For 0 degrees ≤ x < 360 degrees , what are the solutions to sin (x/2) + cos(x) - 1 =0

Recall the double angle identity for cosine:

cos(x) = cos(2×x/2) = 1 - 2 sin²(x/2)

Then the equation can be rewritten as

sin(x/2) + (1 - 2 sin²(x/2)) - 1 = 0

sin(x/2) - 2 sin²(x/2) = 0

sin(x/2) (1 - 2 sin(x/2)) = 0

sin(x/2) = 0   or   1 - 2 sin(x/2) = 0

sin(x/2) = 0   or   sin(x/2) = 1/2

[x/2 = arcsin(0) + 360n °   or   x/2 = 180° - arcsin(0) + 360n °]

… …   or   [x/2 = arcsin(1/2) + 360n °   or   x/2 = 180° - arcsin(1/2) + 360n °]

x/2 = 360n °   or   x/2 = 180° + 360n °

… …   or   x/2 = 30° + 360n °   or   x/2 = 150° + 360n °

x = 720n °   or   x = 360° + 720n °

… …   or   x = 60° + 720n °   or   x = 300° + 720n °

(where n is any integer)

We get only three solutions in 0° ≤ x < 360° :

720×0° = 0°

60° + 720×0° = 60°

300° + 720×0° = 300°

step-by-step explanation: i dont