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好吧问答库 > 在△ABC中，a，b，c分别是A，B，C的对边，且满足（2a-c）cosB=bcosC．（Ⅰ）求角B的大小；（Ⅱ）若b=7，a

在△ABC中，a，b，c分别是A，B，C的对边，且满足（2a-c）cosB=bcosC．（Ⅰ）求角B的大小；（Ⅱ）若b=7，a

2025年04月06日 23:33

有1个网友回答

网友（1）：

（I）在△ABC中，由正弦定理得：
a=2RsinA，b=2RsinB，c=2RsinC代入（2a-c）cosB=bcosC整理得：
2sinAcosB=sinBcosC+sinCcosB
即：2sinAcosB=sin（B+C）=sinA，在三角形中，sinA＞0，2cosB=1，
∵∠B是三角形的内角，∴B=60°．
（II）在△ABC中，由余弦定理得：
b²=a²+c²-2ac?cosB=（a+c）²-2ac-2ac?cosB
将b＝

7

，a+c＝4代入整理得ac=3
故S△ABC＝

1

2

acsinB＝

3

2

sin60°＝

3

3

4

．

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