Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To find the solution for the equation [tex]\(\cot \frac{x}{2} = 0\)[/tex], let's carefully analyze the behavior of the cotangent function.
1. Understanding [tex]\(\cot \theta\)[/tex]:
- The cotangent function, [tex]\(\cot \theta\)[/tex], is defined as [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
- [tex]\(\cot \theta\)[/tex] is zero when [tex]\(\tan \theta\)[/tex] approaches infinity, meaning that [tex]\(\tan \theta\)[/tex] should be undefined or have vertical asymptotes.
2. Setting up the Equation:
- The equation becomes [tex]\(\cot \frac{x}{2} = 0\)[/tex]. This implies that [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex] must be undefined.
- [tex]\(\tan \theta\)[/tex] is undefined when [tex]\(\theta = \frac{\pi}{2} + n\pi\)[/tex], where [tex]\(n\)[/tex] is any integer.
3. Determining Specific Solutions:
- For [tex]\(\cot \frac{x}{2} = 0\)[/tex], [tex]\(\frac{x}{2} = \frac{\pi}{2} + n\pi\)[/tex].
- Solving for [tex]\(x\)[/tex], we get [tex]\(x = \pi + 2n\pi = (2n+1)\pi\)[/tex].
4. Matching Given Options to [tex]\(x\)[/tex]:
- We need to check which of the given options satisfy this:
[tex]\[ \begin{aligned} \text{A. } & \frac{3\pi}{4} \implies \frac{x}{2} = \frac{3\pi}{8} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\ \text{B. } & \frac{3\pi}{2} \implies \frac{x}{2} = \frac{3\pi}{4} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\ \text{C. } & \frac{\pi}{2} \implies \frac{x}{2} = \frac{\pi}{4} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\ \text{D. } & 3\pi \implies \frac{x}{2} = \frac{3\pi}{2} & \text{(Matches \(\frac{3\pi}{2} = \frac{\pi}{2} + \pi\))} \end{aligned} \][/tex]
Choice D ([tex]\(3\pi\)[/tex]) satisfies the given equation. Therefore, the solution to the equation [tex]\(\cot \frac{x}{2} = 0\)[/tex] is:
[tex]\[ \boxed{3\pi} \][/tex]
1. Understanding [tex]\(\cot \theta\)[/tex]:
- The cotangent function, [tex]\(\cot \theta\)[/tex], is defined as [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
- [tex]\(\cot \theta\)[/tex] is zero when [tex]\(\tan \theta\)[/tex] approaches infinity, meaning that [tex]\(\tan \theta\)[/tex] should be undefined or have vertical asymptotes.
2. Setting up the Equation:
- The equation becomes [tex]\(\cot \frac{x}{2} = 0\)[/tex]. This implies that [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex] must be undefined.
- [tex]\(\tan \theta\)[/tex] is undefined when [tex]\(\theta = \frac{\pi}{2} + n\pi\)[/tex], where [tex]\(n\)[/tex] is any integer.
3. Determining Specific Solutions:
- For [tex]\(\cot \frac{x}{2} = 0\)[/tex], [tex]\(\frac{x}{2} = \frac{\pi}{2} + n\pi\)[/tex].
- Solving for [tex]\(x\)[/tex], we get [tex]\(x = \pi + 2n\pi = (2n+1)\pi\)[/tex].
4. Matching Given Options to [tex]\(x\)[/tex]:
- We need to check which of the given options satisfy this:
[tex]\[ \begin{aligned} \text{A. } & \frac{3\pi}{4} \implies \frac{x}{2} = \frac{3\pi}{8} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\ \text{B. } & \frac{3\pi}{2} \implies \frac{x}{2} = \frac{3\pi}{4} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\ \text{C. } & \frac{\pi}{2} \implies \frac{x}{2} = \frac{\pi}{4} & \text{(Does not match \(\frac{\pi}{2} + n\pi\))}\\ \text{D. } & 3\pi \implies \frac{x}{2} = \frac{3\pi}{2} & \text{(Matches \(\frac{3\pi}{2} = \frac{\pi}{2} + \pi\))} \end{aligned} \][/tex]
Choice D ([tex]\(3\pi\)[/tex]) satisfies the given equation. Therefore, the solution to the equation [tex]\(\cot \frac{x}{2} = 0\)[/tex] is:
[tex]\[ \boxed{3\pi} \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.