GCSE数学 三角函数 SOHCAHTOA 正弦余弦定理
三角函数是 GCSE 数学几何与测量部分的常考内容，AQA、Edexcel 与 OCR 的试卷里都会出现，也经常和方位角、三维图形、面积计算混在同一道大题里。这一篇按卷面上真正出现的顺序，把直角三角形里的三个比值、精确值、正弦定理、余弦定理、面积公式和图像方程串成一条线，每个知识点都配一道具体数字的题。很多学生以为三角函数就是背下 SOHCAHTOA，真到卷子上却在判断哪个是邻边这一步卡住，后面所有计算跟着一起错。下面七个部分按卷面顺序走：先把直角三角形的三个比值用熟，再记住几个精确值，然后处理斜三角形上的两条定理，接着是面积、三维和方位角，最后是图像与解方程，以及公式表上到底给了什么。
Trigonometry sits inside the geometry and measures part of every GCSE maths specification in England, and the same handful of ideas keeps coming back in slightly different clothes. This article runs through them in the order they usually appear on a paper, with the numbers you would actually meet in a question. Plenty of students can recite SOHCAHTOA and still lose marks, because deciding which side counts as the adjacent one is where the marks quietly disappear.
SOHCAHTOA 与直角三角形的三个比值
三个比值只在一个三角形含有直角时才成立。斜边永远是直角对面那条最长的边；剩下两条边里，哪条叫对边、哪条叫邻边，取决于你在看哪个角：站在角 θ 的位置，正对着你的那条是对边，仍然挨着角 θ 的另一条是邻边。sin θ 是对边比斜边，cos θ 是邻边比斜边，tan θ 是对边比邻边，三个式子里斜边都只出现在分母。已知一个锐角和一条边、想求另一条边的时候，就挑那个既包含已知边又包含要求边的比值；想求角，就把两条已知边代进去，再用计算器上的 sin⁻¹、cos⁻¹、tan⁻¹ 反解。举一组数字：直角三角形里一个角是 38 度、斜边 12 cm，求它的对边，用 sin 38 = x ÷ 12 得到 x = 12 × 0.6157 = 7.39 cm。求高度和距离的题几乎都用 tan，因为题里给的往往是一条水平距离和一条竖直高度，夹角就藏在直角三角形里面；反过来求角时顺序也一样，先看两条已知边里哪一条挨着要找的角，那一条就是邻边。很多学生以为邻边就是画在下面的那条边，图一旦转 90 度就代错，这是选择题里最常见的失分点。
These three ratios only work inside a right-angled triangle, where the hypotenuse is always the longest side and which of the other two counts as opposite or adjacent depends on the angle you are looking at; from the corner marked theta the side facing you is the opposite, and the one still touching theta is the adjacent. Finding a side means choosing the ratio that contains both the side you know and the side you want, while finding an angle means feeding two known sides into that ratio and pressing the inverse button. Take a triangle with an angle of 38 degrees and a hypotenuse of 12 cm, where sin 38 equals x over 12 and x comes out at 7.39 cm to three significant figures. The examiners see the same mistake every year and it is rarely the arithmetic, because students label the adjacent side by where it sits on the page rather than by where it sits relative to the angle.
精确值 sin cos tan 30 45 60 度
精确值指的是不该用小数表示的那几个角：0°、30°、45°、60°、90°。sin 这一行按 √0/2、√1/2、√2/2、√3/2、√4/2 排下来，化简就是 0、1/2、√2/2、√3/2、1；cos 那一行把顺序倒过来读，0° 处是 1，90° 处是 0；tan 是 sin 除以 cos，所以 30° 是 √3/3，45° 是 1，60° 是 √3。90° 处 cos 等于 0，tan 没有定义，写成 0 或者写成一个很大的数都会扣分，规范答案是 undefined。AQA 8300 的 G21 与 Edexcel 1MA1 的 G21 都把这张表放在两个层级共有的内容里，也就是说 Foundation 卷上同样会考。背下这张表的额外好处是，题目要求用根式表示中间步骤时，可以直接把 √3/2 代进后面的计算，避免每一步四舍五入带来的误差累积。常见的错误是拿计算器算出 0.866 就交卷，题目要求精确值的时候，答案必须是 √3/2。会做的学生都会先把这三行默写出来，再动手处理题目剩下的部分。
The exact values are the angles you should never write as decimals: 0, 30, 45, 60 and 90 degrees. Reading sin across that range gives root zero over two, root one over two, root two over two, root three over two and root four over two, which simplify to 0, a half, root two over two, root three over two and 1; cos is the same list read backwards, and tan comes from dividing the two. At 90 degrees cos is zero so tan has no value at all, and the accepted answer is undefined rather than zero. Both AQA 8300 and Edexcel 1MA1 place these values in content that sits on the foundation tier as well as the higher tier, so a calculator decimal such as 0.866 will not earn the mark when the paper asks for an exact form.
