Let's solve the equation step-by-step.
Given:
[tex]\[ -2 \cos (3 \theta) = 1 \][/tex]
1. Isolate the cosine term:
[tex]\[ \cos (3 \theta) = -\frac{1}{2} \][/tex]
2. Find the principal value:
We need to find the angle whose cosine is [tex]\(-\frac{1}{2}\)[/tex]. The principal value of [tex]\(\theta\)[/tex] for [tex]\(\cos^{-1}(-\frac{1}{2})\)[/tex] is:
[tex]\[ 3 \theta = \frac{2\pi}{3} \quad \text{or} \quad 3 \theta = \frac{4\pi}{3} \][/tex]
These angles are in the second and third quadrants respectively where the cosine function is negative.
3. Solve for [tex]\(\theta\)[/tex]:
Divide by 3 to find [tex]\(\theta\)[/tex]:
[tex]\[ \theta = \frac{\frac{2\pi}{3}}{3} = \frac{2\pi}{9} \quad \text{or} \quad \theta = \frac{\frac{4\pi}{3}}{3} = \frac{4\pi}{9} \][/tex]
4. General solution:
Since the cosine function is periodic with a period of [tex]\(2\pi\)[/tex], the general solutions for [tex]\(\cos (3 \theta) = -\frac{1}{2}\)[/tex] are given by the angles:
[tex]\[ 3 \theta = \frac{2\pi}{3} + 2k\pi \quad \text{and} \quad 3 \theta = \frac{4\pi}{3} + 2k\pi \][/tex]
where [tex]\(k\)[/tex] is any integer. Dividing by 3, we get:
[tex]\[ \theta = \frac{2\pi}{9} + \frac{2k\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{9} + \frac{2k\pi}{3} \][/tex]
5. Calculate specific solutions:
To illustrate the pattern, let’s consider integer values of [tex]\(k\)[/tex] from [tex]\(-5\)[/tex] to [tex]\(5\)[/tex].
For [tex]\(k = -5\)[/tex]:
[tex]\[ \theta = \frac{2\pi}{9} + \frac{2(-5)\pi}{3} = \frac{2\pi}{9} - \frac{10\pi}{3} = \frac{2\pi - 30\pi}{9} = -\frac{28\pi}{9} \][/tex]
[tex]\[ \theta = \frac{4\pi}{9} + \frac{2(-5)\pi}{3} = \frac{4\pi}{9} - \frac{10\pi}{3} = \frac{4\pi - 30\pi}{9} = -\frac{26\pi}{9} \][/tex]
Similarly, for [tex]\(k = 5\)[/tex]:
[tex]\[ \theta = \frac{2\pi}{9} + \frac{2(5)\pi}{3} = \frac{2\pi}{9} + \frac{10\pi}{3} = \frac{2\pi + 30\pi}{9} = \frac{32\pi}{9} \][/tex]
[tex]\[ \theta = \frac{4\pi}{9} + \frac{2(5)\pi}{3} = \frac{4\pi}{9} + \frac{10\pi}{3} = \frac{4\pi + 30\pi}{9} = \frac{34\pi}{9} \][/tex]
6. List of specific solutions:
After calculating specific solutions for [tex]\(k\)[/tex] from [tex]\(-5\)[/tex] to [tex]\(5\)[/tex], the solutions are approximately:
[tex]\[
[(-10.2392649450334, -10.704686078898554), (-8.144869842640205, -8.61029097650536), (-6.050474740247009, -6.5158958741121635), (-3.9560796378538137, -4.421500771718968), (-1.8616845354606182, -2.3271056693257726), (0.2327105669325773, -0.2327105669325773), (2.3271056693257726, 1.8616845354606182), (4.421500771718968, 3.9560796378538137), (6.5158958741121635, 6.050474740247009), (8.61029097650536, 8.144869842640205), (10.704686078898554, 10.2392649450334)]
\][/tex]
So, the solutions to the equation [tex]\(-2 \cos (3 \theta) = 1\)[/tex] are the values listed above.
