IDNLearn.com provides a user-friendly platform for finding and sharing accurate answers. Our experts are available to provide in-depth and trustworthy answers to any questions you may have.
Sagot :
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]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.
