📚 Using the 3-4-5 Right-Angled Triangle to Find Exact Trig Ratios | 利用 3-4-5 直角三角形求精确三角比
A 3-4-5 triangle is the most famous Pythagorean triple, and it gives exact rational values for sine, cosine and tangent without a calculator. At AQA A-Level, you are often expected to draw this triangle to convert one given ratio into another, especially when an angle is acute or when only a reference triangle is needed.
3-4-5 直角三角形是最著名的勾股数组合，它能够让你在没有计算器的情况下，直接写出正弦、余弦和正切的精确有理数值。在 AQA A-Level 考试中，题目经常要求你画出这个三角形，把一个已知的三角比转换成另一个，尤其是在角度为锐角或只需要参考三角形时。
1. Why the 3-4-5 Triangle Matters | 为什么 3-4-5 直角三角形很重要
The sides 3, 4 and 5 satisfy the Pythagorean theorem, so they always form a right-angled triangle. Because all three side lengths are integers, the six standard trigonometric ratios for the two acute angles are exact fractions rather than decimals.
边长 3、4、5 满足勾股定理，因此它们总能构成一个直角三角形。由于三条边都是整数，两个锐角的六个标准三角比都是精确的分数，而不是近似小数。
mmediately label the opposite side as 3 and the hypotenuse as 5, then use Pythagoras to find the missing adjacent side as 4.
在 AQA 考试题中，这个三角形经常被用作捷径。如果你知道一个比值，例如 sin θ = 3/5，你可以立即把对边标为 3，斜边标为 5，然后用勾股定理求出缺失的邻边为 4。
2. Labelling the Triangle and Choosing Angle A | 标记三角形并选择角 A
Draw a right-angled triangle with the right angle at C. Let the side opposite angle A be 3, the side adjacent to angle A be 4, and the hypotenuse be 5. Angle A is the acute angle between the side of length 4 and the hypotenuse of length 5.
画一个直角三角形，直角在 C 处。设角 A 的对边为 3，角 A 的邻边为 4，斜边为 5。角 A 是长度为 4 的边与长度为 5 的斜边之间的锐角。
This labelling is important because sine, cosine and tangent are defined relative to a specific acute angle. The side that is opposite for angle A becomes adjacent for the other acute angle, so labelling first prevents sign and ratio errors.
这种标记非常重要，因为正弦、余弦和正切都是相对于一个特定锐角定义的。对于角 A 来说的对边，对于另一个锐角来说就变成了邻边，因此先标记清楚可以避免符号和比值错误。
For angle A: opposite = 3, adjacent = 4, hypotenuse = 5
对于角 A：对边 = 3，邻边 = 4，斜边 = 5
3. Finding sin, cos and tan for Angle A | 求角 A 的正弦、余弦和正切
e ÷ adjacent. Substituting the side lengths gives exact fractions with no calculator work.
使用标准定义：sin A = 对边 ÷ 斜边，cos A = 邻边 ÷ 斜边，tan A = 对边 ÷ 邻边。代入边长后即可得到精确分数，无需使用计算器。
sin A = 3/5, cos A = 4/5, tan A = 3/4
sin A = 3/5，cos A = 4/5，tan A = 3/4
These values are exact. The angle A is approximately 36.87°, but AQA questions normally want the exact ratios rather than the angle measure unless a calculator section asks for it.
这些值都是精确的。角 A 大约为 36.87°，但 AQA 题目通常要求写出精确比值，而不是角度值，除非计算器部分的题目明确要求角度。
4. Finding Ratios for the Other Acute Angle B | 求另一个锐角 B 的三角比
The other acute angle, call it B, has the opposite side 4 and the adjacent side 3. Its hypotenuse is still 5. Therefore its ratios are the reverse of those for angle A.
另一个锐角记为 B，它的对边为 4，邻边为 3，斜边仍然为 5。因此它的三角比正好与角 A 的比值互换。
sin B = 4/5, cos B = 3/5, tan B = 4/3
sin B = 4/5，cos B = 3/5，tan B = 4/3
check in exam solutions and links the 3-4-5 triangle to the identity sin(90° − θ) = cos θ.
