Applying the Unit Circle in IB Mathematics | IB数学：单位圆的应用方法

📚 Applying the Unit Circle in IB Mathematics | IB数学：单位圆的应用方法

The unit circle is one of the most powerful visual tools in IB Mathematics. It connects angles, coordinates, and trigonometric functions in a single diagram, allowing students to solve problems without memorising hundreds of values.

单位圆是IB数学中最强大的视觉工具之一。它将角度、坐标和三角函数连接在同一个图形中，让学生无需死记硬背数百个数值就能解决问题。

1. Definition and Core Idea | 定义与核心思想

The unit circle is a circle with radius 1 centred at the origin of the coordinate plane. For any angle θ measured from the positive x-axis, the point where the terminal side intersects the circle has coordinates (cos θ, sin θ).

单位圆是以坐标原点为圆心、半径为1的圆。对于从x轴正方向开始测量的任意角度θ，终边与圆的交点坐标为(cos θ, sin θ)。

This definition means that the x-coordinate gives the cosine value, and the y-coordinate gives the sine value. The tangent value is the ratio y/x, provided x ≠ 0.

这一定义意味着x坐标给出余弦值，y坐标给出正弦值。正切值是y与x的比值，前提是x ≠ 0。

(cos θ, sin θ) = (x, y) on the unit circle

单位圆上的点 (cos θ, sin θ) = (x, y)

2. Special Angles and Exact Values | 特殊角与精确值

The unit circle makes it easy to recall exact values for special angles: 0°, 30°, 45°, 60°, 90°, and their multiples in all four quadrants.

单位圆使我们可以轻松回忆特殊角的精确值：0°、30°、45°、60°、90°以及它们在四个象限中的倍数角。

At 0° (0 rad): (1, 0), so sin 0 = 0 and cos 0 = 1.

在0°（0弧度）处：(1, 0)，所以 sin 0 = 0，cos 0 = 1。
At 30° (π/6): (√3/2, 1/2), so sin 30° = 1/2 and cos 30° = √3/2.

在30°（π/6）处：(√3/2, 1/2)，所以 sin 30° = 1/2，cos 30° = √3/2。
At 45° (π/4): (√2/2, √2/2), so both sine and cosine equal √2/2.

在45°（π/4）处：(√2/2, √2/2)，所以正弦和余弦都等于√2/2。
At 60° (π/3): (1/2, √3/2), so sin 60° = √3/2 and cos 60° = 1/2.

在60°（π/3）处：(1/2, √3/2)，所以 sin 60° = √3/2，cos 60° = 1/2。
At 90° (π/2): (0, 1), so sin 90° = 1 and cos 90° = 0.

在90°（π/2）处：(0, 1)，所以 sin 90° = 1，cos 90° = 0。

Using symmetry, these values extend to angles in every quadrant, such as 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 330°.

利用对称性，这些值可以扩展到每个象限的角度，如120°、135°、150°、180°、210°、225°、240°、270°、300°、315°和330°。

3. Quadrant Signs and CAST Rule | 象限符号与CAST法则

The sign of each trigonometric function depends on the quadrant in which the terminal side lies. The unit circle clearly shows this through the signs of x and y coordinates.

每个三角函数的符号取决于终边所在的象限。单位圆通过x和y坐标的符号清晰地展示了这一点。

Quadrant I (0° to 90°): x > 0, y > 0, so sin, cos, and tan are all positive.

第一象限（0°到90°）：x > 0，y > 0，所以sin、cos和tan都是正的。
Quadrant II (90° to 180°): x < 0, y > 0, so sin is positive, while cos and tan are negative.

第二象限（90°到180°）：x < 0，y > 0，所以sin为正，cos和tan为负。
Quadrant III (180° to 270°): x < 0, y < 0, so tan is positive, while sin and cos are negative.

第三象限（180°到270°）：x < 0，y < 0，所以tan为正，sin和cos为负。
Quadrant IV (270° to 360°): x > 0, y < 0, so cos is positive, while sin and tan are negative.

第四象限（270°到360°）：x > 0，y < 0，所以cos为正，sin和tan为负。

A common mnemonic is CAST: Cosine positive in Quadrant IV, All positive in Quadrant I, Sine positive in Quadrant II, Tangent positive in Quadrant III.

常用助记法则是CAST：第四象限余弦为正，第一象限全部为正，第二象限正弦为正，第三象限正切为正。

4. Converting Between Radians and Degrees | 弧度与角度的换算

The unit circle naturally uses radians, where one full revolution equals 2π radians, and 180° equals π radians. Converting between units is straightforward with the proportion:

单位圆天然使用弧度，其中一整圈等于2π弧度，180°等于π弧度。通过比例关系可以轻松进行单位换算：

180° = π rad, so 1° = π/180 rad, and 1 rad = 180°/π

180° = π弧度，因此1° = π/180弧度，1弧度 = 180°/π

For example, 45° is π/4 rad, 90° is π/2 rad, 270° is 3π/2 rad, and 360° is 2π rad. In IB exams, radian measure is assumed unless the degree symbol is explicitly shown.

