urgent!! if x satifies the equation sinx+cosx =√2, find the value of tan (x/2)
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发布时间：2022-04-28 10:32
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热心网友
时间：2023-09-26 10:48
sinx + cosx =√2 = √2·sin[x + (π/4)]，∴sin[x + (π/4)] = 1，∴x + (π/4) = 2kπ + (π/2)，k∈Z
∴x/2 = kπ + (π/8)，而tan(x/2) = tan[kπ + (π/8)] = tan(π/8)>0
∵1 = tan(π/4) = 2tan(π/8)/{1 - [tan(π/8)]^2}，解得tan(π/8) = √2 - 1，负值舍
∴当x满足sinx+cosx =√2时，tan(x/2)值为 √2 - 1
