Question

Find all the solutions of sin x + cos x = 1 in the interval [0, 2pie).

Answer 1

X = 0, π/2 in the interval [0, 2pi).

Explanation:

Use the auxiliary angle method:

R sin(x + a) = Rsin x cos a + Rcos x sin a = 1

sin x + cos x = 1

Comparing coefficients:

R cos a = 1 and R sin a = 1, so

tan a = R sin a / R cos a = 1

So a = π/4 radians.

Also R^2(sin^2 a + cos^2 a) = 1^2 + 1^2 = 2

Therefore R = √2.

So √2 sin (x +π/4 = 1

sin x + π/4 = 1/√2

x + π/4 = π/4

x = 0 radians

Also

x = 0 + π/2 = π/2.