Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
First, let's understand the problem we are dealing with: solving the equation [tex]\(\sin(4x) = \frac{1}{2}\)[/tex] within the interval [tex]\((0, 2\pi)\)[/tex].
1. Identifying Basic Solutions:
The equation [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex] has specific angles that satisfy it. We know that:
[tex]\[ \theta = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2k\pi \][/tex]
where [tex]\(k\)[/tex] is any integer.
2. Relating [tex]\(\theta\)[/tex] to [tex]\(4x\)[/tex]:
We rewrite the given equation in terms of [tex]\(\theta\)[/tex]:
[tex]\[ 4x = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad 4x = \frac{5\pi}{6} + 2k\pi \][/tex]
3. Solving for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we divide both sides by 4:
[tex]\[ x = \frac{\pi}{24} + \frac{k\pi}{2} \quad \text{and} \quad x = \frac{5\pi}{24} + \frac{k\pi}{2} \][/tex]
4. Determining Valid [tex]\(x\)[/tex] Values Within the Interval [tex]\((0, 2\pi)\)[/tex]:
We need to find [tex]\(x\)[/tex] values that lie within the interval [tex]\((0, 2\pi)\)[/tex]. We test various integer values of [tex]\(k\)[/tex].
- For [tex]\(\frac{\pi}{24} + \frac{k\pi}{2}\)[/tex]:
[tex]\[ \begin{aligned} &k = 0:\quad x = \frac{\pi}{24} \\ &k = 1:\quad x = \frac{\pi}{24} + \frac{\pi}{2} = \frac{13\pi}{24} \\ &k = 2:\quad x = \frac{\pi}{24} + \pi = \frac{25\pi}{24} \\ &k = 3:\quad x = \frac{\pi}{24} + \frac{3\pi}{2} = \frac{37\pi}{24} \end{aligned} \][/tex]
All these values are within [tex]\((0, 2\pi)\)[/tex].
- For [tex]\(\frac{5\pi}{24} + \frac{k\pi}{2}\)[/tex]:
[tex]\[ \begin{aligned} &k = 0:\quad x = \frac{5\pi}{24} \\ &k = 1:\quad x = \frac{5\pi}{24} + \frac{\pi}{2} = \frac{17\pi}{24} \\ &k = 2:\quad x = \frac{5\pi}{24} + \pi = \frac{29\pi}{24} \\ &k = 3:\quad x = \frac{5\pi}{24} + \frac{3\pi}{2} = \frac{41\pi}{24} \end{aligned} \][/tex]
All these values are within [tex]\((0, 2\pi)\)[/tex].
5. Counting the Solutions:
Combining all the valid [tex]\(x\)[/tex] values, we have 8 solutions in total for the given equation in the specified interval.
Thus, the number of solutions to the equation [tex]\(\sin(4x) = \frac{1}{2}\)[/tex] within the interval [tex]\((0, 2\pi)\)[/tex] is:
[tex]\[ \boxed{8} \][/tex]
1. Identifying Basic Solutions:
The equation [tex]\(\sin(\theta) = \frac{1}{2}\)[/tex] has specific angles that satisfy it. We know that:
[tex]\[ \theta = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2k\pi \][/tex]
where [tex]\(k\)[/tex] is any integer.
2. Relating [tex]\(\theta\)[/tex] to [tex]\(4x\)[/tex]:
We rewrite the given equation in terms of [tex]\(\theta\)[/tex]:
[tex]\[ 4x = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad 4x = \frac{5\pi}{6} + 2k\pi \][/tex]
3. Solving for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we divide both sides by 4:
[tex]\[ x = \frac{\pi}{24} + \frac{k\pi}{2} \quad \text{and} \quad x = \frac{5\pi}{24} + \frac{k\pi}{2} \][/tex]
4. Determining Valid [tex]\(x\)[/tex] Values Within the Interval [tex]\((0, 2\pi)\)[/tex]:
We need to find [tex]\(x\)[/tex] values that lie within the interval [tex]\((0, 2\pi)\)[/tex]. We test various integer values of [tex]\(k\)[/tex].
- For [tex]\(\frac{\pi}{24} + \frac{k\pi}{2}\)[/tex]:
[tex]\[ \begin{aligned} &k = 0:\quad x = \frac{\pi}{24} \\ &k = 1:\quad x = \frac{\pi}{24} + \frac{\pi}{2} = \frac{13\pi}{24} \\ &k = 2:\quad x = \frac{\pi}{24} + \pi = \frac{25\pi}{24} \\ &k = 3:\quad x = \frac{\pi}{24} + \frac{3\pi}{2} = \frac{37\pi}{24} \end{aligned} \][/tex]
All these values are within [tex]\((0, 2\pi)\)[/tex].
- For [tex]\(\frac{5\pi}{24} + \frac{k\pi}{2}\)[/tex]:
[tex]\[ \begin{aligned} &k = 0:\quad x = \frac{5\pi}{24} \\ &k = 1:\quad x = \frac{5\pi}{24} + \frac{\pi}{2} = \frac{17\pi}{24} \\ &k = 2:\quad x = \frac{5\pi}{24} + \pi = \frac{29\pi}{24} \\ &k = 3:\quad x = \frac{5\pi}{24} + \frac{3\pi}{2} = \frac{41\pi}{24} \end{aligned} \][/tex]
All these values are within [tex]\((0, 2\pi)\)[/tex].
5. Counting the Solutions:
Combining all the valid [tex]\(x\)[/tex] values, we have 8 solutions in total for the given equation in the specified interval.
Thus, the number of solutions to the equation [tex]\(\sin(4x) = \frac{1}{2}\)[/tex] within the interval [tex]\((0, 2\pi)\)[/tex] is:
[tex]\[ \boxed{8} \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
