Question

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

Answer 1

Answer:

All apply values are:

0 ⇒ A

$\frac{\pi }{3}$ ⇒ B

$\frac{5\pi }{3}$ ⇒ E

Step-by-step explanation:

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

∴ 0 ≤ $\frac{x}{2}$ ≤ π is the domain of angle

∵ sin($\frac{x}{2}$) + cos(x) - 1 = 0

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

∵ cos(x) = 1 - 2 sin²($\frac{x}{2}$)

→ Substitute it in the equation above

∴ sin($\frac{x}{2}$) + (1 - 2 sin²($\frac{x}{2}$) = 0

∴ sin($\frac{x}{2}$) + 1 - 2 sin²($\frac{x}{2}$) = 0

→ Add the like terms

∴ sin($\frac{x}{2}$) + (1 - 1) - 2 sin²($\frac{x}{2}$) = 0

∴ sin($\frac{x}{2}$) - 2 sin²($\frac{x}{2}$) = 0

→ Take sin($\frac{x}{2}$) as a common factor

∴ sin($\frac{x}{2}$) [1 - 2sin($\frac{x}{2}$)] = 0

→ Equate each factor by 0

∵ sin($\frac{x}{2}$) = 0

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

∴ $\frac{x}{2}$ = 0, π, 2π

∵ The domain of  $\frac{x}{2}$ is 0 ≤ ($\frac{x}{2}$) ≤ π

∴  $\frac{x}{2}$ = 0 and π ⇒ 2π refused because ∉ the domain

→ Multiply both sides by 2 to find x

∴ x = 0 and 2π

∵ 1 - 2sin($\frac{x}{2}$) = 0

→ Subtract 1 from both sides

∴ - 2sin($\frac{x}{2}$) = -1

→ Divide both sides by -2

∴ sin($\frac{x}{2}$) = $\frac{1}{2}$

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

∴ ($\frac{x}{2}$) lies on the 1st OR 2nd quadrants

∵ ($\frac{x}{2}$) = $sin^{-1}(\frac{1}{2})$

∴ ($\frac{x}{2}$) = $\frac{\pi }{6}$ ⇒ 1st quadrant

→ Multiply both sides by 2

∴ x = $\frac{\pi }{3}$

∵ ($\frac{x}{2}$) = π - $\frac{\pi }{6}$ = $\frac{5\pi }{6}$ ⇒ 2nd quadrant

→ Multiply both sides by 2

∴ x = $\frac{5\pi }{3}$

∴ The values of x are 0, $\frac{\pi }{3}$,  $\frac{5\pi }{3}$, 2π

All apply values are:

0 ⇒ A

$\frac{\pi }{3}$ ⇒ B

$\frac{5\pi }{3}$ ⇒ E

Answer 2

Answer:

0

StartFraction pi Over 3 EndFraction  

StartFraction 5 pi Over 3 EndFraction

Step-by-step explanation:

A,B,E