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Solve for x. sin(x) = -1/2 if the domain is [-π,π]. Please provide full working out and explanation :)



Answer :

Answer:

[tex]\displaystyle x=-\frac{5\pi}{6}\; \;\&\;-\frac{\pi}{6}[/tex]

Step-by-step explanation:

    We will solve the given equation for x with the domain [-π, π].

    In the unit circle, the y-coordinate represents sin(θ). In our case, θ = x. We will use this unit circle, see the attached image, to find when the sin(x) equals negative one-half.

    This equation has infinite solutions, however, the domain narrows it down to just a few.

    We can also graph sin(x) and negative one-half, see attached. Their points of intersection are the solution.

Given:

    [tex]\displaystyle sin(x) = -\frac{1}{2}[/tex]

Utilize the unit circle:

    [tex]\displaystyle x=\frac{7\pi}{6} ,\frac{11\pi}{6}[/tex]

Find the equivalent angles within [-π, π] by subtracting 2π:

    [tex]\displaystyle x=\frac{7\pi}{6}-2\pi ,\frac{11\pi}{6}-2\pi[/tex]

    [tex]\displaystyle x=-\frac{5\pi}{6},-\frac{\pi}{6}[/tex]

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