IGCSE Maths: Trigonometric Ratios in Right-Angled Triangles | IGCSE数学：直角三角形中的三角比

📚 IGCSE Maths: Trigonometric Ratios in Right-Angled Triangles | IGCSE数学：直角三角形中的三角比

In IGCSE Mathematics, trigonometry is introduced through right-angled triangles. It gives you a powerful way to find unknown sides and angles when you know only a few pieces of information. This topic builds directly on Pythagoras’ theorem and prepares you for sine, cosine and tangent functions in later study.

在IGCSE数学中，三角学从直角三角形开始学习。当你只知道少量信息时，它能帮助你求出未知的边长和角度。这一主题直接建立在毕达哥拉斯定理的基础上，并为你以后学习正弦、余弦和正切函数做好准备。

1. Introduction to Right-Angled Triangles | 直角三角形导论

A right-angled triangle has one angle equal to 90°. The side opposite this right angle is always the longest side and is called the hypotenuse. The other two sides are named relative to a chosen acute angle.

直角三角形有一个角等于90°。直角对面的边总是最长的一条，称为斜边。另外两条边则根据所选的锐角来命名。

In any right-angled triangle, the three sides are labelled as follows: the hypotenuse is opposite the right angle, the opposite side is opposite the chosen acute angle, and the adjacent side is next to the chosen acute angle but is not the hypotenuse.

在任何一个直角三角形中，三条边的标记如下：斜边在直角的对面，对边在所选锐角的对面，邻边紧挨着所选锐角但不是斜边。

Hypotenuse: always the longest side, opposite the right angle.
Opposite: the side facing the angle you are working with.
Adjacent: the side touching the angle and the right angle.

斜边：总是最长的一条边，位于直角的对面。对边：面向你正在研究的角的边。邻边：同时接触该角和直角的边。

2. The Three Trigonometric Ratios | 三个三角比

Trigonometric ratios link the angles of a right-angled triangle to the ratios of its side lengths. There are three primary ratios: sine, cosine and tangent. They are usually shortened to sin, cos and tan.

三角比将直角三角形的角与其边长的比值联系起来。主要有三个比：正弦、余弦和正切。它们通常缩写为sin、cos和tan。

For an acute angle θ in a right-angled triangle, the definitions are:

对于直角三角形中的锐角θ，定义如下：

sin θ = opposite ÷ hypotenuse

sin θ = 对边 ÷ 斜边

cos θ = adjacent ÷ hypotenuse

cos θ = 邻边 ÷ 斜边

tan θ = opposite ÷ adjacent

tan θ = 对边 ÷ 邻边

A common memory aid is SOH CAH TOA. SOH means Sine is Opposite over Hypotenuse, CAH means Cosine is Adjacent over Hypotenuse, and TOA means Tangent is Opposite over Adjacent.

一个常用的记忆口诀是SOH CAH TOA。SOH表示正弦等于对边比斜边，CAH表示余弦等于邻边比斜边，TOA表示正切等于对边比邻边。

3. Labelling the Sides Correctly | 正确标记边

Before using any trigonometric ratio, you must label the sides correctly. The hypotenuse is the easiest to find because it is always opposite the right angle. The opposite and adjacent sides depend on the angle you are given or the angle you are trying to find.

在使用任何三角比之前，你必须正确标记各条边。斜边最容易找到，因为它总是在直角的对面。对边和邻边则取决于你已知或要求的角度。

If the angle is marked at the bottom left of the triangle, then the side directly across from it is the opposite side. The side running from the angle to the right angle is the adjacent side.

如果角标记在三角形的左下角，那么该角正对面的边就是对边。从该角连到直角的边就是邻边。

Angle position	Opposite side	Adjacent side
At the bottom left	Vertical side on the right	Horizontal side along the bottom
At the top	Horizontal side along the bottom	Vertical side on the right

标记角度位置：左下角时，对边是右边的竖直边，邻边是底部的水平边。位于顶部时，对边是底部的水平边，邻边是右边的竖直边。

4. Finding an Unknown Side Using Sine | 用正弦求未知边

When you know one acute angle and the length of the hypotenuse, you can find the opposite side using the sine ratio. Write down the formula, substitute the known values, and then solve the equation.

当你已知一个锐角和斜边的长度时，可以用正弦比求出对边。写出公式，代入已知值，然后解方程。

Example: A right-angled triangle has hypotenuse 12 cm and an angle of 30°. Find the length of the side opposite the 30° angle.

例子：一个直角三角形的斜边为12 cm，一个角为30°。求30°角对面的边长。

sin 30° = opposite ÷ 12

sin 30° = 对边 ÷ 12

Since sin 30° = 0.5, this becomes 0.5 = opposite ÷ 12. Multiply both sides by 12: opposite = 0.5 × 12 = 6 cm.

