若sinQ+cosQ=√2,则tan(Q+π/3)的值是?
问题描述：
若sinQ+cosQ=√2,则tan(Q+π/3)的值是?
答
sinQ+cosQ=√2
sin²Q+cos²Q=1
sinQ=cosQ=√2/2
tana=1
tan(Q+π/3)
=(tana+tanπ/3)/(1-tanatanπ/3)
=(1+√3)/(1-√3)
=4√3-8
答
由题目可以转成sin(Q+45度)=1/√2
此时tan(Q+π/4)=1
Tan(Q+π/3)=tan[(Q+π/4)+π/12]=(tan(Q+π/4)+tan(π/12))/1-tan(Q+π/4)tan(π/12)
=(1+tan15°)/(1-tan15°)
答
∵sinQ+cosQ=√2
∴(√2)sin(Q+π/4)=√2
∴sin(Q+π/4)=1
∴Q+π/4=π/2+2kπ(k∈Z)
∴Q=π/4+2kπ(k∈Z)
∴tanQ=tan(π/4+2kπ)=tan(π/4)=1
∴tan(Q+π/3)
=[tanQ+tan(π/3)]/[1-tanQtan(π/3)]
=(1+√3)/(1-√3)
=(1+√3)^2/[(1-√3)(1+√3)]
=(1+2√3+3)/(1-3)
=(4+2√3)/(-2)
=-(2+√3).