All apply values to the equation are:
0 ⇒ A
π/3 ⇒ B
5π/3 ⇒ E
What is an equation?
A formula known as an equation shows the equality of two expressions by joining them with the equals sign =.
Main Body:
∵ 0 ≤ x ≤ 2π is the domain for angle x
∴ 0 ≤x/2 ≤ π is the domain of angle
∵ sin(x/2) + cos(x) - 1 = 0
→ To solve the equation we should use the rule of cosine double angle
∵ cos(x) = 1 - 2 sin²(x/2)
→ Substitute it in the equation above
∴ sin(x/2) + (1 - 2 sin²(x/2) = 0
∴ sin(x/2) + 1 - 2 sin²(x/2) = 0
→ Add the like terms
∴ sin(x/2) + (1 - 1) - 2 sin²(x/2) = 0
∴ sin(x/2) - 2 sin²(x/2) = 0
→ Take sin(x/2) as a common factor
∴ sin(x/2) [1 - 2sin(x/2)] = 0
→ Equate each factor by 0
∵ sin(x/2) = 0
→ The value of sine equal zero on the x-axis
∴ x/2 = 0, π, 2π
∵ The domain of x/2 is 0 ≤ (x/2) ≤ π
∴ x/2 = 0 and π ⇒ 2π refused because ∉ the domain
→ Multiply both sides by 2 to find x
∴ x = 0 and 2π
∵ 1 - 2sin(x/2) = 0
→ Subtract 1 from both sides
∴ - 2sin(x/2) = -1
→ Divide both sides by -2
∴ sin(x/2) = 1/2
→ The sine is positive in the 1st and 2nd quadrants
∴ (x/2) lies on the 1st OR 2nd quadrants
∵ (x/2) = sin⁻¹(1/2)
∴ (x/2) = π/6 = ⇒ 1st quadrant
→ Multiply both sides by 2
∴ x = π/3
∵ (x/2) = π -π/6 = 5π/6 ⇒ 2nd quadrant
→ Multiply both sides by 2
∴ x = 5π/3
∴ The values of x are 0,π/3 , 5π/3 , 2π
Hence apply values are:
0 ⇒ A
π/3⇒ B
5π/3⇒ E
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