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Solve this equation given that 0≤x≤2π.

Provide all the steps and a full explanation please :)

Solve This Equation Given That 0x2π Provide All The Steps And A Full Explanation Please class=

Sagot :

Heather

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]

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