已知tanα+1/tanα=5/2,α属于(π/4,π/2),求cos2α和sin(2α+π/4)
来源:学生作业帮助网 编辑:六六作业网 时间:2024/11/18 14:55:57
已知tanα+1/tanα=5/2,α属于(π/4,π/2),求cos2α和sin(2α+π/4)
已知tanα+1/tanα=5/2,α属于(π/4,π/2),求cos2α和sin(2α+π/4)
已知tanα+1/tanα=5/2,α属于(π/4,π/2),求cos2α和sin(2α+π/4)
∵α∈(π/4,π/2) ∴tanα>1
∵tanα+1/tanα=5/2=2+1/2 ∴tanα=2
∴cos2α=(1-tan²α)/(1+tan²α)=﹣3/5
sin2α=2tanα/(1+tan²α)=4/5
sin(2α+π/4)=√2/2(sin2α+cos2α)=√2/2(4/5-3/5)=√2/10
tanα+1/tanα=5/2
sinα/cosα+cosα/sinα=5/2
(sin²α+cos²α)/sinαcosα=5/2
1/sinαcosα=5/2
sinαcosα=2/5
2sinαcosα=4/5
sin2α=4/5
α∈(π/4,π/2),
2α∈(π/2,π)
所以:cos2α<0<...
全部展开
tanα+1/tanα=5/2
sinα/cosα+cosα/sinα=5/2
(sin²α+cos²α)/sinαcosα=5/2
1/sinαcosα=5/2
sinαcosα=2/5
2sinαcosα=4/5
sin2α=4/5
α∈(π/4,π/2),
2α∈(π/2,π)
所以:cos2α<0
cos2α=-根号(1-sin²2α)
=-根号(9/25)
=-3/5
sin(2α+π/4)
=sin2αcosπ/4+cos2αsinπ/4
=√2/2×(4/5-3/5)
=√2/10
收起
tanα+1/tanα=5/2可得tanα=1/2或2
而α属于(π/4,π/2)知α属于[1,+∝),所以tanα=2
cos2α={(cosα)^2-(sinα)^2}/{(sinα)^2+(cosα)^2}
={1-(tanα)^2}/{1+(tanα)^2}= -3/5
sin2α=2sinαcosα=4/5
sin(2α+π/4)=√2/2(sin2α+cos2α)= √2/10