📚 Circular Functions: Definitions and Core Properties | 圆函数的定义与核心性质
The circular functions, also known as trigonometric functions, are defined using the unit circle and provide a fundamental link between angles and ratios. They are essential for modelling periodic behaviour in physics, engineering, and the natural sciences.
圆函数，也称三角函数，基于单位圆定义，是角度与比值之间的基本联系。它们在物理、工程和自然科学中对周期性行为的建模至关重要。
1. The Unit Circle and Radian Measure | 单位圆与弧度制
clockwise from the positive x-axis, the point where the terminal side of the angle intersects the circle is denoted P(x, y).
单位圆是以直角坐标系原点为圆心、半径为1的圆，其方程为 x² + y² = 1。对于从正 x 轴逆时针旋转的任意角度 θ，其终边与圆的交点记为 P(x, y)。
Angles are commonly measured in radians in mathematics. One radian is the angle subtended by an arc whose length equals the radius. Since the total circumference is 2π, a full revolution corresponds to 2π radians, and 180° equals π radians.
数学中常用弧度作为角度单位。1弧度是弧长等于半径时对应的圆心角。由于圆周长为 2π，因此一整圈对应 2π 弧度，而 180° 等于 π 弧度。
θ (radians) = θ (degrees) × π / 180
This conversion is essential when working with circular functions, as their derivatives and series expansions are naturally expressed in radians.
这一换算在处理圆函数时至关重要，因为其导数与级数展开均以弧度自然表达。
2. Definitions of Sine and Cosine | 正弦函数与余弦函数的定义
For a point P(x, y) on the unit circle corresponding to angle θ, the cosine and sine functions are defined as the x- and y-coordinates of P respectively:
对于单位圆上与角度 θ 对应的点 P(x, y)，余弦函数和正弦函数分别定义为 P 的横坐标与纵坐标：
cos θ = x, sin θ = y
Thus P can be written as P(cos θ, sin θ). These definitions extend the trigonometric ratios from right-angled triangles to all real angles, including negative angles and angles greater than 360°.
因此 P 可写为 P(cos θ, sin θ)。这一定义将三角函数比从直角三角形推广到所有实数角度，包括负角和大于 360° 的角。
osite / hypotenuse, cos θ = adjacent / hypotenuse, provided the hypotenuse is scaled to 1.
对于直角三角形，这些定义与常见比值一致：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，前提是将斜边缩放为1。
3. The Other Four Circular Functions | 其他四个圆函数
From sine and cosine, four further circular functions are defined as ratios:
由正弦和余弦还可定义另外四个圆函数之比：
- Tangent: tan θ = sin θ / cos θ, where cos θ ≠ 0.
- Cotangent: cot θ = cos θ / sin θ, where sin θ ≠ 0.
- Secant: sec θ = 1 / cos θ, where cos θ ≠ 0.
- Cosecant: csc θ = 1 / sin θ, where sin θ ≠ 0.
- 正切：tan θ = sin θ / cos θ，其中 cos θ ≠ 0。
- 余切：cot θ = cos θ / sin θ，其中 sin θ ≠ 0。
- 正割：sec θ = 1 / cos θ，其中 cos θ ≠ 0。
- 余割：csc θ = 1 / sin θ，其中 sin θ ≠ 0。
minators vanish. For example, tan θ is undefined at θ = π/2 + nπ, because cos θ = 0 there.
当分母为零时这些函数无定义。例如，tan θ 在 θ = π/2 + nπ 处无定义，因为此时 cos θ = 0。
4. Periodicity | 周期性
. The circular functions exhibit natural periodic behaviour due to the rotation of the unit circle.
若函数 f 满足 f(θ + T) = f(θ) 对所有定义域内的 θ 成立，则称 T 为周期。由于单位圆的旋转，圆函数表现出天然的周期性。
- sin θ and cos θ have period 2π: sin(θ + 2π) = sin θ, cos(θ + 2π) = cos θ.
- tan θ and cot θ have period π: tan(θ + π) = tan θ, cot(θ + π) = cot θ.
- sin θ 与 cos θ 的周期为 2π：sin(θ + 2π) = sin θ，cos(θ + 2π) = cos θ。
- tan θ 与 cot θ 的周期为 π：tan(θ + π) = tan θ，cot(θ + π) = cot θ。
Because of this repetition, circular functions are ideal for modelling alternating currents, sound waves, and tidal patterns.
由于这种重复性，圆函数非常适合模拟交流电、声波和潮汐模式。
5. Parity Properties | 奇偶性
Parity describes whether a function is symmetric about the y-axis (even) or about the origin (odd). For circular functions:
奇偶性描述函数是关于 y 轴对称（偶函数）还是关于原点对称（奇函数）。对于圆函数：
- sin θ is odd: sin(−θ) = −sin θ.
- cos θ is even: cos(−θ) = cos θ.
- tan θ is odd: tan(−θ) = −tan θ.
- csc θ, cot θ are odd; sec θ is even.
- sin θ 是奇函数：sin(−θ) = −sin θ。
- cos θ 是偶函数：cos(−θ) = cos θ。
- tan θ 是奇函数：tan(−θ) = −tan θ。
- csc θ、cot θ 是奇函数；sec θ 是偶函数。
These symmetry properties simplify evaluating functions at negative angles and help when sketching graphs.
