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In the Warring States Period (zhàn guó 战国,481 - 386BC),the King of the State of Zhao (zhào guó 赵国) got an extraordinarily fabulous jade from someone. The ruler of Qin (qín guó 秦国),who reigned a far more powerful country and wanted the rare treasure badly,promising to exchange it with 15 cities. So,Lin Xiangru (lín xiàng rú 蔺相如),an bold and talent minister of the State of Zhao, was then appointed by the king as an envoy to the State of Qin to negotiate with the emperor of Qin. But the ruler of Qin treated him very impolitely and were not attempt to give 15 cities as he previously pledged to do. Lin was pretty angry and humiliated,his hair stood up so stiffly on his head that it lifted up his hat.
Since then,This idiom came to be used to mean being extremely angry.
故事发生在战国时代，赵国的国王得到了一块漂亮罕见的玉。秦王听说了这个消息后，也非常想要这件宝物，并假惺惺地承诺愿意用他的15座城池来交换。于是，赵王便派遣聪明勇敢的大臣蔺相如出使到秦国。当他把玉交给秦王后，秦王态度恶劣，蛮不讲理，并不打算兑现他送城池的诺言。蔺相如非常气愤，气得连头发都竖起来,把帽子冲开了。
至此以后，这个成语就用来形容一个人愤怒到了极点。
怒发冲冠 (nù fà chōng guān)
【翻译】So Angry That One' s Hair Lifts Up One' s Hat.
【释义】愤怒得头发直竖，顶着帽子。形容一个人出于极端愤怒的状态。
【例句】老师这次被这个调皮的学生气坏了，可以说是怒发冲冠啊!
【近义词】大发雷霆、怒形于色、怒火中烧
【反义词】喜不自胜、欣喜若狂、喜形于色