今天我们继续进行十年级GCSE几何部分的总结。在九年级学习完锐角三角函数之后，十年级，我们将进行更为实用的三角函数公式的学习，正余弦定理首先就将我们的三角函数的角度取值范围从原先直角三角形中的0-90°拓展到了0-180°，让我们能够在更加通用的所有三角形中去使用三角函数。下面为大家总结正余弦定理中的重难点内容：

Today we continue with the wrap-up of the Year 10 GCSE Geometry section. After learning acute trigonometric functions in the ninth grade, in the tenth grade, we will learn more practical trigonometric function formulas. The law of sine cosine first defines the angle value range of our trigonometric functions from 0-90 in the original right triangle. ° is extended to 0-180°, allowing us to use trigonometric functions in all triangles that are more general. The following summarizes the important and difficult contents of the sine and cosine theorem:

正弦定理公式总结：

在使用正弦定理时，首先，我们要注意，角和边的对应关系。角和其对面边的名字是一样的，R是这个三角形的外接圆半径。并且，最重要的一点是，要明确这个待求解三角形的基本形状，它是锐角三角形或者是钝角三角形。因为由于正弦函数的性质，在某些情况例如sin x=0.98时，我们如果使用计算器的反三角函数功能，求得 (sin-1) 0.98= 78.5°，这个解是没有问题的，但是计算器往往会忽略掉另一个解，180-78.5=101.5°。那么，这两个解就分别对应了两个完全不同的三角形，也希望大家注意。

When using the law of sine, first of all, we have to pay attention to the correspondence between angles and sides. A corner has the same name as its opposite side, and R is the radius of the circumcircle of the triangle. And, the most important point is to clarify the basic shape of the triangle to be solved, whether it is an acute-angled triangle or an obtuse-angled triangle. Because due to the nature of the sine function, in some cases, such as sin x=0.98, if we use the inverse trigonometric function of the calculator to obtain (sin-1) 0.98= 78.5°, this solution is no problem, but the calculation Calculators tend to ignore another solution, 180-78.5=101.5°. Then, these two solutions correspond to two completely different triangles, and I hope everyone pays attention.

余弦定理公式总结：

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同样，在使用余弦定理时，我们仍然首先需要弄清楚边与角的对应关系。与正弦定理不同的是，由于余弦函数的性质，cos 在0-180 之间的每一个值都是独一无二的，不存在像正弦函数那样，存在着一个值对应两个角度的情况。但是，这就导致了，cos在90-180之间时，它的值会出现负数，同学们在计算时要尤其注意这一点。与此同时，使用余弦定理求边长时，要注意最后记得等式两边同时求根号，取正值作为最后答案，还要将所求出的二次根式化简到最简形式。

Likewise, when using the cosine law, we still first need to figure out the correspondence between sides and angles. Unlike the law of sine, due to the nature of the cosine function, each value of cos between 0-180 is unique, and there is no such thing as a sine function, where one value corresponds to two angles. However, this leads to the fact that when cos is between 90 and 180, its value will appear negative. Students should pay special attention to this when calculating. At the same time, when using the cosine theorem to find the length of a side, pay attention to remember to find the square root of both sides of the equation at the same time, take the positive value as the final answer, and also simplify the quadratic surds to the simplest form.