试题与答案

下列命题中正确的命题是( ) A.若点P(a,2a)(a≠0)为角a终边上一点,则

题型:选择题

题目:

下列命题中正确的命题是(  )
A.若点P(a,2a)(a≠0)为角a终边上一点,则sina=
2
5
5
B.同时满足sina=
1
2
,cosa=
3
2
的角a有且只有一个
C.当|a|<1时,tan(arcsina)的值恒正
D.三角方程tan(x+
π
3
)=
3
的解集为{x|x=kπ,k∈Z}

答案:

A.若P(a,2a)(a≠0)为角a终边上一点,则r=

a2+4a2
=
5
|a|,所以sinα=
2a
r
=
2a
5
|a|

所以当a>0时,sinα=

2
5
5
,当a<0时,sinα=-
2
5
5
,所以A错误.

B.由于三角函数y=sinx,y=cosx都是周期函数,所以同时满足sina=

1
2
,cosa=
3
2
的角a有无穷多个,所以B错误.

C.当|a|<1时,arcsina∈(-

π
2
π
2
),所以tan(arcsina)∈R,所以C错误.

D.由tan(x+

π
3
)=
3
x+
π
3
=kπ+
π
3
,即x=kπ,k∈z,所以解集为{x|x=kπ,k∈Z},所以D正确.

故选D.

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