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已知a是第二相象限角,且tana=-12/5,求(1)sina,cosa(2)sin(a+派／4)/sin(派/2+2a)
题目详情
已知a是第二相象限角,且tana=-12/5,求(1)sina,cosa
(2)sin(a+派／4)/sin(派/2+2a)
(2)sin(a+派／4)/sin(派/2+2a)
▼优质解答
答案和解析
（1）∵a是第二相象限角
∴sina>0,cosasina/cosa=-12/5
==>sina=-12cosa/5
∴由sin²a+cos²a=1,得(-12cosa/5)²+cos²a=1
==>144cos²a/25+cos²a=1
==>169cos²a/25=1
==>cos²a=25/169
==>cosa=-5/13
==>sina=12/13
（2）sin(a+π/4)/sin(π/2+2a)=sin(a+π/4)/sin(2(a+π/4))
=sin(a+π/4)/(2sin(a+π/4)cos(a+π/4))
=(1/2)/cos(a+π/4)
=(1/2)/(cos(π/4)cosa-sin(π/4)sina)
=(1/2)/(cosa/√2-sina/√2)
=(√2/2)/(cosa-sina)
=(√2/2)/(-5/13-12/13)
=(√2/2)/(-17/13)
=-13√2/34.
∴sina>0,cosasina/cosa=-12/5
==>sina=-12cosa/5
∴由sin²a+cos²a=1,得(-12cosa/5)²+cos²a=1
==>144cos²a/25+cos²a=1
==>169cos²a/25=1
==>cos²a=25/169
==>cosa=-5/13
==>sina=12/13
（2）sin(a+π/4)/sin(π/2+2a)=sin(a+π/4)/sin(2(a+π/4))
=sin(a+π/4)/(2sin(a+π/4)cos(a+π/4))
=(1/2)/cos(a+π/4)
=(1/2)/(cos(π/4)cosa-sin(π/4)sina)
=(1/2)/(cosa/√2-sina/√2)
=(√2/2)/(cosa-sina)
=(√2/2)/(-5/13-12/13)
=(√2/2)/(-17/13)
=-13√2/34.
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