Find all the solutions in the interval [0, 2pi): sin [x + (pi/4)] + sin [x - (pi/4)] = 1.?

Question

a. pi/4, 3pi/4
b. pi/4, pi/2
c. pi/6, 5pi/6
d. pi/4
e. none of these here

Answer 1

sin(x+PI/4) + sin(x - PI/4) = 1
sin x cos (PI/4) + cos x sin (PI/4) + sin x cos (PI/4) - cos x sin (PI/4) = 1

2 sin x cos (PI/4) = 1 (This is a standard result)

cos (PI/4) = 1/sqrt(2)

so

sin x = sqrt(2)/2

x = pi/4, 3pi/4

Hence, a. and d.

Answer 2

Using: sin(a)+sin(b) = 2 sin((a+b)/2)*sin((a-b)/2) you get:
sin(x)*sin(pi/4) = 1/2
sin(x) = 1/sqrt(2).
Therefore the answer is a.