已知△ABC中,内角A,B,C所对的边分别为a,b,c,若a,b,c成等比数列,且cosB=34则cotA+cotC等于______.-数学

题目简介

已知△ABC中,内角A,B,C所对的边分别为a,b,c,若a,b,c成等比数列,且cosB=34则cotA+cotC等于______.-数学

题目详情

已知△ABC中,内角A,B,C所对的边分别为a,b,c,若a,b,c成等比数列,且cosB=
3
4
则cotA+cotC等于 ______.
题型:填空题难度:中档来源:不详

答案

因为cosB=class="stub"3
4
>0,所以sinB=
1-cos2B
=
7
4

由a,b,c成等比数列得到b2=ac,根据正弦定理得:class="stub"a
sinA
=class="stub"b
sinB
=class="stub"c
sinC

而cotA+cotC=class="stub"cosA
sinA
+class="stub"cosC
sinC
=
sin(A+C)
sinAsinC
=
sin(π-B)
sinAsinC

=class="stub"sinB
sinAsinC
=
sin2B
sinAsinC
class="stub"1
sinB
=class="stub"sinB
sinA
class="stub"sinB
sinC
class="stub"1
sinB
=class="stub"b
a
class="stub"b
c
•class="stub"1
sinB
=class="stub"1
sinB
=class="stub"1
7
4
=
4
7
7

故答案为:
4
7
7

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