ESAT Mathematics: Geometry & Trigonometry | ESAT 数学：几何与三角学专题

📚 ESAT Mathematics: Geometry & Trigonometry | ESAT 数学：几何与三角学专题

The ESAT (Engineering and Science Admissions Test) demands a solid command of geometry and trigonometry — two interconnected pillars of pre-university mathematics. This article breaks down the essential concepts, formulas, and problem-solving strategies you need to maximise your score.

ESAT（工程与科学入学考试）要求考生扎实掌握几何与三角学——这是大学预科数学中两个紧密相连的支柱。本文将系统梳理核心概念、公式及解题策略，助你冲刺高分。

1. Coordinate Geometry: Lines and Gradients | 坐标几何：直线与斜率

The equation of a straight line can be written in the form y = mx + c, where m is the gradient and c is the y-intercept. The gradient between two points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ − y₁) / (x₂ − x₁).

直线方程可写作 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率公式为 m = (y₂ − y₁) / (x₂ − x₁)。

Two lines are parallel if they have equal gradients. Two lines are perpendicular if the product of their gradients is −1; that is, m₁ × m₂ = −1. This fact is frequently tested in ESAT coordinate geometry questions.

两条直线平行当且仅当它们的斜率相等。两条直线垂直时，它们的斜率之积为 −1，即 m₁ × m₂ = −1。这是 ESAT 坐标几何题目中的高频考点。

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2) | Distance = √((x₂ − x₁)² + (y₂ − y₁)²)

For the general form ax + by + c = 0, the gradient is −a/b. When a question gives a line in this form, always convert to y = mx + c first to avoid sign errors.

对于一般式 ax + by + c = 0，斜率为 −a/b。当题目以这种形式给出直线时，务必先转换为 y = mx + c，以避免符号错误。

2. Circles: Equations and Tangents | 圆：方程与切线

The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². When the centre is at the origin, this simplifies to x² + y² = r².

圆心为 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。当圆心在原点时，方程简化为 x² + y² = r²。

To find where a line intersects a circle, substitute the line equation into the circle equation to obtain a quadratic. The discriminant Δ = b² − 4ac tells you the number of intersections: Δ > 0 means two intersections (secant), Δ = 0 means one intersection (tangent), and Δ < 0 means no intersection.

要求直线与圆的交点，将直线方程代入圆方程得到一个二次方程。判别式 Δ = b² − 4ac 决定交点数量：Δ > 0 表示两个交点（割线），Δ = 0 表示一个交点（切线），Δ < 0 表示无交点。

A tangent to a circle is always perpendicular to the radius at the point of contact. Thus, if you know the centre and the point of tangency, the tangent’s gradient is the negative reciprocal of the radius’s gradient.

圆的切线在切点处始终垂直于半径。因此，若已知圆心和切点，切线的斜率就是半径斜率的负倒数。

3. Circle Theorems | 圆的基本定理

ESAT frequently tests the standard circle theorems. Memorise these relationships — they form the backbone of many geometry problems.

ESAT 经常考查标准的圆定理。牢记以下关系——它们是许多几何问题的核心基础。

The angle at the centre is twice the angle at the circumference subtended by the same arc.

圆心角等于同弧所对圆周角的二倍。
The angle in a semicircle is a right angle (90°).

半圆内的圆周角是直角（90°）。
Angles in the same segment are equal.

同弧上的圆周角相等。
The opposite angles of a cyclic quadrilateral sum to 180°.

圆内接四边形的对角之和为 180°。
The tangent-chord theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

弦切角定理：切线与弦的夹角等于其所夹弧对应的圆周角。

When solving circle theorem questions, always mark the centre, radii, and any equal angles on the diagram before attempting the problem. This visual approach significantly reduces mistakes.

在解答圆定理题目时，先标出圆心、半径和图中相等的角，再进行求解。这种图示化方法能显著减少错误。

4. Trigonometric Ratios: SOH CAH TOA | 三角函数比：SOH CAH TOA

For a right-angled triangle, the three primary trigonometric ratios are defined as follows, where θ is an acute angle:

对于直角三角形，三个基本三角比定义如下，其中 θ 为锐角：

sin θ = opposite/hypotenuse | cos θ = adjacent/hypotenuse | tan θ = opposite/adjacent

Remember the exact values for common angles: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. These appear constantly in ESAT questions.

记住常见角的精确值：sin 30° = ½，sin 45° = √2/2，sin 60° = √3/2，cos 30° = √3/2，cos 45° = √2/2，cos 60° = ½，tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。这些在 ESAT 题目中反复出现。

5. Sine Rule and Cosine Rule | 正弦定理与余弦定理

For any triangle with sides a, b, c opposite angles A, B, C respectively, the sine rule states:

对于任意三角形，设边 a、b、c 分别对角 A、B、C，正弦定理为：

a/sin A = b/sin B = c/sin C

The sine rule is ideal when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA — beware of the ambiguous case).

