Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
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]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