因为sin 30° = 0.5，式子变为0.5 = 对边 ÷ 12。两边同时乘以12：对边 = 0.5 × 12 = 6 cm。

Always check that your answer is shorter than the hypotenuse. In this case 6 cm is shorter than 12 cm, so the answer is sensible.

始终检查你的答案是否小于斜边。在这个例子中，6 cm小于12 cm，因此答案是合理的。

5. Finding an Unknown Side Using Cosine | 用余弦求未知边

The cosine ratio is used when you know the hypotenuse and need the adjacent side, or when you know the adjacent side and need the hypotenuse. The angle must be the included acute angle.

当你已知斜边并需要求邻边，或已知邻边并需要求斜边时，使用余弦比。该角必须是相关的锐角。

Example: A right-angled triangle has a hypotenuse of 15 cm and an angle of 40°. Find the length of the side adjacent to the 40° angle.

例子：一个直角三角形的斜边为15 cm，一个角为40°。求40°角的邻边长度。

cos 40° = adjacent ÷ 15

cos 40° = 邻边 ÷ 15

Using a calculator, cos 40° ≈ 0.7660. Therefore adjacent = 0.7660 × 15 ≈ 11.49 cm. Give your answer to a sensible degree of accuracy, such as 11.5 cm.

用计算器计算，cos 40° ≈ 0.7660。因此邻边 = 0.7660 × 15 ≈ 11.49 cm。答案给出合理的精确度，例如11.5 cm。

6. Finding an Unknown Side Using Tangent | 用正切求未知边

The tangent ratio connects the opposite and adjacent sides without involving the hypotenuse. This is useful when you know one of those two sides and an acute angle, and need to find the other.

正切比将对边和邻边联系起来，不涉及斜边。当你已知其中一条边和一个锐角，并需要求另一条边时，正切非常有用。

Example: A right-angled triangle has an angle of 35° and the side opposite this angle is 8 cm. Find the length of the adjacent side.

例子：一个直角三角形有一个35°角，这个角的对边为8 cm。求邻边的长度。

tan 35° = 8 ÷ adjacent

tan 35° = 8 ÷ 邻边

Using a calculator, tan 35° ≈ 0.7002. So 0.7002 = 8 ÷ adjacent. Rearranging gives adjacent = 8 ÷ 0.7002 ≈ 11.43 cm.

用计算器计算，tan 35° ≈ 0.7002。所以0.7002 = 8 ÷ 邻边。移项得邻边 = 8 ÷ 0.7002 ≈ 11.43 cm。

When rearranging, multiply both sides by adjacent first, then divide by tan 35°. This avoids mistakes with division.

移项时，先两边乘以邻边，再除以tan 35°。这样可以避免除法错误。

7. Choosing the Correct Ratio | 选择正确的比

To decide which ratio to use, first label the sides relative to the given angle. Then ask yourself which two sides are involved in the problem. If the hypotenuse is one of them and the opposite side is the other, use sine. If the hypotenuse and adjacent side are involved, use cosine. If the opposite and adjacent sides are involved, use tangent.

要决定使用哪个比，首先根据已知角标记各条边。然后问自己题目中涉及哪两条边。如果斜边是其中一条，对边是另一条，就用正弦。如果涉及斜边和邻边，就用余弦。如果涉及对边和邻边，就用正切。

Opposite and Hypotenuse → sin θ
Adjacent and Hypotenuse → cos θ
Opposite and Adjacent → tan θ

对边和斜边 → sin θ。邻边和斜边 → cos θ。对边和邻边 → tan θ。

Drawing a small diagram and labelling the sides before starting the calculation will help you avoid selecting the wrong ratio.

在开始计算之前，画一个小图并标记各条边，有助于避免选错比。

8. Finding an Unknown Angle | 求未知角

If you know the lengths of two sides and need to find an angle, use the inverse trigonometric functions. On a calculator these are usually shown as sin⁻¹, cos⁻¹ and tan⁻¹.

如果你已知两条边的长度并需要求一个角，就使用反三角函数。在计算器上，它们通常显示为sin⁻¹、cos⁻¹和tan⁻¹。

Example: A right-angled triangle has opposite side 5 cm and hypotenuse 9 cm. Find the angle θ opposite the 5 cm side.

例子：一个直角三角形的对边为5 cm，斜边为9 cm。求5 cm边对面的角θ。

sin θ = 5 ÷ 9

θ = sin⁻¹(5 ÷ 9)

Using a calculator, θ ≈ 33.75°. You should round this to a suitable degree of accuracy, for example 33.8° or 34° depending on the question.

