Right-Angled Triangle Trigonometry: Sine, Cosine and Tangent | 直角三角形三角学：正弦、余弦与正切

📚 Right-Angled Triangle Trigonometry: Sine, Cosine and Tangent | 直角三角形三角学：正弦、余弦与正切

Right-angled triangle trigonometry connects side lengths and acute angles through three ratios: sine, cosine and tangent. These ratios appear in IGCSE examinations in questions on missing sides, missing angles, bearings, elevation and depression, and real-life contexts such as ramps, ladders and navigation.

直角三角形三角学通过正弦、余弦和正切这三个比将边长与锐角联系起来。这些比出现在 IGCSE 考试中，涉及求未知边、未知角、方位角、仰角与俯角，以及斜坡、梯子和导航等实际情境。

Mastering this topic requires accurate labelling, correct ratio selection and confidence with inverse trigonometric functions. This revision guide covers the essential skills and common exam techniques you need.

掌握这一主题需要准确标记边、正确选择三角比以及熟练使用反三角函数。本复习指南涵盖你所需的基本技能和常见考试技巧。

1. Labelling the Right-Angled Triangle | 标记直角三角形

Correct labelling is the first step in any trigonometric solution. The hypotenuse is always the longest side and is directly opposite the right angle. When you choose an acute angle, usually labelled θ, the side opposite θ is the opposite side, and the remaining side next to θ is the adjacent side.

正确标记是任何三角解题的第一步。斜边总是最长的边，并且正对直角。当你选定一个锐角，通常记为 θ 时，对着 θ 的边是对边，剩下与 θ 相邻的边就是邻边。

The labels depend on which angle you are using. If you switch from one acute angle to the other, the opposite and adjacent sides swap, but the hypotenuse stays the same.

这些标签取决于你使用哪个角。如果从一个锐角切换到另一个锐角，对边和邻边会互换，但斜边保持不变。

For example, in a 5-12-13 triangle, if θ is formed by the 5 and 13 sides, then the adjacent side is 5, the opposite side is 12 and the hypotenuse is 13.

例如，在一个 5-12-13 的三角形中，如果 θ 由 5 和 13 两条边构成，那么邻边是 5，对边是 12，斜边是 13。

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2. The Three Trigonometric Ratios | 三个三角比

For any right-angled triangle and an acute angle θ, three ratios are defined:

对于任何直角三角形和锐角 θ，定义三个比：

sin θ = opposite ÷ hypotenuse

cos θ = adjacent ÷ hypotenuse

tan θ = opposite ÷ adjacent

These ratios depend only on the angle θ, not on the size of the triangle. This is why they are so powerful: if you know one side and one acute angle, the other sides can be found.

这些比只取决于角 θ，而不取决于三角形的大小。这就是它们如此有用的原因：如果你知道一条边和一个锐角，就可以求出其他边。

For acute angles, sine and cosine always give values between 0 and 1. Tangent can be greater than 1 because it compares the opposite side with the adjacent side, and either side can be longer.

对于锐角，正弦和余弦的值总是在 0 和 1 之间。正切可以大于 1，因为它比较的是对边和邻边，而这两条边中任何一条都可能更长。

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3. SOH CAH TOA Memory Aid | SOH CAH TOA 记忆法

The acronym SOH CAH TOA helps you remember which ratio uses which sides. SOH tells you that sine uses the opposite and hypotenuse. CAH tells you that cosine uses the adjacent and hypotenuse. TOA tells you that tangent uses the opposite and adjacent.

缩写 SOH CAH TOA 可以帮助你记住哪个比使用哪些边。SOH 告诉你正弦使用对边和斜边。CAH 告诉你余弦使用邻边和斜边。TOA 告诉你正切使用对边和邻边。

Write SOH CAH TOA at the side of your working in the exam. This small memory aid can prevent many errors when you are under pressure.

考试时把 SOH CAH TOA 写在运算步骤旁边。这个小小的记忆方法可以在你紧张时避免许多错误。

To choose the correct ratio, first circle the angle you are using, then label the sides, and finally underline the two sides given or required. The underlined sides will match one of the three ratios.

要选择正确的比，先圈出你正在使用的角，然后标记各边，最后在你已知或要求的两条边下面画线。画线的边会对应三个比中的一个。

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4. Finding a Missing Side | 求未知边

To find a missing side, choose the ratio that includes the known side and the side you want. Substitute the known values, then solve the equation.

要求未知边，先选择一个包含已知边和所求边的比。代入已知值，然后解方程。

Example 1: A triangle has hypotenuse 10 cm and angle 30°. Find the side opposite 30°.

例题 1：一个三角形的斜边为 10 cm，角为 30°。求 30° 角的对边。

Since the opposite and hypotenuse are involved, use sine:

因为涉及对边和斜边，所以使用正弦：

sin 30° = opposite ÷ 10

opposite = 10 × sin 30° = 10 × 0.5 = 5 cm

Example 2: The angle is 40° and the adjacent side is 7 m. Find the hypotenuse.

例题 2：角为 40°，邻边为 7 m。求斜边。

Use cosine because adjacent and hypotenuse are involved:

因为涉及邻边和斜边，所以使用余弦：

cos 40° = 7 ÷ hypotenuse

hypotenuse = 7 ÷ cos 40° ≈ 9.14 m

Notice that when the unknown side is in the denominator, you divide by the trigonometric value. Many students incorrectly multiply instead of divide here.

注意，当未知边在分母上时，你要除以三角函数值。许多学生在这里容易错误地相乘而不是相除。

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5. Finding a Missing Angle | 求未知角

If you know two sides and need an angle, use the inverse trigonometric functions. On a calculator these are usually labelled sin⁻¹, cos⁻¹ and tan⁻¹.

如果你知道两条边而要求一个角，就使用反三角函数。在计算器上，它们通常标记为 sin⁻¹、cos⁻¹ 和 tan⁻¹。

Example: The opposite side is 4 and the adjacent side is 5. Find θ.

例题：对边为 4，邻边为 5。求 θ。

Use tangent because opposite and adjacent are given:

因为已知对边和邻边，所以使用正切：

tan θ = 4 ÷ 5

θ = tan⁻¹(4 ÷ 5) ≈ 38.7°

Always check that your calculator is in degree mode when the question uses degrees. If the calculator is in radian mode, the numerical answer will be very different.

当题目使用度数时，始终检查计算器是否处于度数模式。如果计算器处于弧度模式，数值答案会截然不同。

Write the inverse function step clearly.

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