | 三角图像的变换

The general sine function is written as y = A sin(B(x − C)) + D, where |A| is the amplitude, the period is 2π/|B|, C is the horizontal phase shift and D is the vertical translation. Similar forms hold for cosine and tangent (though amplitude is not defined for tangent). Combining these parameters allows you to model real‑world oscillations such as tides, sound waves or temperature variations.

一般正弦函数写作 y = A sin(B(x − C)) + D，其中 |A| 为振幅，周期为 2π/|B|，C 为水平相位位移，D 为垂直平移。余弦函数同样如此（正切无振幅定义）。组合使用这些参数，你可以对现实世界中的振荡现象建模，如潮汐、声波或温度变化。

When analysing a transformed graph, work from the inside out: factor B, identify the base function, apply horizontal shift/scale, then vertical scale and shift. This systematic approach prevents sign errors and mislabelling of axes.

在分析变换后的图像时，由内向外逐步进行：提取因子 B，确定基函数，然后依次进行水平移位与缩放，最后是纵向缩放与平移。这种系统的方法能防止符号错误和坐标轴标注错误。

9. Inverse Circular Functions | 反圆函数

To define inverse functions, the domains of sine, cosine and tangent must be restricted so they become one‑to‑one. The principal branches are:

arcsin x, y ∈ [−π/2, π/2]
arccos x, y ∈ [0, π]
arctan x, y ∈ (−π/2, π/2)

These inverse functions are tested in equations like sin⁻¹(x) + cos⁻¹(x) = π/2 and in calculus contexts. IB students should be comfortable using them in both exact‑value problems and when solving for an angle given a trigonometric ratio.

要定义反函数，必须限制正弦、余弦和正切的定义域，使其成为一一映射。主值分支为：

arcsin x，y ∈ [−π/2, π/2]
arccos x，y ∈ [0, π]
arctan x，y ∈ (−π/2, π/2)

这些反函数在诸如 sin⁻¹(x) + cos⁻¹(x) = π/2 的方程以及微积分情境中都会考查。IB学生应能熟练运用它们处理精确值问题，并在已知三角比时求解角度。

Remember that arcsin(sin(θ)) only equals θ if θ lies within the principal range. Always check the domain before cancelling a trigonometric function with its inverse.

请记住，仅当 θ 落在主值范围内时，arcsin(sin(θ)) 才等于 θ。在消去一个三角函数与其反函数之前，务必检查定义域。

10. Solving Trigonometric Equations | 解三角方程

IB trigonometric equations often require a multi‑step strategy: use identities to rewrite the equation in terms of a single circular function, find the principal solution using the unit circle or a calculator, and then generate all solutions within the specified interval using periodicity and symmetry. For example, to solve 2 sin²x − sin x − 1 = 0, treat it as a quadratic in sin x, factor it to (2 sin x + 1)(sin x − 1) = 0, and solve sin x = −1/2 and sin x = 1.

IB三角方程经常需要多步策略：利用恒等式将方程写成只含一个圆函数的形式，用单位圆或计算器求出主解，然后利用周期性和对称性在所给区间内生成所有解。例如，解方程 2 sin²x − sin x − 1 = 0，可将其视为 sin x 的二次方程，分解为 (2 sin x + 1)(sin x − 1) = 0，再分别解 sin x = −1/2 和 sin x = 1。

When solving equations like sin 2x = cos x, apply double‑angle identities to reduce the argument and then use zero‑product property. Always state the general solution set before listing specific answers in the required domain, and check for extraneous solutions introduced by squaring or by undefined values in tangent or reciprocal functions.

在求解诸如 sin 2x = cos x 的方程时，先应用倍角恒等式统一变量，然后利用零积性质。在给出指定区间内的具体答案之前，应先写出通解，并检查因平方或因正切与倒数函数无定义而引入的增根。

11. Modelling with Circular Functions | 用圆函数建模

Periodic behaviour appears in countless real‑life contexts: daylight hours, ferris wheel motion, alternating current, pendulum swings. The sinusoidal model y = A sin(B(t − C)) + D (or cosine) is the standard tool. The parameter D is the equilibrium level, |A| the amplitude, and 2π/|B| the period. IB problems often give data points and ask you to determine these parameters and predict future values.

周期性行为出现在无数现实情境中：日照时长、摩天轮运动、交流电、摆锤摆动等。正弦模型 y = A sin(B(t − C)) + D（或余弦）是标准工具。参数 D 为平衡位置，|A| 为振幅，2π/|B| 为周期。IB问题通常会给出数据点，要求你确定这些参数并预测未来数值。

When modelling, choose the offset C so that the graph passes through a given point at t = 0, and always verify that your model yields values within a realistic range. Always interpret your results in the context of the problem, including units and meaningful rounding.

建模时，应选取相位偏移 C 使图像在 t = 0 时经过给定点，并确保模型产生的数值在合理范围内。务必结合问题背景解释结果，包括单位与合理的舍入。

12. Summary and Common Pitfalls | 总结与常见错误

Circular functions unify geometry, algebra and analysis through the unit circle. Mastery requires fluency with radian measure, exact values, identities and graph transformations. Common mistakes include confusing degrees and radians, forgetting to check the quadrant for sign, applying identities incorrectly (e.g. assuming sin(A+B) = sin A + sin B), and losing solutions when solving equations. Using a structured approach — sketch the circle or graph, note symmetries, and verify domain restrictions — will dramatically reduce errors.

圆函数通过单位圆将几何、代数与分析统一起来。若要精通，需熟练掌握弧度制、精确值、恒等式和图像变换。常见错误包括混淆度与弧度、忘记检查象限以确定符号、错误应用恒等式（例如误认为 sin(A+B) = sin A + sin B）以及在解方程时遗漏解。采用有条理的方法——画圆或图像草图、注意对称性、验证定义域限制——将显著减少错误。

Consistent practice with IB‑style questions, particularly those that combine algebraic manipulation with graphical interpretation, is the best way to build confidence. Remember that every circular function problem, no matter how complex, can be traced back to the simple geometry of a circle of radius 1.

持续练习IB风格的题目，尤其是那些将代数运算与图像解读相结合的题目，是建立信心的最佳途径。请记住，无论多么复杂的圆函数问题，都可追溯到半径为1的圆的简单几何。

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Circular Functions: The Unit Circle and Trigonometric Graphs | 圆函数：单位圆与三角图像