已知正实数a,b满足asinπ5+bcosπ5acosπ5-bsinπ5=tan8π15,则log3ba的值为()A.13B.12C.33D.22-数学

题目简介

已知正实数a,b满足asinπ5+bcosπ5acosπ5-bsinπ5=tan8π15,则log3ba的值为()A.13B.12C.33D.22-数学

题目详情

已知正实数a,b满足
asin
π
5
+bcos
π
5
acos
π
5
-bsin
π
5
=tan
15
,则log3
b
a
的值为(  )
A.
1
3
B.
1
2
C.
3
3
D.
2
2
题型:解答题难度:中档来源:不详

答案

由于a•sinclass="stub"π
5
+b•cosclass="stub"π
5
=
a2+b2
sin(class="stub"π
5
+α),a•sinclass="stub"π
5
-b•cosclass="stub"π
5
=
a2+b2
cos(class="stub"π
5
+α),且tanα=class="stub"b
a

则由 
asinclass="stub"π
5
+bcosclass="stub"π
5
acosclass="stub"π
5
-bsinclass="stub"π
5
=tanclass="stub"8π
15
,可得tan(class="stub"π
5
+α)=tanclass="stub"8π
15
,∴class="stub"π
5
+α=kπ+class="stub"8π
15
,k∈z.
解得 α=kπ+class="stub"π
3
,∴tanα=
3
,即 class="stub"b
a
=
3

log3class="stub"b
a
=log3
3
=class="stub"1
2

故选B.

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