IB Mathematics: Trigonometry and Circular Functions | IB数学：三角学与圆函数

📚 IB Mathematics: Trigonometry and Circular Functions | IB数学：三角学与圆函数

Trigonometry is fundamentally the study of relationships between angles and side lengths in triangles, while circular functions extend these ideas via the unit circle.

三角学是研究三角形中角度与边长关系的学科，而圆函数通过单位圆将这些概念进行了拓展。

In IB Mathematics, mastery of radian measure, the unit circle, graphing, identities and equation solving is essential for both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses.

在IB数学中，掌握弧度制、单位圆、图像、恒等式与方程求解，对于分析与方法（AA）和应用与解释（AI）两门课程都至关重要。

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

A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. In many IB questions, radian measure is the default unit for trigonometric calculations.

弧度定义为圆上长度等于半径的弧所对的圆心角。在IB考试中，弧度制通常是三角计算的默认单位。

To convert between degrees and radians: 360° = 2π rad, hence 1° = π/180 rad and 1 rad = 180/π degrees.

弧度和角度换算：360° = 2π rad，因此 1° = π/180 rad，1 rad = 180/π 度。

arc length: s = rθ, area of sector: A = ½r²θ

These formulas only work when θ is measured in radians. If degrees are given, convert first.

这些公式仅在θ以弧度为单位时成立。如果题目给出角度，需要先转换为弧度。

2. The Unit Circle and Circular Functions | 2. 单位圆与圆函数

The unit circle is a circle of radius 1 centred at the origin. For any real number θ, we define cos θ as the x-coordinate and sin θ as the y-coordinate of the point where the terminal side of the angle meets the circle.

单位圆是以原点为圆心、半径为1的圆。对于任意实数θ，我们定义cos θ为角的终边与圆交点处点的x坐标，sin θ为y坐标。

From the unit circle equation x² + y² = 1, we immediately get the fundamental Pythagorean identity sin² θ + cos² θ = 1.

由单位圆方程 x² + y² = 1，我们立即得到基本毕达哥拉斯恒等式 sin² θ + cos² θ = 1。

The other circular functions are defined as ratios:

其他圆函数定义为比值：

tan θ = sin θ / cos θ

tan θ = sin θ / cos θ
sec θ = 1 / cos θ

sec θ = 1 / cos θ
csc θ = 1 / sin θ

csc θ = 1 / sin θ
cot θ = cos θ / sin θ

cot θ = cos θ / sin θ

Remembering the signs of these functions in the four quadrants is often tested. The acronym ASTC (All, Sine, Tangent, Cosine) helps: in the first quadrant all are positive; in the second sine and cosecant are positive; in the third tangent and cotangent are positive; in the fourth cosine and secant are positive.

记住四个象限中这些函数的正负号常被考查。口诀“ASTC”（All, Sine, Tangent, Cosine）可帮助记忆：第一象限全部为正；第二象限正弦和余割为正；第三象限正切和余切为正；第四象限余弦和正割为正。

3. Graphs of Sine, Cosine and Tangent | 3. 正弦、余弦和正切函数图像

Understanding the graphs of circular functions is essential for solving equations and modelling periodic phenomena. Sine and cosine have amplitude 1 and period 2π, while tangent has period π and vertical asymptotes.

理解圆函数的图像对解方程和建立周期性模型至关重要。正弦和余弦的振幅为1，周期为2π；正切的周期为π，并具有垂直渐近线。

Key points to sketch y = sin x: maximum at x = π/2, 5π/2, …, minimum at x = 3π/2, 7π/2, …, and zeros at x = nπ. The cosine graph is a horizontal translation of the sine graph by π/2.

绘制y = sin x的关键点：在x = π/2, 5π/2, …处取最大值，在x = 3π/2, 7π/2, …处取最小值，在x = nπ处为零。余弦图像是正弦图像水平平移π/2的结果。

For y = tan x, there are vertical asymptotes at x = π/2 + nπ, and zeros at x = nπ. The graph has no amplitude; it increases without bound between asymptotes.

对于y = tan x，垂直渐近线位于x = π/2 + nπ，零点在x = nπ。图像没有振幅，在渐近线之间无限增涨。

4. Transformations of Trigonometric Graphs | 4. 三角函数的图像变换

In IB, you are expected to describe and construct transformations of trigonometric functions. The general form is y = a sin(bx – c) + d, where a affects amplitude, b affects period, c causes horizontal shift, and d shifts vertically.

IB考试要求你描述和构造三角函数的变换。一般形式为y = a sin(bx – c) + d，其中a影响振幅，b影响周期，c产生水平位移，d产生垂直位移。

Amplitude = |a|. If a is negative, the graph is reflected in the x-axis.

振幅 = |a|。若a为负，则图像关于x轴反射。
Period = 2π/|b| for sine and cosine; for tangent it is π/|b|.

正弦和余弦的周期 = 2π/|b|；正切的周期为 π/|b|。
Phase shift = c/b. The graph is shifted to the right if c > 0 and to the left if c < 0 in the standard form y = a sin(bx - c) + d.

相位位移 = c/b。在标准形式y = a sin(bx – c) + d中，若c > 0向右平移，若c < 0向左平移。
Vertical shift = d, moving the midline to y = d.

垂直位移 = d，中心线移到y = d。

For example, y = 2 sin(3x – π) + 1 has amplitude 2, period 2π/3, phase shift π/3 to the right, and midline y = 1.

例如，y = 2 sin(3x – π) + 1 的振幅为2，周期为2π/3，相位位移为π/3向右，中心线为y = 1。

5. Fundamental Identities | 5. 基本恒等式

Trigonometric identities are equations that hold for all values of the variable. They are used to simplify expressions and prove other identities.

三角恒等式是对变量所有取值都成立的等式，常用于化简表达式和证明其他恒等式。

The three Pythagorean identities are central:

三个毕达哥拉斯恒等式是关键：

sin² θ + cos² θ = 1

sin² θ + cos² θ = 1
1 + tan² θ = sec² θ

1 + tan² θ = sec² θ
1 + cot² θ = csc² θ

1 + cot² θ = csc² θ

These identities follow from dividing sin² θ + cos² θ = 1 by cos² θ and sin² θ respectively.

这些恒等式分别通过将 sin² θ + cos² θ = 1 除以 cos² θ 和 sin² θ 得到。

You should also know the negative-angle identities: sin(-θ) = -sin θ, cos(-θ) = cos θ, tan(-θ) = -tan θ.

你还应知道负角恒等式：sin(-θ) = -sin θ，cos(-θ) = cos θ，tan(-θ) = -tan θ。

6. Compound Angle Identities | 6. 复角恒等式

Compound angle identities express trigonometric functions of sums or differences of angles.

复角恒等式表达了两个角和或差的三角函数。

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

tan(A ± B) = (tan A ± tan B) /

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IB Mathematics: Trigonometry and Circular Functions | IB数学：三角学与圆函数