问题 填空题
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 ______.
答案

因为cosB=

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:

a
sinA
=
b
sinB
=
c
sinC

而cotA+cotC=

cosA
sinA
+
cosC
sinC
=
sin(A+C)
sinAsinC
=
sin(π-B)
sinAsinC

=

sinB
sinAsinC
=
sin2B
sinAsinC
1
sinB
=
sinB
sinA
sinB
sinC
1
sinB
=
b
a
b
c
1
sinB
=
1
sinB
=
1
7
4
=
4
7
7

故答案为:

4
7
7

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