5.1 Angles, Radians, and Unit Circle Definitions

5.1 Angles, Radians, and Unit Circle Definitions

Trigonometry in upper secondary mathematics extends far beyond the right-triangle ratios ($\text{opposite}/\text{hypotenuse}$, etc.) encountered in middle school geometry. In the Gaokao examination framework, trigonometric functions are formulated as real-valued functions of real variables defined on the Cartesian coordinate plane via the unit circle. This section establishes the fundamental concepts of general angles, radian measure, sector optimization, and unit circle definitions.

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1. Extension of the Angle Concept (任意角的概念)

In elementary geometry, an angle is restricted to the interval $[0^\circ, 360^\circ]$. However, describing continuous phenomena such as rotation, harmonic oscillation, and wave motion requires angles to encompass arbitrary real numbers $\alpha \in \mathbb{R}$.

Definition of General Angles

An angle is generated by rotating a ray about its endpoint (the vertex) from an initial side (始边) to a terminal side (终边):

• Positive Angles (正角): Generated by counterclockwise rotation.
• Negative Angles (负角): Generated by clockwise rotation.
• Zero Angle (零角): Generated when the ray undergoes no rotation at all.

Standard Position and Co-Terminal Angles

When an angle $\alpha$ is placed in standard position on a Cartesian coordinate system, its vertex is at the origin $O(0,0)$ and its initial side coincides with the positive $x$-axis.

• Co-Terminal Angles (终边相同的角): Two angles $\alpha$ and $\beta$ share the exact same terminal side if and only if their difference is an integer multiple of $360^\circ$ (or $2\pi$ radians). The set of all angles co-terminal with $\alpha$ is expressed as: $$S = \{\beta \mid \beta = \alpha + k \cdot 360^\circ, k \in \mathbb{Z}\} = \{\beta \mid \beta = \alpha + 2k\pi, k \in \mathbb{Z}\}$$

Quadrant Angles and Axis-Bound Angles

Depending on where its terminal side lies, an angle in standard position is classified as:

1. Quadrant Angle (象限角): The terminal side lies strictly inside one of the four quadrants.
   • Quadrant I: $\{\alpha \mid 2k\pi < \alpha < 2k\pi + \frac{\pi}{2}, k \in \mathbb{Z}\}$
   • Quadrant II: $\{\alpha \mid 2k\pi + \frac{\pi}{2} < \alpha < 2k\pi + \pi, k \in \mathbb{Z}\}$
   • Quadrant III: $\{\alpha \mid 2k\pi + \pi < \alpha < 2k\pi + \frac{3\pi}{2}, k \in \mathbb{Z}\}$
   • Quadrant IV: $\{\alpha \mid 2k\pi + \frac{3\pi}{2} < \alpha < 2k\pi + 2\pi, k \in \mathbb{Z}\}$
2. Axis-Bound Angle (轴线角 / 象限界角): The terminal side coincides with one of the coordinate axes (e.g., $0, \pm\frac{\pi}{2}, \pm\pi, \pm\frac{3\pi}{2}, \dots$).

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2. Radian Measure and Sector Formulas (弧度制与扇形公式)

Definition of Radian

One radian ($1\text{ rad}$) is defined as the measure of a central angle that subtends an arc whose length is equal to the radius of the circle. Because radian measure represents a ratio of two lengths (arc length divided by radius), radians are dimensionless real numbers.

Conversion Between Degrees and Radians

Since a full circle of radius $r$ has circumference $2\pi r$ and corresponds to $360^\circ$, we have:

$$360^\circ = 2\pi \text{ rad} \implies 180^\circ = \pi \text{ rad}$$

• Degree to Radian: $\alpha_{\text{rad}} = \alpha^\circ \times \frac{\pi}{180^\circ}$
• Radian to Degree: $\alpha^\circ = \alpha_{\text{rad}} \times \frac{180^\circ}{\pi}$
• Common special values: $30^\circ = \frac{\pi}{6}$, $45^\circ = \frac{\pi}{4}$, $60^\circ = \frac{\pi}{3}$, $90^\circ = \frac{\pi}{2}$, $120^\circ = \frac{2\pi}{3}$, $135^\circ = \frac{3\pi}{4}$, $150^\circ = \frac{5\pi}{6}$, $180^\circ = \pi$.

Arc Length and Sector Area Formulas

For a circle of radius $r$ and a central angle $\alpha$ measured in radians ($|\alpha|$ being its absolute magnitude):

• Arc Length ($l$): $$l = |\alpha| r$$
• Sector Area ($S$): $$S = \frac{1}{2} l r = \frac{1}{2} |\alpha| r^2$$

Sector Area Optimization (Gaokao High-Frequency Problem Type)

A recurring problem on the Gaokao involves finding the maximum area of a sector given a fixed perimeter $C$. Let $C = 2r + l$ be the fixed perimeter. Substituting $l = C - 2r$ into the sector area formula yields:

$$S(r) = \frac{1}{2} (C - 2r) r = \frac{C}{2}r - r^2 = -\left(r - \frac{C}{4}\right)^2 + \frac{C^2}{16}$$

Thus, the sector achieves its maximum area $S_{\max} = \frac{C^2}{16}$ when $r = \frac{C}{4}$ and arc length $l = \frac{C}{2}$. The corresponding central angle is:

$$\alpha = \frac{l}{r} = \frac{C/2}{C/4} = 2 \text{ rad}$$

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3. Unit Circle Definition of Trigonometric Functions (三角函数的单位圆定义)

To define trigonometric functions for any angle $\alpha \in \mathbb{R}$, we construct a unit circle centered at the origin with radius $r = 1$, given by the equation $x^2 + y^2 = 1$.

