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To solve the equation [tex]\( 4 \cos (2\theta) = -12 \cos(\theta) - 8 \)[/tex] on the interval [tex]\( 0 \leq \theta < 2\pi \)[/tex], follow these steps:
1. Use a Double-Angle Identity: Recall that [tex]\( \cos(2\theta) \)[/tex] can be expressed in terms of [tex]\( \cos(\theta) \)[/tex]:
[tex]\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \][/tex]
Substitute this identity into the equation:
[tex]\[ 4(2\cos^2(\theta) - 1) = -12 \cos(\theta) - 8 \][/tex]
2. Expand and Simplify:
[tex]\[ 8\cos^2(\theta) - 4 = -12\cos(\theta) - 8 \][/tex]
Bring all terms to one side of the equation:
[tex]\[ 8\cos^2(\theta) + 12\cos(\theta) - 4 + 8 = 0 \][/tex]
Combine like terms to simplify:
[tex]\[ 8\cos^2(\theta) + 12\cos(\theta) + 4 = 0 \][/tex]
3. Solve the Quadratic Equation: Let [tex]\( x = \cos(\theta) \)[/tex]. The equation becomes a standard quadratic equation:
[tex]\[ 8x^2 + 12x + 4 = 0 \][/tex]
This can be solved using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 8 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = 4 \)[/tex]:
[tex]\[ x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 8 \cdot 4}}{2 \cdot 8} \][/tex]
Simplify within the square root:
[tex]\[ x = \frac{-12 \pm \sqrt{144 - 128}}{16} \][/tex]
[tex]\[ x = \frac{-12 \pm \sqrt{16}}{16} \][/tex]
[tex]\[ x = \frac{-12 \pm 4}{16} \][/tex]
Therefore, we have:
[tex]\[ x = \frac{-12 + 4}{16} = \frac{-8}{16} = -\frac{1}{2} \][/tex]
[tex]\[ x = \frac{-12 - 4}{16} = \frac{-16}{16} = -1 \][/tex]
So, [tex]\( x \)[/tex] has two solutions:
[tex]\[ x = -\frac{1}{2} \quad \text{and} \quad x = -1 \][/tex]
4. Determine [tex]\( \theta \)[/tex] values:
- For [tex]\( \cos(\theta) = -\frac{1}{2} \)[/tex]:
[tex]\[ \theta = \cos^{-1}\left(-\frac{1}{2}\right) \][/tex]
The values of [tex]\( \theta \)[/tex] in the interval [tex]\( 0 \leq \theta < 2\pi \)[/tex] are:
[tex]\[ \theta = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3} \][/tex]
- For [tex]\( \cos(\theta) = -1 \)[/tex]:
[tex]\[ \theta = \cos^{-1}(-1) \][/tex]
The value of [tex]\( \theta \)[/tex] in the interval [tex]\( 0 \leq \theta < 2\pi \)[/tex] is:
[tex]\[ \theta = \pi \][/tex]
5. Compile the Solutions:
The solutions to the equation [tex]\( 4 \cos(2\theta) = -12 \cos(\theta) - 8 \)[/tex] on the interval [tex]\( 0 \leq \theta < 2\pi \)[/tex] are:
[tex]\[ \theta = \frac{2\pi}{3}, \quad \theta = \frac{4\pi}{3}, \quad \theta = \pi \][/tex]
1. Use a Double-Angle Identity: Recall that [tex]\( \cos(2\theta) \)[/tex] can be expressed in terms of [tex]\( \cos(\theta) \)[/tex]:
[tex]\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \][/tex]
Substitute this identity into the equation:
[tex]\[ 4(2\cos^2(\theta) - 1) = -12 \cos(\theta) - 8 \][/tex]
2. Expand and Simplify:
[tex]\[ 8\cos^2(\theta) - 4 = -12\cos(\theta) - 8 \][/tex]
Bring all terms to one side of the equation:
[tex]\[ 8\cos^2(\theta) + 12\cos(\theta) - 4 + 8 = 0 \][/tex]
Combine like terms to simplify:
[tex]\[ 8\cos^2(\theta) + 12\cos(\theta) + 4 = 0 \][/tex]
3. Solve the Quadratic Equation: Let [tex]\( x = \cos(\theta) \)[/tex]. The equation becomes a standard quadratic equation:
[tex]\[ 8x^2 + 12x + 4 = 0 \][/tex]
This can be solved using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 8 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = 4 \)[/tex]:
[tex]\[ x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 8 \cdot 4}}{2 \cdot 8} \][/tex]
Simplify within the square root:
[tex]\[ x = \frac{-12 \pm \sqrt{144 - 128}}{16} \][/tex]
[tex]\[ x = \frac{-12 \pm \sqrt{16}}{16} \][/tex]
[tex]\[ x = \frac{-12 \pm 4}{16} \][/tex]
Therefore, we have:
[tex]\[ x = \frac{-12 + 4}{16} = \frac{-8}{16} = -\frac{1}{2} \][/tex]
[tex]\[ x = \frac{-12 - 4}{16} = \frac{-16}{16} = -1 \][/tex]
So, [tex]\( x \)[/tex] has two solutions:
[tex]\[ x = -\frac{1}{2} \quad \text{and} \quad x = -1 \][/tex]
4. Determine [tex]\( \theta \)[/tex] values:
- For [tex]\( \cos(\theta) = -\frac{1}{2} \)[/tex]:
[tex]\[ \theta = \cos^{-1}\left(-\frac{1}{2}\right) \][/tex]
The values of [tex]\( \theta \)[/tex] in the interval [tex]\( 0 \leq \theta < 2\pi \)[/tex] are:
[tex]\[ \theta = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3} \][/tex]
- For [tex]\( \cos(\theta) = -1 \)[/tex]:
[tex]\[ \theta = \cos^{-1}(-1) \][/tex]
The value of [tex]\( \theta \)[/tex] in the interval [tex]\( 0 \leq \theta < 2\pi \)[/tex] is:
[tex]\[ \theta = \pi \][/tex]
5. Compile the Solutions:
The solutions to the equation [tex]\( 4 \cos(2\theta) = -12 \cos(\theta) - 8 \)[/tex] on the interval [tex]\( 0 \leq \theta < 2\pi \)[/tex] are:
[tex]\[ \theta = \frac{2\pi}{3}, \quad \theta = \frac{4\pi}{3}, \quad \theta = \pi \][/tex]
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