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Solve [tex]\(\sin \left(\frac{\theta}{2}\right) = -\frac{1}{2}\)[/tex] exactly over all real values of [tex]\(\theta\)[/tex].

A. [tex]\(\theta = \frac{(7 + 6n) \pi}{3}, \frac{(11 + 6n) \pi}{3}\)[/tex], where [tex]\(n\)[/tex] is an integer

B. [tex]\(\theta = \frac{(7 + 4n) \pi}{3}, \frac{(11 + 4n) \pi}{3}\)[/tex], where [tex]\(n\)[/tex] is an integer

C. [tex]\(\theta = \frac{(7 + 12n) \pi}{6}, \frac{(11 + 12n) \pi}{6}\)[/tex], where [tex]\(n\)[/tex] is an integer

D. [tex]\(\theta = \frac{(7 + 12n) \pi}{3}, \frac{(11 + 12n) \pi}{3}\)[/tex], where [tex]\(n\)[/tex] is an integer

E. [tex]\(\theta = \frac{7\pi}{3}, \frac{11\pi}{3}\)[/tex]



Answer :

GinnyAnswer
To solve the equation [tex]\(\sin\left(\frac{\theta}{2}\right) = -\frac{1}{2}\)[/tex] over all real values of [tex]\(\theta\)[/tex], we proceed as follows:

1. Understand the basic trigonometric equation:
We start with [tex]\(\sin x = -\frac{1}{2}\)[/tex]. We know that sine is negative in the third and fourth quadrants. The reference angle for [tex]\(\sin x = \frac{1}{2}\)[/tex] is [tex]\(\frac{\pi}{6}\)[/tex], thus the solutions for [tex]\(\sin x = -\frac{1}{2}\)[/tex] are:

[tex]\[ x = -\frac{\pi}{6} + 2k\pi \quad \text{and} \quad x = \frac{7\pi}{6} + 2k\pi \][/tex]
for any integer [tex]\(k\)[/tex].

2. Apply the solutions to [tex]\(\sin\left(\frac{\theta}{2}\right)\)[/tex]:
Let [tex]\(x = \frac{\theta}{2}\)[/tex]. Substitute [tex]\(x\)[/tex] with [tex]\(\frac{\theta}{2}\)[/tex] in the general solutions:

[tex]\[ \frac{\theta}{2} = -\frac{\pi}{6} + 2k\pi \quad \text{and} \quad \frac{\theta}{2} = \frac{7\pi}{6} + 2k\pi \][/tex]

3. Solve for [tex]\(\theta\)[/tex]:
Multiply both sides of each equation by 2 to solve for [tex]\(\theta\)[/tex]:

[tex]\[ \theta = -\frac{\pi}{3} + 4k\pi \quad \text{and} \quad \theta = \frac{7\pi}{3} + 4k\pi \][/tex]

4. Rewrite the solutions:
Using [tex]\(n\)[/tex] as any integer (where [tex]\(n = k\)[/tex]):

[tex]\[ \theta = (-\frac{\pi}{3}) + 4n\pi \quad \text{and} \quad \theta = (\frac{7\pi}{3}) + 4n\pi \][/tex]

5. Additional equivalent form:
Noting that [tex]\(\frac{7\pi}{3}\)[/tex] can be expressed as [tex]\(\frac{4\pi}{3} + \pi\)[/tex], we get another commonly used form:

[tex]\[ \theta = (\frac{4\pi}{3}) + 4n\pi \][/tex]

Thus, the general solutions for [tex]\(\theta\)[/tex] are:

[tex]\[ \theta = (-\frac{\pi}{3}) + 4n\pi \quad \text{and} \quad \theta = (\frac{4\pi}{3}) + 4n\pi \][/tex]

So the correct multiple-choice answer is:

[tex]\[ \theta = \frac{7\pi}{3}, \frac{11\pi}{3} \][/tex]

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