例如，45°是π/4弧度，90°是π/2弧度，270°是3π/2弧度，360°是2π弧度。在IB考试中，除非明确显示角度符号，否则默认使用弧度制。

5. Finding Coordinates of Points | 求点的坐标

A direct application of the unit circle is finding the coordinates of a point on a circle of radius r. If a point P lies on a circle of radius r and the angle from the positive x-axis is θ, then:

单位圆的一个直接应用是求半径为r的圆上点的坐标。如果点P位于半径为r的圆上，且与x轴正方向的夹角为θ，则：

P = (r cos θ, r sin θ)

This formula is essential for solving many IB problems involving circular motion, coordinates on a Ferris wheel, or points on a rotating object.

这个公式对于解决许多IB问题至关重要，包括圆周运动、摩天轮上的坐标或旋转物体上的点。

Example: A point is 5 units from the origin and makes an angle of 60° with the positive x-axis. Its coordinates are (5 cos 60°, 5 sin 60°) = (5 × 1/2, 5 × √3/2) = (2.5, 4.33).

示例：一个点距离原点5个单位，与x轴正方向成60°角。其坐标为(5 cos 60°, 5 sin 60°) = (5 × 1/2, 5 × √3/2) = (2.5, 4.33)。

6. Solving Trigonometric Equations | 解三角方程

The unit circle is the most reliable method for solving trigonometric equations over a given interval. By visualising where the angle lies, students can find all solutions without losing roots.

单位圆是求解给定区间内三角方程最可靠的方法。通过可视化角度所在的位置，学生可以找到所有解而不遗漏根。

For example, solve sin θ = 1/2 for 0 ≤ θ < 2π.

例如，求解 sin θ = 1/2，其中 0 ≤ θ < 2π。

Sine equals 1/2 at two points on the unit circle: θ = π/6 and θ = 5π/6. The first is in Quadrant I, and the second is in Quadrant II, where sine is also positive.

正弦等于1/2在单位圆上有两个点：θ = π/6和θ = 5π/6。第一个在第一象限，第二个在第二象限，正弦在第二象限也为正。

sin θ = 1/2 ⇒ θ = π/6, 5π/6

Similarly, cos θ = -√2/2 has solutions at θ = 3π/4 and θ = 5π/4, because x is negative in Quadrants II and III.

类似地，cos θ = -√2/2的解在θ = 3π/4和θ = 5π/4处，因为x在第二和第三象限为负。

7. Trigonometric Identities from the Circle | 由单位圆推导三角恒等式

The unit circle provides immediate visual proof of the Pythagorean identity. Since the point (cos θ, sin θ) lies on the circle x² + y² = 1, we substitute to obtain:

单位圆为勾股恒等式提供了直观的视觉证明。因为点(cos θ, sin θ)位于圆x² + y² = 1上，我们代入得到：

cos²θ + sin²θ = 1

Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives cot²θ + 1 = csc²θ. These identities are essential in IB calculus and proof questions.

除以cos²θ得到1 + tan²θ = sec²θ，除以sin²θ得到cot²θ + 1 = csc²θ。这些恒等式在IB微积分和证明题中至关重要。

The unit circle also demonstrates that cos(-θ) = cos θ and sin(-θ) = -sin θ, because reflecting an angle across the x-axis changes only the sign of y.

单位圆还展示了cos(-θ) = cos θ和sin(-θ) = -sin θ，因为将角度关于x轴反射只会改变y的符号。

8. Reference Angles | 参考角

A reference angle is the acute angle between the terminal side and the x-axis. The unit circle shows that trigonometric values for any angle are equal in magnitude to those of its reference angle, with signs determined by the quadrant.

参考角是终边与x轴之间的锐角。单位圆表明，任意角的三角函数值与其参考角的函数值在绝对值上相等，符号由象限决定。

For an angle θ in standard position, the reference angle α is calculated as:

对于标准位置的角θ，参考角α的计算方法如下：

Quadrant I: α = θ

第一象限：α = θ
Quadrant II: α = π – θ

第二象限：α = π – θ
Quadrant III: α = θ – π

第三象限：α = θ – π
Quadrant IV: α = 2π – θ

第四象限：α = 2π – θ

For example, sin 150° = sin 30° = 1/2, and cos 240° = -cos 60° = -1/2. This technique reduces every trigonometric evaluation to a known special angle.

