Find the solution(s) to the equation Sine (StartFraction x Over 2 EndFraction) + cosine x minus 1 = 0 on the interval 0 ≤ x 2π. Check all that apply

All apply values are:

0 ⇒ A

⇒ B

⇒ E

Step-by-step explanation:

∵ 0 ≤ x ≤ 2π is the domain for angle x

∴ 0 ≤ ≤ π is the domain of angle

∵ sin() + cos(x) - 1 = 0

→ To solve the equation we should use the rule of cosine double angle

∵ cos(x) = 1 - 2 sin²()

→ Substitute it in the equation above

∴ sin() + (1 - 2 sin²() = 0

∴ sin() + 1 - 2 sin²() = 0

→ Add the like terms

∴ sin() + (1 - 1) - 2 sin²() = 0

∴ sin() - 2 sin²() = 0

→ Take sin() as a common factor

∴ sin() [1 - 2sin()] = 0

→ Equate each factor by 0

∵ sin() = 0

→ The value of sine equal zero on the x-axis

∴ = 0, π, 2π

∵ The domain of   is 0 ≤ () ≤ π

∴   = 0 and π ⇒ 2π refused because ∉ the domain

→ Multiply both sides by 2 to find x

∴ x = 0 and 2π

∵ 1 - 2sin() = 0

→ Subtract 1 from both sides

∴ - 2sin() = -1

→ Divide both sides by -2

∴ sin() =

→ The sine is positive in the 1st and 2nd quadrants

∴ () lies on the 1st OR 2nd quadrants

∵ () =

∴ () = ⇒ 1st quadrant

→ Multiply both sides by 2

∴ x =

∵ () = π - = ⇒ 2nd quadrant

→ Multiply both sides by 2

∴ x =

∴ The values of x are 0, ,  , 2π

All apply values are:

0 ⇒ A

⇒ B

⇒ E