0= sin (pi)x + cos (pi)x

Question

Solve 0= sin (pi)x + cos(pi)x for the interval (-1,3)

Answer 1

sinΠx+cosΠx=0

sinΠx=-cosΠx

tanΠx=-1

Πx=arctan(-1)

πx=-π/4, 3π/4, 7π/4, 11π/4

x=-1/4, 3/4, 7/4, 11/4

Answer 2

Another solution.

sinΠx+cosΠx=0

sinΠx+cosΠx=sinΠx+sin(Πx+Π/2)

=2sin(Πx+Π/4)cos(-Π/4)=0

or sin(Πx+Π/4)=0 or Πx+Π/4=k∏ or x+1/4=k an integer

x=k-1/4 which falls in the interval (-1,3) for k=0,1,2,3 giving

x=-1/4, 3/4, 1 3/4, 2 3/4