Answer:
[tex]\displaystyle x=\frac{\pi}{4}[/tex]
Step-by-step explanation:
We will solve the given equation for x, knowing that 0 ≤ x ≤ 2π. To solve this sine function equation, we will need to utilize the unit circle.
Given:
[tex]\displaystyle sin(2x+\frac{\pi}{3} )=0.5[/tex]
Find when sinθ = 0.5:
➜ In the unit circle, the y-coordinate represents sine.
[tex]\displaystyle 2x+\frac{\pi}{3}=\frac{\pi}{6}[/tex]
[tex]\displaystyle 2x+\frac{\pi}{3}=\frac{5\pi}{6}[/tex]
Multiply both sides of both equations by 6:
[tex]\displaystyle 12x+2\pi=\pi[/tex]
[tex]\displaystyle 12x+2\pi=5\pi[/tex]
Subtract 5π from both sides of both equations:
[tex]\displaystyle 12x=-\pi[/tex]
[tex]\displaystyle 12x=3\pi[/tex]
Divide both sides of both equations by 12:
[tex]\displaystyle x=-\frac{\pi}{12}[/tex]
[tex]\displaystyle x=\frac{3\pi}{12}[/tex]
Reduce the second fraction:
[tex]\displaystyle x=-\frac{\pi}{12},\;\;\frac{\pi}{4}[/tex]
Apply the domain of 0 ≤ x ≤ 2π:
[tex]\displaystyle x=\frac{\pi}{4}[/tex]