例如，sin 150° = sin 30° = 1/2，cos 240° = -cos 60° = -1/2。这种技巧将每个三角函数的求值简化为已知的特殊角。

9. Periodic Properties | 周期性质

The unit circle has circumference 2π, so after a full rotation, the coordinates repeat. This means sine and cosine are periodic with period 2π:

单位圆的周长为2π，因此经过一整圈旋转后坐标会重复。这意味着正弦和余弦是周期为2π的周期函数：

sin(θ + 2πn) = sin θ, cos(θ + 2πn) = cos θ, n ∈ ℤ

sin(θ + 2πn) = sin θ，cos(θ + 2πn) = cos θ，n ∈ ℤ

Tangent has period π because tan(θ + π) = tan θ, which follows from the fact that (x, y) and (-x, -y) have the same ratio y/x.

正切的周期为π，因为tan(θ + π) = tan θ，这源于(x, y)和(-x, -y)具有相同的比值y/x。

These periodic properties are used to simplify angles greater than 2π, such as sin(10π/3), by subtracting 2π to find an equivalent angle within one revolution.

这些周期性质用于简化大于2π的角度，例如sin(10π/3)，通过减去2π找到一个等效角度。

10. Composite Angle Problems | 复合角问题

IB questions often ask for the exact value of expressions involving sums or differences of angles. The unit circle provides the values needed to apply the compound angle identities correctly.

IB题目经常要求涉及角和或差的表达式的精确值。单位圆提供了正确应用复合角恒等式所需的值。

For example, find cos(75°) using cos 75° = cos(45° + 30°). From the unit circle, cos 45° = √2/2, sin 45° = √2/2, cos 30° = √3/2, and sin 30° = 1/2.

例如，使用cos 75° = cos(45° + 30°)求cos 75°。从单位圆可得cos 45° = √2/2，sin 45° = √2/2，cos 30° = √3/2，sin 30° = 1/2。

cos(45° + 30°) = cos 45° cos 30° – sin 45° sin 30° = (√2/2)(√3/2) – (√2/2)(1/2) = (√6 – √2)/4

This approach is faster and more reliable than memorising separate tables, because the unit circle values are all derived consistently.

这种方法比记忆单独的表格更快、更可靠，因为单位圆的值都是一致推导出来的。

11. Coordinate Geometry Applications | 坐标几何应用

The unit circle can be used to find the equation of a circle, the distance between points, and the location of points after rotation.

单位圆可用于求圆的方程、点之间的距离以及旋转后点的位置。

The standard equation of a circle centred at the origin with radius r is x² + y² = r². When the centre moves to (h, k), the equation becomes (x – h)² + (y – k)² = r².

以原点为圆心、半径为r的圆的标准方程为x² + y² = r²。当圆心移动到(h, k)时，方程变为(x – h)² + (y – k)² = r²。

If a point is rotated by angle θ about the origin, its new coordinates are found by matrix multiplication or by applying the unit circle definitions in reverse.

如果一个点绕原点旋转角度θ，其新坐标可通过矩阵乘法或反向应用单位圆定义来求得。

Rotation by θ: (x’, y’) = (x cos θ – y sin θ, x sin θ + y cos θ)

旋转θ角：(x’, y’) = (x cos θ – y sin θ, x sin θ + y cos θ)

12. Common IB Exam Pitfalls | IB考试常见陷阱

Many students lose marks in IB exams because of small mistakes with the unit circle. Knowing these pitfalls helps you avoid them.

许多学生在IB考试中因为单位圆的小错误而丢分。了解这些陷阱可以帮助你避免它们。

Using degrees in a radians-only question: always check the calculator setting and question wording.

在仅使用弧度的问题中使用角度模式：始终检查计算器设置和题目措辞。
Forgetting negative signs in Quadrants II, III, and IV: always sketch the quadrant.

忘记第二、三、四象限中的负号：始终画出象限。
Mixing up sine and cosine: remember cos is x (horizontal) and sin is y (vertical).

混淆正弦和余弦：记住cos是x（水平），sin是y（垂直）。
Finding only one solution for an equation that has two solutions in a period.

对于在一个周期内有两个解的方程只找到一个解。
Using x/y instead of y/x for tangent: tangent is opposite over adjacent, which is y/x.

使用x/y而不是y/x来计算正切：正切是对边比邻边，即y/x。

Always draw a rough unit circle for any trigonometric question. The visual representation prevents sign errors and missing solutions.

在任何三角函数问题中都要画一个粗略的单位圆。视觉化的表示可以防止符号错误和遗漏解。

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Applying the Unit Circle in IB Mathematics | IB数学：单位圆的应用方法