已知△ABC的三个内角A、B、C的对边分别为a、b、c,若a、b、c成等差数列,且2cos2B-8cosB+5=0,求角B的大小
已知△ABC的三个内角A、B、C的对边分别为a、b、c,若a、b、c成等差数列,且2cos2B-8cosB+5=0,求角B的大小并判断△ABC的形状....
已知△ABC的三个内角A、B、C的对边分别为a、b、c,若a、b、c成等差数列,且2cos2B-8cosB+5=0,求角B的大小并判断△ABC的形状.
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卧聋的人627
推荐于2016-08-15
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B= ![](https://iknow-pic.cdn.bcebos.com/bba1cd11728b47104350027ec0cec3fdfc032301?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,△ABC是等边三角形 |
方法一 ∵2cos2B-8cosB+5=0, ∴2(2cos 2 B-1)-8cosB+5=0. ∴4cos 2 B-8cosB+3=0, 即(2cosB-1)(2cosB-3)=0. 解得cosB= ![](https://iknow-pic.cdn.bcebos.com/a08b87d6277f9e2f4d2ce2ba1c30e924b899f363?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) 或cosB= ![](https://iknow-pic.cdn.bcebos.com/f3d3572c11dfa9ecd637a9a561d0f703918fc163?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) (舍去).∴cosB= ![](https://iknow-pic.cdn.bcebos.com/a08b87d6277f9e2f4d2ce2ba1c30e924b899f363?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . ∵0<B< ![](https://iknow-pic.cdn.bcebos.com/b8389b504fc2d5623c772f5fe41190ef76c66c01?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∴B= ![](https://iknow-pic.cdn.bcebos.com/bba1cd11728b47104350027ec0cec3fdfc032301?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . ∵a,b,c成等差数列,∴a+c=2b. ∴cosB= ![](https://iknow-pic.cdn.bcebos.com/902397dda144ad34f9c07835d3a20cf431ad8501?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) = ![](https://iknow-pic.cdn.bcebos.com/b3b7d0a20cf431ad6eadbc8b4836acaf2edd9801?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) = ![](https://iknow-pic.cdn.bcebos.com/a08b87d6277f9e2f4d2ce2ba1c30e924b899f363?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , 化简得a 2 +c 2 -2ac=0,解得a=c. 又∵B= ![](https://iknow-pic.cdn.bcebos.com/bba1cd11728b47104350027ec0cec3fdfc032301?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∴△ABC是等边三角形. 方法二 ∵2cos2B-8cosB+5=0, ∴2(2cos 2 B-1)-8cosB+5=0. ∴4cos 2 B-8cosB+3=0, 即(2cosB-1)(2cosB-3)=0. 解得cosB= ![](https://iknow-pic.cdn.bcebos.com/a08b87d6277f9e2f4d2ce2ba1c30e924b899f363?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) 或cosB= ![](https://iknow-pic.cdn.bcebos.com/f3d3572c11dfa9ecd637a9a561d0f703918fc163?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) (舍去). ∴cosB= ![](https://iknow-pic.cdn.bcebos.com/a08b87d6277f9e2f4d2ce2ba1c30e924b899f363?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∵0<B< ![](https://iknow-pic.cdn.bcebos.com/b8389b504fc2d5623c772f5fe41190ef76c66c01?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∴B= ![](https://iknow-pic.cdn.bcebos.com/bba1cd11728b47104350027ec0cec3fdfc032301?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∵a,b,c成等差数列,∴a+c=2b. 由正弦定理得sinA+sinC=2sinB=2sin ![](https://iknow-pic.cdn.bcebos.com/bba1cd11728b47104350027ec0cec3fdfc032301?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) = ![](https://iknow-pic.cdn.bcebos.com/908fa0ec08fa513d42b3936c3e6d55fbb2fbd963?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . ∴sinA+sin ![](https://iknow-pic.cdn.bcebos.com/4d086e061d950a7b562cb17609d162d9f2d3c963?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) = ![](https://iknow-pic.cdn.bcebos.com/908fa0ec08fa513d42b3936c3e6d55fbb2fbd963?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴sinA+sin ![](https://iknow-pic.cdn.bcebos.com/0b7b02087bf40ad135fd3851542c11dfa9ecce63?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) -cos ![](https://iknow-pic.cdn.bcebos.com/03087bf40ad162d9aaf79dae12dfa9ec8a13cd63?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) = ![](https://iknow-pic.cdn.bcebos.com/908fa0ec08fa513d42b3936c3e6d55fbb2fbd963?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . 化简得 ![](https://iknow-pic.cdn.bcebos.com/f3d3572c11dfa9ecd637a9a561d0f703918fc163?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) sinA+ ![](https://iknow-pic.cdn.bcebos.com/b58f8c5494eef01ff26be777e3fe9925bc317d01?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) cosA= ![](https://iknow-pic.cdn.bcebos.com/908fa0ec08fa513d42b3936c3e6d55fbb2fbd963?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∴sin ![](https://iknow-pic.cdn.bcebos.com/58ee3d6d55fbb2fb2a32a7a04c4a20a44623dc63?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) =1. ∴A+ ![](https://iknow-pic.cdn.bcebos.com/2cf5e0fe9925bc319122fe2a5ddf8db1cb137001?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) = ![](https://iknow-pic.cdn.bcebos.com/4a36acaf2edda3ccd7fd479202e93901213f9263?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∴A= ![](https://iknow-pic.cdn.bcebos.com/bba1cd11728b47104350027ec0cec3fdfc032301?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴C= ![](https://iknow-pic.cdn.bcebos.com/bba1cd11728b47104350027ec0cec3fdfc032301?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∴△ABC为等边三角形. |
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