已知锐角α,β,γ满足cosα=tanβ;cosβ=tanγ;cosγ=tanα, 求sinα的值
1个回答
展开全部
cosα=tanβ;cosβ=tanγ;两式相乘得cosαtanγ=tanβ×cosβ=sinβ
所以(sinβ)^2=(cosαtanγ)^2; (cosβ)^2=(tanγ)^2
则1=(cosβ)^2+(sinβ)^2=(tanγ)^2[1+(cosα)^2]
=[(sinγ)^2/(cosγ)^2][1+(cosα)^2]
=[(1-(cosγ)^2)/(cosγ)^2][1+(cosα)^2]
=[(1-(tanα)^2)/(tanα)^2][1+(cosα)^2]
=[(1-2(sinα)^2)/(sinα)^2)][2-(sinα)^2]
所以[(1-2(sinα)^2)][2-(sinα)^2]=(sinα)^2
(sinα)^4-3(sinα)^2+1=0
(sinα)^2=(3-√5)/2
sinα=(√5-1)/2
所以(sinβ)^2=(cosαtanγ)^2; (cosβ)^2=(tanγ)^2
则1=(cosβ)^2+(sinβ)^2=(tanγ)^2[1+(cosα)^2]
=[(sinγ)^2/(cosγ)^2][1+(cosα)^2]
=[(1-(cosγ)^2)/(cosγ)^2][1+(cosα)^2]
=[(1-(tanα)^2)/(tanα)^2][1+(cosα)^2]
=[(1-2(sinα)^2)/(sinα)^2)][2-(sinα)^2]
所以[(1-2(sinα)^2)][2-(sinα)^2]=(sinα)^2
(sinα)^4-3(sinα)^2+1=0
(sinα)^2=(3-√5)/2
sinα=(√5-1)/2
11
2024-08-28 广告
2024-08-28 广告
北京索莱宝科技有限公司(Beijing Solarbio Science & Technology Co.,Ltd.)是一家从事生物学试剂及试剂盒的高科技生物企业。总部位于北京,并在全国设有经销或代理机构。产品因可靠而稳定的质量和完...
点击进入详情页
本回答由11提供
推荐律师服务:
若未解决您的问题,请您详细描述您的问题,通过百度律临进行免费专业咨询