已知△ABC的内角A,B,C成等差数列,则cos2A+cos2C的取值范围是______.-数学

题目简介

已知△ABC的内角A,B,C成等差数列,则cos2A+cos2C的取值范围是______.-数学

题目详情

已知△ABC的内角A,B,C成等差数列,则cos2A+cos2C的取值范围是______.
题型:填空题难度:中档来源:不详

答案

∵A,B,C成等差数列,
∴2B=A+C,又A+B+C=π,
∴B=60°,即A+C=120°,
cos2A+cos2C
=class="stub"1+cos2A
2
+class="stub"1+cos2c
2

=1+class="stub"cos2A+cos2C
2

=1+cos(A+C)cos(A-C)
=1-class="stub"1
2
cos(A-C),
∵-1≤cos(A-C)≤1,
class="stub"1
2
≤1-class="stub"1
2
cos(A-C)≤class="stub"3
2

则cos2A+cos2C的取值范围是[class="stub"1
2
class="stub"3
2
].
故答案为:[class="stub"1
2
class="stub"3
2
]

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