当已知两角一边（AAS 或 ASA），或两边及其中一边的对角（SSA——注意多解情况）时，使用正弦定理最为合适。

The cosine rule relates sides and an included angle:

余弦定理将三边与一个夹角联系起来：

a² = b² + c² − 2bc·cos A

Use the cosine rule when you know two sides and the included angle (SAS), or all three sides (SSS) to find an angle: cos A = (b² + c² − a²) / (2bc).

已知两边及其夹角（SAS）或三边（SSS）求角时使用余弦定理：cos A = (b² + c² − a²) / (2bc)。

The area of a triangle using trigonometry is Area = ½ab·sin C. This formula is particularly useful when combined with the sine and cosine rules in multi-step problems.

使用三角函数求三角形面积的公式为：面积 = ½ab·sin C。该公式与正余弦定理结合时，在解决多步骤问题中尤为实用。

6. Trigonometric Identities | 三角恒等式

The two fundamental identities you must know perfectly are:

你必须熟练掌握的两个基本恒等式是：

sin² θ + cos² θ = 1 | tan θ = sin θ / cos θ

From sin² θ + cos² θ = 1, you can derive two useful forms by dividing by cos² θ and sin² θ respectively:

由 sin² θ + cos² θ = 1，分别除以 cos² θ 和 sin² θ 可导出两个有用形式：

tan² θ + 1 = sec² θ | 1 + cot² θ = cosec² θ

These identities allow you to simplify expressions, prove equations, and solve trigonometric equations. When you see sin² θ + cos² θ in an ESAT problem, immediately consider replacing it with 1.

这些恒等式可用于化简表达式、证明等式和解三角方程。当在 ESAT 题目中看到 sin² θ + cos² θ 时，立即考虑将其替换为 1。

7. Graphs of Trigonometric Functions | 三角函数的图像

The graph of y = sin θ is a wave that oscillates between −1 and 1, with a period of 360° (or 2π radians). The graph of y = cos θ has the same shape but is shifted 90° to the left. The graph of y = tan θ has vertical asymptotes at θ = 90° + 180°k, where k is an integer.

y = sin θ 的图像是在 −1 和 1 之间振荡的波形，周期为 360°（或 2π 弧度）。y = cos θ 的图像形状相同，但向左平移 90°。y = tan θ 的图像在 θ = 90° + 180°k（k 为整数）处有垂直渐近线。

Transformations of trig graphs follow the same rules as other functions: y = a·sin(bx) has amplitude |a| and period 360°/b. The value of a stretches or compresses the graph vertically, while b affects the horizontal scale.

三角函数图像的变换遵循与其他函数相同的规则：y = a·sin(bx) 的振幅为 |a|，周期为 360°/b。a 的值垂直拉伸或压缩图像，而 b 影响水平缩放。

To solve equations like sin θ = 0.5 for 0° ≤ θ < 360°, find the principal solution using a calculator, then use symmetry: the second solution is 180° − θ for sine and 360° − θ for cosine.

求解如 sin θ = 0.5（0° ≤ θ < 360°）的方程时，先用计算器找出主解，再利用对称性：正弦的第二个解为 180° − θ，余弦的第二个解为 360° − θ。

8. Radians and Circular Measure | 弧度制与圆弧度量

Radians provide an alternative way to measure angles. One full revolution equals 2π radians, so 180° = π radians. The conversion factors are: radians = degrees × π/180, and degrees = radians × 180/π.

弧度是度量角的另一种方式。一整圈等于 2π 弧度，所以 180° = π 弧度。换算关系为：弧度 = 角度 × π/180，角度 = 弧度 × 180/π。

When working in radians, the arc length of a sector is s = rθ, and the area of a sector is A = ½r²θ. These formulas are much cleaner in radians than in degrees.

使用弧度时，扇形弧长为 s = rθ，扇形面积为 A = ½r²θ。这些公式在弧度制下远比角度制简洁。

Arc length: s = rθ | Sector area: A = ½r²θ | Segment area: ½r²(θ − sin θ)

The segment area formula (a sector minus the triangle) is a favourite in ESAT. Remember that θ must be in radians for these formulas to work correctly.

弓形面积公式（扇形减去三角形）是 ESAT 的常用考点。注意 θ 必须使用弧度，这些公式才能正确成立。

9. Solving Trigonometric Equations | 解三角方程

Solving trig equations requires careful attention to the domain. For example, solve 2cos θ + 1 = 0 for 0 ≤ θ < 2π. Rearrange to cos θ = −½. The reference angle is π/3, and since cosine is negative in quadrants II and III, the solutions are θ = 2π/3 and θ = 4π/3.

