化简求值①tan70°cos10°(3tan20°-1)②已知sin(α+π3)+sinα=-435,(-π2<α<0),求cosα的值.-数学

题目简介

化简求值①tan70°cos10°(3tan20°-1)②已知sin(α+π3)+sinα=-435,(-π2<α<0),求cosα的值.-数学

题目详情

化简求值
tan70°cos10°(
3
tan20°-1)

②已知sin(α+
π
3
)+sinα=-
4
3
5
(-
π
2
<α<0)
,求cosα的值.
题型:解答题难度:中档来源:不详

答案

①tan70°cos10°( 
3
tan20°-1)
=cot20°cos10°( 
3
sin20°
cos20°
-1)
=cot20°cos10°(
3
sin20°-cos20°
cos20°

=class="stub"cos20°
sin20°
×cos10°×(
2(
3
2
sin20°-class="stub"1
2
cos20°)
cos20°

=class="stub"cos20°
sin20°
×cos10°×(
2sin(20°-30°)
cos20°

=class="stub"cos20°
sin20°
×(-class="stub"sin20°
cos20°

=-1
②∵sin(α+class="stub"π
3
)+sinα=-
4
3
5

class="stub"1
2
sinα+
3
2
cosα+sinα=-
4
3
5

3
sin(α+class="stub"π
6
)=-
4
3
5

∴sin(α+class="stub"π
6
)=-class="stub"4
5
,又∵-class="stub"π
2
<α<0

∴cos(α+class="stub"π
6
)=class="stub"3
5

∴cosα=cos(α+class="stub"π
6
-class="stub"π
6
)=
3
2
cos(α+class="stub"π
6
)+class="stub"1
2
sin(α+class="stub"π
6
)=
3
2
×class="stub"3
5
+class="stub"1
2
×(-class="stub"4
5
)=
3
3
-4
10

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