Question

满足sin（x+sinx）=cos（x-cosx）的锐角x为几

Answer 1

cos(X-cosX)=sin(π/2-X+cosX)

带入原式

sin(x+sinx)=sin(π/2-x+cosx)

x+sinx=π/2-x+cosx

sinx-cosx=π/2-2x

对于原式左侧有

sinx-cosx

=√2(√2/2*sinx-√2/2cosx)

=√2(sinxcosπ/4-cosxsinπ/4)

=√2sin(x-π/4)

∴ √2sin(x-π/4)=π/2-2x

将x=π/4带入,易得原方程两侧都等于0

∴满足原方程的x∈(0,π/2)存在且x=π/4