解三角方程需要仔细关注定义域。例如，求解 2cos θ + 1 = 0（0 ≤ θ < 2π）。移项得 cos θ = −½。参考角为 π/3，由于余弦在第二和第三象限为负，因此解为 θ = 2π/3 和 θ = 4π/3。

For equations involving a transformed angle like sin(2θ) = ½ with 0 ≤ θ < 2π, first change the domain: multiply the bounds by 2, giving 0 ≤ 2θ < 4π. Then find all solutions for 2θ, and finally divide by 2.

对于涉及变换角如 sin(2θ) = ½（0 ≤ θ < 2π）的方程，先变换定义域：将边界乘以 2，得到 0 ≤ 2θ < 4π。然后找出 2θ 的所有解，最后除以 2。

Quadratic forms such as 2sin² θ − sin θ − 1 = 0 can be factorised just like an ordinary quadratic. Let x = sin θ, solve 2x² − x − 1 = 0 = (2x + 1)(x − 1), giving sin θ = −½ or sin θ = 1.

二次形式如 2sin² θ − sin θ − 1 = 0 可以像普通二次方程一样因式分解。设 x = sin θ，解 2x² − x − 1 = 0 = (2x + 1)(x − 1)，得到 sin θ = −½ 或 sin θ = 1。

10. Three-Dimensional Geometry | 三维几何

ESAT often includes 3D geometry problems that require you to find angles between lines and planes, or the shortest distance between points. The key strategy is to identify a suitable right-angled triangle within the 3D figure.

ESAT 常包含三维几何问题，要求求解线与平面之间的夹角，或两点间的最短距离。关键策略是在三维图形中找出合适的直角三角形。

To find the angle between a line and a plane, project the line onto the plane and measure the angle between the line and its projection. This angle is always the smallest angle between the line and any line in the plane.

要求一条直线与一个平面的夹角，需将直线投影到平面上，然后测量直线与其投影之间的角度。这个角始终是直线与平面内任意直线所成的最小角。

For the angle between two planes, take a cross-section perpendicular to both planes. The angle between the two intersection lines in this cross-section equals the dihedral angle between the planes.

要求两个平面之间的夹角，取垂直于两个平面的截面。该截面中两条交线之间的夹角即为二面角。

11. Problem-Solving Strategies | 解题策略

ESAT geometry and trigonometry questions reward a systematic approach. Always start by drawing a clear diagram — even if one is given, redraw it with all known values labelled. Next, identify which formula connects the given information to the unknown.

ESAT 几何与三角题青睐系统化的解题方法。始终先画清晰的示意图——即使题目已给出图形，也要重画并标注所有已知值。接下来，确定哪个公式将已知信息与未知量联系起来。

Look for right angles: they trigger Pythagoras, SOH CAH TOA, or the angle-in-semicircle theorem.

寻找直角：直角可触发勾股定理、SOH CAH TOA 或半圆圆周角定理。
If a triangle has no right angle, apply the sine rule, cosine rule, or area formula.

如果三角形没有直角，则应用正弦定理、余弦定理或面积公式。
Check your calculator is in the correct mode (degrees or radians) before every calculation.

每次计算前检查计算器是否处于正确的模式（角度制或弧度制）。
For multi-step problems, write down intermediate results — ESAT rewards method marks.

对于多步骤问题，写下中间结果——ESAT 按步骤给分。

Time management is critical: aim to spend no more than 90 seconds per multiple-choice question. If a geometry problem seems intractable, move on and return to it with fresh eyes.

时间管理至关重要：每道选择题的用时尽量不超过 90 秒。如果某道几何题一时无法攻克，先跳过，稍后带着新的思路回头解答。

12. Common Pitfalls | 常见易错点

Even strong students lose marks on avoidable errors. Here are the most frequent traps in ESAT geometry and trigonometry:

即使是优秀学生也会因可避免的错误而失分。以下是 ESAT 几何与三角中最常见的陷阱：

Using the sine rule with the ambiguous case (SSA): check whether two triangles are possible.

使用正弦定理时遇到多解情况（SSA）：检查是否可能存在两个三角形。
Forgetting that sin θ = sin(180° − θ), so two angles share the same sine value.

忘记 sin θ = sin(180° − θ)，即两个角可能共享同一个正弦值。
Mixing up the period of tan (180°) with sin and cos (360°).

混淆 tan 的周期（180°）与 sin、cos 的周期（360°）。
Using degree-mode values in radian formulas: arc length s = rθ requires θ in radians.

在弧度公式中使用角度制数值：弧长 s = rθ 要求 θ 以弧度为单位。
Sign errors when applying the cosine rule to find an obtuse angle: cos is negative in quadrant II.

用余弦定理求解钝角时出现符号错误：余弦在第二象限为负值。

Practice with past papers and time yourself under exam conditions. Review every mistake to identify whether it was a conceptual gap or a careless slip.

用历年真题练习，并在考试条件下计时。复盘每一个错误，判断它是概念漏洞还是粗心失误。

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ESAT Mathematics: Geometry & Trigonometry | ESAT 数学：几何与三角学专题