由于 A 和 B 是直角三角形中的互余角，所以 sin A = cos B，cos A = sin B。这在考试解题中是一个很有用的检验，并且将 3-4-5 三角形与恒等式 sin(90° − θ) = cos θ 联系了起来。
5. Verifying with Pythagoras and Identifying the Hypotenuse | 用勾股定理验证并识别斜边
otenuse is always the longest side, opposite the right angle, so it must be labelled as 5.
3-4-5 三角形之所以成立，是因为 3² + 4² = 9 + 16 = 25，且 5² = 25。斜边始终是最长边，位于直角对面，因此必须标为 5。
3² + 4² = 5²
3² + 4² = 5²
If you are given two shorter sides, Pythagoras finds the hypotenuse. If you are given the hypotenuse and one leg, subtracting the squares finds the other leg. In the 3-4-5 triangle, this calculation is clean and exact.
如果已知两条较短边，用勾股定理可以求出斜边。如果已知斜边和一条直角边，用平方相减可以求出另一条直角边。在 3-4-5 三角形中，这个计算过程十分整洁且精确。
6. Using the Triangle When a Ratio Is Given | 已知一个三角比时使用该三角形
act value of cos θ and tan θ. Draw a right triangle, label the opposite side as 3 and the hypotenuse as 5, then use Pythagoras to find the adjacent side.
AQA 的典型题目是：已知 θ 为锐角且 sin θ = 3/5，求 cos θ 和 tan θ 的精确值。画一个直角三角形，把对边标为 3，斜边标为 5，然后用勾股定理求出邻边。
Adjacent = √5² − 3² = √25 − 9 = √16 = 4
邻边 = √5² − 3² = √25 − 9 = √16 = 4
d tan θ = 3/4 immediately. The 3-4-5 triangle is the hidden structure behind the question.
一旦求出邻边为 4，你就可以立刻写出 cos θ = 4/5，tan θ = 3/4。3-4-5 三角形正是题目背后的隐藏结构。
7. Working Backwards from tan θ = 3/4 | 从 tan θ = 3/4 反推其他比值
e, label the opposite side as 3 and the adjacent side as 4. Since the hypotenuse is unknown, use Pythagoras to find it.
如果已知 tan θ = 3/4 且 θ 为锐角，把对边标为 3，邻边标为 4。由于斜边未知，用勾股定理求出斜边。
Hypotenuse = √3² + 4² = √9 + 16 = √25 = 5
斜边 = √3² + 4² = √9 + 16 = √25 = 5
rse process is very common in AQA papers because tangent is often given in the form of a fraction.
于是 sin θ = 3/5，cos θ = 4/5。这种反推过程在 AQA 试卷中非常常见，因为正切值常常以分数形式给出。
8. Exact Values and Rational Trig Ratios | 精确值与有理三角比
Unlike the special angles 30°, 45° and 60°, the angles in a 3-4-5 triangle are not standard exam angles. However, their trig ratios are rational numbers, which makes them ideal for exact-value questions.
与特殊角 30°、45° 和 60° 不同，3-4-5 三角形中的角并不是标准的考试角。但是它们的三角比是有理数，因此非常适合出精确值题目。
| Angle | sin | cos | tan |
|---|---|---|---|
| A (opposite 3) | 3/5 | 4/5 | 3/4 |
| B (opposite 4) | 4/5 | 3/5 | 4/3 |
The table summarises all six basic ratios. Memorising them can save time, but it is safer to draw the triangle in the exam to avoid swapping sine and cosine accidentally.
上表总结了六个基本比值。记住它们可以节省时间，但在考试中更稳妥的做法是画出三角形，以免不小心把正弦和余弦搞混。
9. Extending to Other Quadrants: Signs and Related Angles | 扩展到其他象限：符号与相关角
ut θ is obtuse, the reference triangle is still a 3-4-5 triangle, but cosine and tangent change sign.
3-4-5 三角形为不同象限中的一组角提供了参考角。如果 sin θ = 3/5 但 θ 为钝角，参考三角形仍然是 3-4-5 三角形，但余弦和正切的符号会发生变化。
le angles between 0° and 180°: the acute angle approximately 36.87° and the obtuse angle approximately 143.13°. In the second quadrant, sine is positive but cosine and tangent are negative.