正弦定理：一组对边对角加一边
正弦定理写成一串等式 a ÷ sin A = b ÷ sin B = c ÷ sin C，求角的时候把它翻过来用更顺手，也就是 sin A ÷ a = sin B ÷ b = sin C ÷ c。它能用的前提是你手里至少有一组配好的边和角，也就是某个角和它正对着的那条边，再搭上一条别的边或者一个别的角。三个信息一旦凑齐，剩下的边和角可以一步步解出来。举个例子：三角形 ABC 里 A = 35°，a = 7 cm，C = 105°，那么 B = 180° - 35° - 105° = 40°，再用 b ÷ sin 40° = 7 ÷ sin 35°，算出 b = 7 × 0.6428 ÷ 0.5736 = 7.84 cm。错得最多的是把边和角配错，比如把 b 配到角 A 上，式子看上去很整齐，答案一定错。已知两边和其中一边的对角时，sin 反解出来的角可能是锐角也可能是钝角，题目通常会给条件排除其中一个，读完题再决定要不要检查第二个解。正弦定理也常用来做反过来的一步：知道两条边和其中一边的对角，先求出另一个角，再用 180° 减去两个已知角得到第三个角，最后回到边长的计算，整道题就是三次代入。
The sine rule is usually written as a over sin A equals b over sin B equals c over sin C, and flipping it upside down makes finding an angle much quicker. It needs one complete side and angle pair, meaning an angle together with the side directly opposite it, plus either one more side or one more angle; once those three pieces are in place the rest of the triangle follows in a few lines. With A equal to 35 degrees, a equal to 7 cm and C equal to 105 degrees, the third angle is 40 degrees and b works out at 7.84 cm from b over sin 40 equals 7 over sin 35. The error that costs most marks is mismatching a side with the wrong angle, since pairing b with angle A still produces a tidy-looking equation and a completely wrong answer, and when two sides and a non-included angle are given the inverse sine can return both an acute and an obtuse possibility.
余弦定理：三边求角与两边夹角求边
余弦定理 a² = b² + c² - 2bc cos A 解决的是正弦定理管不了的两类题。手里有三个边、想求某个角的时候，把式子变形为 cos A = (b² + c² - a²) ÷ 2bc，其中 a 必须是你要求的那个角正对着的边。手里有两个边和它们的夹角、想求第三条边的时候，直接用原式，减号后面的 2bc cos A 那一项不能漏。算一个例子：两边长 6 cm 和 9 cm、夹角 62°，第三条边 a² = 36 + 81 - 2 × 6 × 9 × 0.4695 = 66.30，开方得 8.14 cm。求角的例子：三边长 5、7、9，求 9 对面那个角，cos A = (25 + 49 - 81) ÷ 70 = -0.1，所以这个角是 95.7°，余弦是负数就说明这个角比 90° 大，这一步可以当自查用。常见错误是求角时把分母写成 2ab 或者 2ac，边长代进去的顺序一乱，结果就完全跑偏。判断该用哪条定理有个很快的办法：题里出现两个角加一条边，或者一组完整的对边对角，走正弦定理；题里出现三条边，或者两条边加夹角，就走余弦定理，而那个夹角在这类题里通常画在两条已知边之间。
The cosine rule, a squared equals b squared plus c squared minus two b c cos A, covers the two situations the sine rule cannot reach. When all three sides are known the rule is rearranged into cos A equals b squared plus c squared minus a squared, all over two b c, and the letter a has to be the side opposite the angle you are after; when two sides and the angle between them are known you use the original form and the minus two b c cos A term has to stay in. Two sides of 6 cm and 9 cm with 62 degrees between them give a squared of 66.30 and a third side of 8.14 cm, while sides of 5, 7 and 9 give cos A equal to minus 0.1 for the angle opposite 9, which is 95.7 degrees and a useful check because a negative cosine always means an obtuse angle. Most lost marks here come from putting the wrong pair of sides into the denominator, and the arithmetic that follows cannot rescue a swapped letter.
