3D Trigonometry | 三维三角学

📚 3D Trigonometry | 三维三角学

Three-dimensional trigonometry extends the familiar rules of right-angled triangles and the sine, cosine and tangent ratios from flat, two-dimensional diagrams into solid shapes such as cuboids, pyramids and prisms. This topic is a core part of the Edexcel IGCSE Mathematics syllabus, and it tests your ability to visualise depth, identify the correct right-angled triangle inside a 3D figure, and apply Pythagoras’ theorem and trigonometric ratios accurately.

三维三角学将平面直角三角形中的勾股定理以及正弦、余弦和正切比值扩展到立方体、棱锥和棱柱等立体图形中。这一主题是 Edexcel IGCSE 数学大纲的核心内容，考查你在三维图形中想象深度、识别正确的直角三角形，并准确运用勾股定理和三角比的能力。

1. The 3D Pythagorean Theorem | 三维勾股定理

In two dimensions, the distance between two points can be found using a² + b² = c². In three dimensions, this extends naturally. If a cuboid has edge lengths a, b and c, then the length of its space diagonal (from one vertex to the opposite vertex) is given by:

在二维平面中，两点之间的距离可以用 a² + b² = c² 求出。在三维空间中，这一公式可以自然推广。如果一个长方体有三条棱长 a、b 和 c，那么它的空间对角线（从一个顶点到对顶的顶点）长度为：

d² = a² + b² + c²

For example, in a cuboid measuring 6 cm × 8 cm × 10 cm, the space diagonal is √(6² + 8² + 10²) = √(36 + 64 + 100) = √200 = 10√2 cm. This is the single most important formula in 3D trigonometry, because almost every problem reduces to applying Pythagoras twice: once to find a face diagonal, and once more to find the space diagonal.

例如，在一个尺寸为 6 cm × 8 cm × 10 cm 的长方体中，空间对角线长度为 √(6² + 8² + 10²) = √(36 + 64 + 100) = √200 = 10√2 cm。这是三维三角学中最重要的一条公式，因为几乎所有问题都可以归结为两次运用勾股定理：先用一次求出面对角线，再用一次求出空间对角线。

2. Identifying Right-Angled Triangles in 3D | 在三维图形中识别直角三角形

The key skill in 3D trigonometry is finding the correct right-angled triangle to work with. Every face of a cuboid, prism or pyramid is either a rectangle, a triangle or another polygon, and the edges of a cuboid meet at right angles. When you are asked to find an angle or a length, first sketch the triangle formed by the three relevant points, then look for the right angle. Often the right angle is hidden: it lies where a vertical edge meets a horizontal face, or where two perpendicular edges meet.

三维三角学的关键技能是找到正确的直角三角形。长方体、棱柱或棱锥的每个面都是矩形、三角形或其他多边形，而长方体的棱在顶点处互相垂直。当你需要求某个角或长度时，先画出由三个相关点构成的三角形，然后寻找直角。直角往往是隐藏的：它位于一条竖直棱与一个水平面的交汇处，或者两条互相垂直的棱的交点处。

A helpful method is to trace the triangle in the air: hold a pencil along each of the three edges that form the triangle. If two edges are perpendicular (they form an L-shape), that is your right angle. For instance, to find the angle between the diagonal AG and the base ABCD of a cuboid, the correct triangle is triangle ACG, where AC is the diagonal of the base, and CG is the vertical edge. The right angle is at C, because CG is perpendicular to the base plane.

一个实用的方法是用手”在空中”画出三角形的三条边：沿着构成三角形的三条边移动笔尖。如果两条边互相垂直（形成 L 形），那就是你要找的直角。例如，要求长方体中对角线 AG 与底面 ABCD 之间的夹角，正确的三角形是三角形 ACG，其中 AC 是底面的对角线，CG 是竖直棱。直角在 C 点，因为 CG 垂直于底面。

3. Angle Between a Line and a Plane | 直线与平面的夹角

The angle between a line and a plane is defined as the angle between the line and its projection (shadow) onto the plane. If you imagine shining a torch directly down onto a plane, the projection of a slanted line is the shadow it casts. The angle is always measured as the smaller angle between the line and the plane.

直线与平面的夹角定义为该直线与其在平面上的投影（影子）之间的夹角。想象用手电筒垂直照向一个平面，一条斜线的投影就是它投下的影子。该夹角总是取直线与平面之间的较小角。

To find this angle, follow three steps. Step 1: identify the point where the line meets the plane. Step 2: drop a perpendicular from the other endpoint of the line to the plane — this is usually a vertical edge. Step 3: join the foot of this perpendicular to the point where the line meets the plane. The right-angled triangle is now complete, and you can use sin θ = opposite ÷ hypotenuse, or tan θ = opposite ÷ adjacent.

要求这个角，按三步进行。第一步：找到直线与平面的交点。第二步：从直线的另一个端点向平面作垂线——这通常是一条竖直棱。第三步：把垂足与直线和平面的交点连接起来。这样直角三角形就完整了，你可以使用 sin θ = 对边 ÷ 斜边，或 tan θ = 对边 ÷ 邻边。

sin θ = opposite ÷ hypotenuse = vertical height ÷ length of the line

sin θ = 对边 ÷ 斜边 = 垂直高度 ÷ 直线长度

For example, in a cuboid ABCDEFGH with AB = 6, BC = 8, and CG = 10, the angle between the diagonal AG and the base ABCD is found using triangle ACG. First compute AC = √(6² + 8²) = 10. Then tan θ = CG ÷ AC = 10 ÷ 10 = 1, so θ = 45°.

例如，在长方体 ABCDEFGH 中，AB = 6，BC = 8，CG = 10。要求对角线 AG 与底面 ABCD 的夹角，使用三角形 ACG。先计算 AC = √(6² + 8²) = 10，然后 tan θ = CG ÷ AC = 10 ÷ 10 = 1，因此 θ = 45°。

4. Angle Between Two Planes | 两平面之间的夹角

The angle between two planes is the angle between two lines, each drawn in one of the planes, both perpendicular to the line of intersection of the planes. This is called the “dihedral angle”. A simpler way to think about it: imagine opening a book — the

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