已知cos(x-π/4)=√2/10,x∈(π/2,3π/4),求(1)sinx的值,(2)求sin(2x+π/3)的值.

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/29 07:46:02
已知cos(x-π/4)=√2/10,x∈(π/2,3π/4),求(1)sinx的值,(2)求sin(2x+π/3)的值.
x1N1E"ދD)&Q,WHC@ ܁;Rr[, iq?f* S mCw n 8f1 Y}GmSw;ed,mJ[5BsT%aSxKI\-#EQ#J$E᝞1k2qsb4;""$爫B"%K2!D3Q -׳iri&Q

已知cos(x-π/4)=√2/10,x∈(π/2,3π/4),求(1)sinx的值,(2)求sin(2x+π/3)的值.
已知cos(x-π/4)=√2/10,x∈(π/2,3π/4),求(1)sinx的值,(2)求sin(2x+π/3)的值.

已知cos(x-π/4)=√2/10,x∈(π/2,3π/4),求(1)sinx的值,(2)求sin(2x+π/3)的值.
∵x∈(π/2,3π/4),
∴x-π/4∈(π/4,π/2)
∵cos(x-π/4)=√2/10
∴sin(x-π/4)=√[1-cos²(x-π/4)]=7√2/10
∴sinx=sin[(x-π/4)+π/4]
=sin(x-π/4)cosπ/4+cos(x-π/4)sinπ/4
=7√2/10*√2/2+√2/10*√2/2
=7/10+1/10=4/5
(2)
∵x∈(π/2,3π/4),
∴cosx=-3/5
∴sin2x=2sinxcosx=-24/25
cos2x=1-2sin²x=-7/25
∴sin(2x+π/3)=sin2xcosπ/3+cos2xsinπ/3
=(-24/25)*1/2-7/25*√3/2
=-(24+7√3)/50