面积公式 1/2 ab sin C 与三维方位角
面积 = 1/2 ab sin C 里的 C 必须是两条已知边夹着的那个角，不是三角形里随便一个角。相比底乘高除以二，它的好处是不用先找垂足，题目只给两条边和一个夹角的时候就直接能用。算一下：两条边 8 cm 和 11 cm，夹角 47°，面积 = 0.5 × 8 × 11 × 0.7314 = 32.2 cm²。夹角是钝角也不用担心符号，sin 在 0° 到 180° 之间始终是正的，算出来的面积照样是正数。三维题的做法是在立体图形里先找出一个直角三角形，再用勾股定理和三个比值把棱长换出来，AQA 8300 的 G20 明确写了要把这些比值用在二维和三维图形上，Edexcel 1MA1 的 Higher 内容是同样的口径。方位角题的角从正北开始顺时针量，写的时候永远是三位数，比如 045°、120°；两条航程加一个夹角求直线距离，就是余弦定理的标准用法，夹角往往要用 180° 减去图上给的那两个角才能得到。立体几何最容易丢分的地方在于标高和底面对角线并不在同一个平面上，很多学生把两条不同平面上的长度直接相减，画辅助线之前先确认这些点到底共不共面。
Area equals one half a b sin C, where C has to be the angle between the two known sides rather than any angle in the triangle, and the advantage over base times height is that no perpendicular height is needed. Sides of 8 cm and 11 cm with 47 degrees between them give an area of 32.2 square centimetres, and an obtuse included angle causes no trouble because sine stays positive between 0 and 180 degrees. Three dimensional questions are solved by locating a right-angled triangle inside the solid and converting lengths with Pythagoras and the three ratios, which is exactly what AQA 8300 asks for under G20 and what the higher tier of Edexcel 1MA1 expects. Bearings are measured clockwise from north and always written with three digits, so a route with two legs and an included angle becomes a straightforward cosine rule problem once you work out that the included angle is often the supplement of the two angles drawn on the diagram.
sin cos tan 图像与方程解的个数
y = sin x 和 y = cos x 的图像是波浪形，值域在 -1 到 1 之间，周期是 360°；y = tan x 的周期只有 180°，并且在 90° 和 270° 处断开，画图时要用虚线标出渐近线。这几个性质直接决定方程有几个解：在 0° 到 360° 的范围内，sin x = 0.5 有两个解，30° 和 150°；tan x = 1 也有两个解，45° 和 225°；cos x = 0.8 的两个解是 36.9° 和 323.1°，它们关于 0° 对称。计算器只会给出一个答案，通常落在 -90° 到 90° 之间，剩下的一定要自己在图上补出来。阅卷时经常看到只写一个解的卷子，分数直接砍掉一半。Edexcel 1MA1 的 Higher 内容要求会画 y = sin x、y = cos x、y = tan x 这三个函数在任意角度下的图像，AQA 8300 则把图像题和后面的解方程放在同一个考点里出。判断解的个数完全可以靠图像：把 y = sin x 的曲线和 y = 0.5 的水平线画在一起，交点有几个答案就有几个，这个方法在 tan 上尤其省事，因为两条渐近线之间的一段只会有一个交点。
The graphs of sin x and cos x are waves that run between minus one and one with a period of 360 degrees, while tan x repeats every 180 degrees and breaks at 90 and 270 degrees where the vertical asymptotes belong. Those shapes decide how many answers an equation has: sin x equal to 0.5 gives 30 and 150 degrees, tan x equal to 1 gives 45 and 225 degrees, and cos x equal to 0.8 gives 36.9 and 323.1 degrees sitting symmetrically either side of zero. A calculator returns only the angle between minus 90 and 90 degrees, and papers that show a single answer are the ones where half the marks disappear, which is why the higher tier of Edexcel 1MA1 asks students to sketch all three graphs for angles of any size.
公式表与考纲层级 AQA 8300 与 Edexcel 1MA1
从 2022 年的考试开始，英格兰的考试局都会随卷附上一张公式表，三角函数这一块也在上面。AQA 8300 的 Higher 公式表印着勾股定理、sin A = a/c、cos A = b/c、tan A = a/b，紧跟下面就是正弦定理、余弦定理和 Area = 1/2 ab sin C；Foundation 的公式表短一截，只有勾股定理和三个比值，正弦定理与余弦定理要上到 Higher 才有。Edexcel 1MA1 的做法一样，Paper 1 是非计算器卷，公式表照样随卷发下来。Ofqual 在 2026 年 5 月 5 日确认，这张表会一直发到现行考纲结束，从 2028 年的考试起继续提供。很多学生以为公式表上有的东西就等于会做，真实的差距在于选题：拿到两条边和一个夹角，是先用余弦定理还是先算面积，判断错了后面全白写。公式表解决的是记忆问题，没有解决判断问题：同一道题里往往先用余弦定理求出边长、再用面积公式收尾，两个公式都要动手，漏掉任何一步都拿不到满分。
Every exam board in England has attached a formulae sheet to the mathematics papers since the 2022 exams, and the trigonometry sits on it in full. On the AQA 8300 higher tier sheet, Pythagoras and the three right-angled ratios appear directly above the sine rule, the cosine rule and the area formula, whereas the foundation sheet stops at Pythagoras and the ratios; Edexcel 1MA1 follows the same pattern and still issues the sheet with the non-calculator Paper 1. Ofqual confirmed on 5 May 2026 that the sheet stays for the lifetime of the current specifications, so it will keep arriving with papers from 2028 onwards, which shifts the difficulty from remembering formulas to choosing between them.
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