Question

已知函数f(x)满足f(x)=tan(2x+π/4),

已知函数f(x)满足f(x)=tan(2x+π/4), (1)求y的定义域和最小正周期 (2)设a属于(0,π/4),若f（a/2）=2cos2a ，求a

Answer 1

1.
最小正周期T=π/2
2x+π/4∈(2kπ-π/2,
2kπ+π/2)
定义域：x∈(kπ-3π/8,
kπ+π/8)
2.
∵
f(α/2)
=tan(α+π/4)
=[tanα+tan(π/4)]/[1-tanα*tan(π/4)]
=(tanα+1)(1-tanα)
=(sinα+cosα)/(cosα-sinα)
f(α/2)
=2cos2α
=2(cos²α-sin²α)
=2(cosα+sinα)(cosα-sinα)
∴(sinα+cosα)/(cosα-sinα)=2(cosα+sinα)(cosα-sinα)
整理得：2(cosα-sinα)²=1
∴cosα-sinα=±√2/2
即：√2sin(π/4-α)=±√2/2
解得：α=5π/12+2kπ
或
π/12+2kπ
(k∈Z)
∵α∈(0，π/4)
∴α=π/12
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