用计算器计算，θ ≈ 33.75°。你应根据题目要求将其舍入到合适的精确度，例如33.8°或34°。

Always check that the angle is less than 90° because it is an acute angle in a right-angled triangle. If your answer is larger than 90°, you have probably used the wrong ratio or made a calculator error.

始终检查角度是否小于90°，因为它是直角三角形中的锐角。如果你的答案大于90°，你可能用错了比或计算器输入有误。

9. Angles of Elevation and Depression | 仰角与俯角

Trigonometry is often used to solve problems involving angles of elevation and depression. The angle of elevation is the angle measured upwards from the horizontal. The angle of depression is the angle measured downwards from the horizontal.

三角学常用于解决涉及仰角和俯角的问题。仰角是从水平线向上测量的角。俯角是从水平线向下测量的角。

These angles are always measured with respect to the horizontal line, not the vertical line. In a diagram, draw the horizontal line first, then mark the angle above or below it.

这些角总是相对于水平线测量的，而不是相对于竖直线。在图中，先画水平线，再标记该水平线上方或下方的角。

Example: A ladder of length 6 m leans against a wall. The ladder makes an angle of 65° with the horizontal ground. Find the height of the top of the ladder above the ground.

例子：一架长6 m的梯子靠在墙上。梯子与水平地面成65°角。求梯子顶端离地面的高度。

sin 65° = height ÷ 6

sin 65° = 高度 ÷ 6

So height = 6 × sin 65° ≈ 6 × 0.9063 ≈ 5.44 m.

所以高度 = 6 × sin 65° ≈ 6 × 0.9063 ≈ 5.44 m。

In many exam questions, the angle of depression from a cliff to a boat is equal to the angle of elevation from the boat to the cliff because they are alternate angles formed by parallel horizontal lines.

在许多考试题中，从悬崖到船的俯角等于从船到悬崖的仰角，因为它们是由平行水平线形成的错角。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

One common mistake is labelling the opposite and adjacent sides incorrectly when the given angle is not at the bottom. Always re-read the question and draw the triangle with the angle in a convenient position.

一个常见错误是当已知角不在底部时，把对边和邻边标错。始终重新读题，并把角画在方便的位置。

Another frequent error is using the right-angle instead of the given acute angle. The right angle is 90° and trigonometric ratios are never applied to the right angle. Use the acute angle that is either given or being asked for.

另一个常见错误是使用直角而不是给定的锐角。直角是90°，三角比绝不能用于直角。要使用题目给出或要求的锐角。

Set your calculator to degree mode, not radian mode.
Write down the formula before substituting numbers.
Check that the answer for a side is positive and less than the hypotenuse when applicable.
Round only the final answer, not intermediate steps.

将计算器设置为角度模式，而不是弧度模式。在代入数字之前写出公式。检查边长答案是否为正且小于斜边（当适用时）。只对最终答案进行舍入，不要对中间步骤舍入。

In IGCSE exams, marks are often awarded for the correct method even if the final answer is wrong. Always show your working clearly, including the ratio you chose and the substituted values.

在IGCSE考试中，即使最终答案错误，正确的解题方法通常也会得分。始终清晰展示你的解题过程，包括你选择的比和代入的值。

11. Summary and Practice | 总结与练习

Trigonometry in right-angled triangles uses three ratios: sine, cosine and tangent. Remember SOH CAH TOA. Label the sides first, choose the correct ratio, substitute the known values, and solve the equation.

直角三角形中的三角学使用三个比：正弦、余弦和正切。记住SOH CAH TOA。先标记边，选择正确的比，代入已知值，然后解方程。

To find an unknown side, use the normal ratio. To find an unknown angle, use the inverse ratio such as sin⁻¹, cos⁻¹ or tan⁻¹. Practise with a variety of diagrams so that labelling becomes automatic.

求未知边时使用正常三角比。求未知角时使用反三角比，如sin⁻¹、cos⁻¹或tan⁻¹。通过多种图形进行练习，使标记边变得熟练。

Practice question: A right-angled triangle has an adjacent side of 7 cm and an opposite side of 4 cm. Find the acute angle opposite the 4 cm side, correct to one decimal place.

练习题：一个直角三角形的邻边为7 cm，对边为4 cm。求4 cm边对面的锐角，精确到一位小数。

tan θ = 4 ÷ 7, so θ = tan⁻¹(4 ÷ 7) ≈ 29.7°

tan θ = 4 ÷ 7，所以 θ = tan⁻¹(4 ÷ 7) ≈ 29.7°

Work through similar questions until you can confidently identify the correct ratio without hesitation.

练习类似的题目，直到你能毫不犹豫地正确识别该用的比。

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IGCSE Maths: Trigonometric Ratios in Right-Angled Triangles | IGCSE数学：直角三角形中的三角比