这些对称性可简化负角函数值的计算，也有助于绘制函数图像。
6. Pythagorean Identities | 毕达哥拉斯恒等式
x² + y² = 1 and the definitions x = cos θ, y = sin θ, we obtain the fundamental identity:
由单位圆方程 x² + y² = 1 以及 x = cos θ、y = sin θ 的定义，可得基本恒等式：
sin²θ + cos²θ = 1
Dividing this identity by cos²θ (where cos θ ≠ 0) yields:
将该恒等式两边除以 cos²θ（其中 cos θ ≠ 0），得到：
1 + tan²θ = sec²θ
Similarly, dividing by sin²θ (where sin θ ≠ 0) gives:
同理，两边除以 sin²θ（其中 sin θ ≠ 0），得到：
cot²θ + 1 = csc²θ
These identities are indispensable for simplifying expressions, proving other identities, and solving trigonometric equations.
这些恒等式在化简表达式、证明其他恒等式以及求解三角方程时不可或缺。
7. Exact Values at Key Angles | 特殊角的精确值
The coordinates of the unit circle provide exact values for sine, cosine, and tangent at commonly used angles. The following table lists the principal values:
单位圆上的坐标给出了常用角度的正弦、余弦和正切的精确值。下表列出了一系列主要角的值：
| θ (radians) | sin θ | cos θ | tan θ |
| 0 | 0 | 1 | 0 |
| π/6 | 1/2 | √3/2 | 1/√3 |
| π/4 | √2/2 | √2/2 | 1 |
| π/3 | √3/2 | 1/2 | √3 |
| π/2 | 1 | 0 | undefined |
These values are derived directly from the symmetry of the unit circle and are expected to be memorised for IB examinations.
这些值可直接由单位圆的对称性得出，在 IB 考试中要求熟记。
8. Signs in the Four Quadrants | 四个象限中的符号
The sign of each circular function depends on the quadrant in which the terminal side of the angle lies. The mnemonic ASTC ('All Students Take Calculus') summarises this:
各圆函数的符号取决于角终边所在的象限。助记口诀“ASTC”（All Students Take Calculus，全部为正、正弦为正、正切为正、余弦为正）可概括如下：
- Quadrant I: all functions are positive.
- Quadrant II: only sin (and csc) are positive.
- Quadrant III: only tan (and cot) are positive.
- Quadrant IV: only cos (and sec) are positive.
- 第一象限：所有函数为正。
- 第二象限：仅 sin（及 csc）为正。
- 第三象限：仅 tan（及 cot）为正。
- 第四象限：仅 cos（及 sec）为正。
Knowing the signs is crucial when solving trigonometric equations over a specified interval, as it helps identify valid solutions.
在指定区间内求解三角方程时，了解符号至关重要，可帮助判断有效解。
9. Graphs and Transformations | 图像与变换
graph of y = tan θ consists of repeated branches with vertical asymptotes at θ = π/2 + nπ.
y = sin θ 的图像是介于 −1 和 1 之间平滑的波浪线，并通过原点。y = cos θ 的图像形状相同，但向左平移 π/2。y = tan θ 的图像由重复的分支组成，在 θ = π/2 + nπ 处有垂直渐近线。
A general sinusoidal function can be written in the form:
一般正弦型函数可写作如下形式：
y = a sin(bx + c) + d
- The amplitude is |a|, half the vertical distance between maximum and minimum.
- The period is 2π / |b| (for sine and cosine).
- The phase shift is −c / b (horizontal shift).
- The vertical shift is d, moving the midline to y = d.
- 振幅 |a|，为最大值与最小值之间垂直距离的一半。
- 周期为 2π / |b|（适用于正弦和余弦）。
- 相移为 −c / b（水平平移）。
- 垂直平移为 d，将中轴线移至 y = d。
These transformations allow us to model real-world oscillatory motion, such as sound waves and alternating current.
这些变换使我们能够模拟真实世界中的振荡运动，如声波与交流电。
10. Inverse Circular Functions | 反圆函数
Because circular functions are periodic, they are not one-to-one over their entire domains. To define inverses, we restrict each function to a specific interval:
由于圆函数具有周期性，它们在其整个定义域上不是一一对应的。为了定义反函数，需将每个函数限制在特定区间内：
- y = arcsin x has domain [−1, 1] and range [−π/2, π/2].
- y = arccos x has domain [−1, 1] and range [0, π].
- y = arctan x has domain ℝ and range (−π/2, π/2).
- y = arcsin x 的定义域为 [−1, 1]，值域为 [−π/2, π/2]。
- y = arccos x 的定义域为 [−1, 1]，值域为 [0, π]。
- y = arctan x 的定义域为 ℝ，值域为 (−π/2, π/2)。
These principal values ensure that each inverse function is single-valued. In IB mathematics, both degrees and radians may be used, but radian measure is preferred in calculus.
这些主值确保每个反函数是单值的。在 IB 数学中，度和弧度都可使用，但在微积分中优先使用弧度。
11. Applications and Problem Solving | 应用与解题
ly many solutions, but within the interval [0, 2π) the solutions are θ = π/6 and θ = 5π/6.
圆函数用于求解三角方程、分析周期数据以及模拟波动。例如，方程 sin θ = 1/2 有无数解，但在区间 [0, 2π) 内解为 θ = π/6 和 θ = 5π/6。
(2x − 1) + 4, where we combine amplitude, period, phase shift, and vertical shift to describe a physical scenario, such as the height of a tide over time.
另一个重要应用是复合函数，如 y = 3 sin(2x − 1) + 4，其中综合振幅、周期、相移和垂直平移来描述实际情境，例如潮汐随时间的变化高度。
Mastery of circular functions is essential for success in IB Mathematics, as they appear in calculus, vectors, complex numbers, and probability.
掌握圆函数是 IB 数学取得好成绩的基础，它们在微积分、向量、复数和概率中均有出现。
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