Definition via Unit Circle Coordinates

Let $\alpha$ be an angle in standard position. Its terminal side intersects the unit circle at a unique point $P(x, y)$. The three basic trigonometric functions are defined as:

$$\sin\alpha = y, \quad \cos\alpha = x, \quad \tan\alpha = \frac{y}{x} \quad (x \neq 0)$$

General Point Definition

If $P(x, y)$ is any arbitrary point (other than the origin) on the terminal side of angle $\alpha$, and its distance to the origin is $r = \sqrt{x^2 + y^2} > 0$, then:

$$\sin\alpha = \frac{y}{r}, \quad \cos\alpha = \frac{x}{r}, \quad \tan\alpha = \frac{y}{x} \quad (x \neq 0)$$

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4. Quadrant Signs and Trigonometric Line Representations (象限符号与三角函数线)

Quadrant Signs of Trigonometric Functions

Because $r > 0$, the algebraic sign of $\sin\alpha$, $\cos\alpha$, and $\tan\alpha$ depends strictly on the coordinates $(x, y)$ of point $P$:

Mnemonic Rule: "一全正，二正弦，三正切，四余弦" (Quadrant I: all positive; II: sine positive; III: tangent positive; IV: cosine positive).

Trigonometric Lines (三角函数线)

In the unit circle, directed line segments represent the numeric values and signs of trigonometric functions for an angle $\alpha$:

1. Sine Line ($MP$): The directed segment from the projection $M(x, 0)$ on the $x$-axis to point $P(x, y)$. Its directed magnitude equals $\sin\alpha$.
2. Cosine Line ($OM$): The directed segment from the origin $O(0, 0)$ to projection $M(x, 0)$. Its directed magnitude equals $\cos\alpha$.
3. Tangent Line ($AT$): The directed segment along the line $x = 1$ from $A(1, 0)$ to the intersection $T(1, \tan\alpha)$ of the terminal side (or its backward extension) with $x = 1$. Its directed magnitude equals $\tan\alpha$.

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5. Worked Gaokao Exam Examples (高考典型例题精讲)

Worked Example 1: Sector Optimization Problem

Problem: A sector has a fixed perimeter of $16\text{ cm}$. Find the radius $r$, arc length $l$, and central angle $\alpha$ that maximize the area of the sector. Calculate the maximum area $S_{\max}$.

Solution:

1. Formulate the perimeter relation: $$2r + l = 16 \implies l = 16 - 2r$$
2. Express sector area $S$ in terms of $r$: $$S = \frac{1}{2} l r = \frac{1}{2} (16 - 2r) r = 8r - r^2$$
3. Complete the square: $$S(r) = -(r^2 - 8r + 16) + 16 = -(r - 4)^2 + 16$$
4. Determine optimal values: Since $r > 0$ and $l = 16 - 2r > 0$, we have $0 < r < 8$. The parabola opens downward, achieving its maximum at $r = 4\text{ cm}$.
   • Arc length: $l = 16 - 2(4) = 8\text{ cm}$.
   • Central angle: $\alpha = \frac{l}{r} = \frac{8}{4} = 2\text{ rad}$.
   • Maximum area: $S_{\max} = 16\text{ cm}^2$.

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Worked Example 2: Point on Terminal Side

Problem: Suppose the terminal side of an angle $\alpha$ in standard position passes through the point $P(-3, y)$ where $y \neq 0$. If $\sin\alpha = -\frac{4}{5}$, determine the value of $y$ and evaluate $\cos\alpha$ and $\tan\alpha$.

Solution:

1. Express distance $r$ from origin: $$r = \sqrt{(-3)^2 + y^2} = \sqrt{9 + y^2}$$
2. Apply unit/general circle definition of sine: $$\sin\alpha = \frac{y}{r} = \frac{y}{\sqrt{9 + y^2}} = -\frac{4}{5}$$
3. Solve for $y$: Since $\sin\alpha < 0$, $y$ must be negative ($y < 0$). Squaring both sides yields: $$\frac{y^2}{9 + y^2} = \frac{16}{25} \implies 25y^2 = 16(9 + y^2) = 144 + 16y^2$$ $$9y^2 = 144 \implies y^2 = 16 \implies y = -4 \quad (\text{since } y < 0)$$
4. Calculate $r$, $\cos\alpha$, and $\tan\alpha$:
   • Distance: $r = \sqrt{9 + (-4)^2} = 5$.
   • Cosine: $\cos\alpha = \frac{x}{r} = \frac{-3}{5} = -\frac{3}{5}$.
   • Tangent: $\tan\alpha = \frac{y}{x} = \frac{-4}{-3} = \frac{4}{3}$.