tan 2a=-1时,为什么a是3π/8以知a为锐角,且tan a+b=3,tana-b=2,则角a等于?

tan 2a=-1时,为什么a是3π/8以知a为锐角,且tan a+b=3,tana-b=2,则角a等于?
tan 2a=-1时,为什么a是3π/8
以知a为锐角,且tan a+b=3,tana-b=2,则角a等于?

tan 2a=-1时,为什么a是3π/8以知a为锐角,且tan a+b=3,tana-b=2,则角a等于?
tan2a=tan(a+b+a-b)
=[tan(a+b)+tan(a-b)]/[1-tan(a+b)tan(a-b)]
=(3+2)/(1-6)
=-1
∴2a=kπ-π/4
a=kπ/2-π/8
∵a为锐角,
∴当k=1时,a=3π/8

tan2α=-1==>2α=3π/4==>α=3π/8
(2)
tan2a=tan[(a+b)+(a-b)]=[3+2]/[1-6]= - 1
由（1）知α=3π/8

tan2a=tan(a+b+a-b)=(tan(a+b)+tan(a-b))/(1-tan(a+b)tan(a-b))
=(3+2)/(1-3*2)=-1
a为锐角, 0<2a<π
所以：2a=3π/4
则：a=3π/8

tan(a+b)=3，tan(a-b)=2，
——》tan2a=tan[(a+b)+(a-b)]
=[tan(a+b)+tan(a-b)]/[1-tan(a+b)tan(a-b)]
=-1
=tan(3π/4)
——》2a=3π/4+kπ，k∈Z，
又a为锐角，a∈（0，π/2)
——》2a∈（0，π)
——》2a=3π/4
——》a=3π/8.

tan(2α)=2tanα/[1-(tanα)^2]=-1
(tanα)^2-2tanα-1=0
(tanα-1)^2=0
tanα=1,-1
tanα=1 tanα=-1
...

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tan(2α)=2tanα/[1-(tanα)^2]=-1
(tanα)^2-2tanα-1=0
(tanα-1)^2=0
tanα=1,-1
tanα=1 tanα=-1
α=π/4 α=3π/4
所以
tan 2a = -1 = tanα
2a = α= 3π/4
a = 3π/8
根据公式
tan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)
tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)
(tanα+tanβ)/(1-tanα·tanβ)=3 (tanα-tanβ)/(1+tanα·tanβ)=2
tanα·tanβ=(3-tanα-tanβ)/3 tanα·tanβ=(tanα-tanβ-2)/2

所以 (3-tanα-tanβ)/3=(tanα-tanβ-2)/2
tanβ=5tanα-12
把tanβ=5tanα-12代入任意一个等式：tanα·tanβ=(3-tanα-tanβ)/3
3tanα·(5tanα-12)=3-tanα-(5tanα-12)
15(tanα)^2-36tanα=3-6tanα+12
15(tanα)^2-30tanα-15=0
(tanα)^2-2tanα-1=1
tanα=1,-1
当tanα=1 当tanα=-1
α=π/4 α=3π/4
已知 a < π/2
所以 α=π/4

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