例如，sin θ = 3/5 在 0° 到 180° 之间有两个可能的角度：锐角约为 36.87°，钝角约为 143.13°。在第二象限中，正弦为正，但余弦和正切为负。
For obtuse θ with sin θ = 3/5: cos θ = −4/5, tan θ = −3/4
对于满足 sin θ = 3/5 的钝角 θ：cos θ = −4/5，tan θ = −3/4
Always check the given interval or quadrant before writing your final answer. The 3-4-5 triangle supplies the magnitudes, while the quadrant supplies the signs.
在写出最终答案之前，一定要检查题目给定的区间或象限。3-4-5 三角形提供数值大小，而象限决定正负号。
10. Common Exam Mistakes and How to Avoid Them | 常见考试错误及避免方法
One common mistake is labelling the side of length 4 as the hypotenuse because it is larger than 3. However, the hypotenuse must be opposite the right angle and is always the longest side, so it must be 5.
一个常见错误是把长度为 4 的边标为斜边，因为它比 3 大。但是斜边必须在直角对面，并且始终是最长边，因此斜边必须是 5。
Another mistake is swapping sine and cosine when using the triangle for the complementary angle. Drawing a clear diagram and writing opposite, adjacent and hypotenuse next to the angle prevents this error.
另一个错误是在求互余角的三角比时把正弦和余弦搞混。画一个清晰的图，并在角旁边标明对边、邻边和斜边，可以避免这种错误。
Also, do not convert the ratios to decimals unless the question specifically asks for a decimal answer. Exact-value questions require fractions such as 3/5 and 4/5.
此外，除非题目明确要求写出小数答案，否则不要把比值转换成小数。精确值题目要求写出 3/5、4/5 这样的分数。
11. Worked AQA-Style Example | AQA 风格例题精讲
Question: Given that θ is obtuse and sin θ = 3/5, find the exact value of cos θ and tan θ.
题目：已知 θ 为钝角且 sin θ = 3/5，求 cos θ 和 tan θ 的精确值。
potenuse = 3/5, label the opposite side as 3 and the hypotenuse as 5. The adjacent side is √5² − 3² = 4.
第一步：画出参考三角形。由于 sin θ = 对边 ÷ 斜边 = 3/5，把对边标为 3，斜边标为 5。邻边为 √5² − 3² = 4。
nce θ is obtuse, it lies in quadrant II, where cosine is negative and tangent is negative.
第二步：对于锐角参考角，cos = 4/5，tan = 3/4。由于 θ 为钝角，它位于第二象限，因此余弦为负，正切为负。
Answer: cos θ = −4/5, tan θ = −3/4
答案：cos θ = −4/5，tan θ = −3/4
This example shows how the 3-4-5 triangle provides the reference values, while the quadrant information gives the final signs.
这个例子说明，3-4-5 三角形提供了参考值，而象限信息给出了最终的正负号。
12. Practice Questions and Exam Tips | 练习题与考试技巧
Try these questions to test your understanding. Draw a 3-4-5 triangle each time rather than relying on memory alone.
尝试以下问题来检验你的理解。每次都要画出 3-4-5 三角形，而不是只依赖记忆。
- Given that θ is acute and cos θ = 4/5, find sin θ and tan θ.
- Given that θ is reflex and tan θ = 3/4, find sin θ and cos θ.
- Given that sin α = 4/5 and α is acute, write down cos α and tan α without calculation.
- Given that θ is obtuse and cos θ = −4/5, find sin θ and tan θ.
- 已知 θ 为锐角且 cos θ = 4/5，求 sin θ 和 tan θ。
- 已知 θ 为优角且 tan θ = 3/4，求 sin θ 和 cos θ。
- 已知 sin α = 4/5 且 α 为锐角，直接写出 cos α 和 tan α，无需计算。
- 已知 θ 为钝角且 cos θ = −4/5，求 sin θ 和 tan θ。
In the exam, always ask yourself three things: Where is the right angle? What is the hypotenuse? Which quadrant is the angle in? Answering these three questions will make 3-4-5 triangle problems much easier.
在考试中，始终问自己三件事：直角在哪里？斜边是哪条？角在哪个象限？回答这三个问题会让 3-4-5 三角形问题变得容易很多。
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