Apply the trig identity:
sin a + cos a = sqrt2 cos (a - pi/4) (found in any trig book)
sin x + cos x = sqrt2cos (x - pi/4) = -1
Trig table, and unit circle give -->
cos (x - pi/4) = -1/sqrt2 = - sqrt2/2 --> x = +- (3pi)/4
Two solutions:
a. (x - pi/4) = (3pi)/4 --> x = (3pi)/4 + pi/4 = pi
b. Since the arc -(3pi)/4 --> arc (5pi)/4 (co-terminal arcs), therefor:
cos (x - pi/4) = (5pi)/4 --> x = ((5pi)/4 + pi/4) = (6pi)/4 = (3pi)/2
Answers for (0, 2pi)
pi, and (3pi)/2
Check:
x = pi --> sin x = 0 --> cos x = -1 --> -1 = -1 .OK
x = (3pi)/2 --> sin x = -1 --> cos x = 0 --> -1 = -1 .OK
