Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To answer whether the given statements are true (T) or false (F), we need to analyze each statement individually:
### Statement 1: The domain of [tex]\(\tan(x)\)[/tex] is all real numbers.
The function [tex]\(\tan(x)\)[/tex] is defined as [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex]. The domain of [tex]\(\tan(x)\)[/tex] will thus be all real numbers for which [tex]\(\cos(x) \neq 0\)[/tex]. The values of [tex]\(x\)[/tex] where [tex]\(\cos(x) = 0\)[/tex] are [tex]\(x = \frac{(2n+1)\pi}{2}\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, [tex]\(\tan(x)\)[/tex] is not defined at these points, and the domain of [tex]\(\tan(x)\)[/tex] is not all real numbers. Thus, the statement is False.
- Answer: [tex]\( F \)[/tex]
### Statement 2: [tex]\( y = \tan(2x) \)[/tex] has period [tex]\( \pi \)[/tex].
The period of the standard tangent function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \pi \)[/tex], which means it repeats its values every [tex]\( \pi \)[/tex] units. However, for [tex]\( y = \tan(2x) \)[/tex], the period will be scaled by a factor of [tex]\(\frac{1}{2}\)[/tex]. This results in a period of [tex]\( \frac{\pi}{2} \)[/tex]. Hence, the statement that the period is [tex]\( \pi \)[/tex] is False.
- Answer: [tex]\( F \)[/tex]
### Statement 3: [tex]\(\tan(x)\)[/tex] and [tex]\(\cot(x)\)[/tex] are odd functions.
A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex]. For [tex]\(\tan(x)\)[/tex],
[tex]\[ \tan(-x) = -\tan(x) \][/tex]
and for [tex]\(\cot(x)\)[/tex],
[tex]\[ \cot(-x) = -\cot(x) \][/tex]
These properties confirm that both [tex]\(\tan(x)\)[/tex] and [tex]\(\cot(x)\)[/tex] are odd functions. Thus, the statement is True.
- Answer: [tex]\( T \)[/tex]
### Statement 4: [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex] are undefined for the same values of [tex]\(x\)[/tex].
The secant function [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex] is undefined where [tex]\(\cos(x) = 0\)[/tex]. Similarly, as previously discussed, [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex] is also undefined where [tex]\(\cos(x) = 0\)[/tex]. These points are [tex]\(x = \frac{(2n+1)\pi}{2}\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex] are indeed undefined for the same values of [tex]\(x\)[/tex]. Thus, the statement is True.
- Answer: [tex]\( T \)[/tex]
Therefore, the final answers to the statements are:
1. [tex]\( F \)[/tex]
2. [tex]\( F \)[/tex]
3. [tex]\( T \)[/tex]
4. [tex]\( T \)[/tex]
### Statement 1: The domain of [tex]\(\tan(x)\)[/tex] is all real numbers.
The function [tex]\(\tan(x)\)[/tex] is defined as [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex]. The domain of [tex]\(\tan(x)\)[/tex] will thus be all real numbers for which [tex]\(\cos(x) \neq 0\)[/tex]. The values of [tex]\(x\)[/tex] where [tex]\(\cos(x) = 0\)[/tex] are [tex]\(x = \frac{(2n+1)\pi}{2}\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, [tex]\(\tan(x)\)[/tex] is not defined at these points, and the domain of [tex]\(\tan(x)\)[/tex] is not all real numbers. Thus, the statement is False.
- Answer: [tex]\( F \)[/tex]
### Statement 2: [tex]\( y = \tan(2x) \)[/tex] has period [tex]\( \pi \)[/tex].
The period of the standard tangent function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \pi \)[/tex], which means it repeats its values every [tex]\( \pi \)[/tex] units. However, for [tex]\( y = \tan(2x) \)[/tex], the period will be scaled by a factor of [tex]\(\frac{1}{2}\)[/tex]. This results in a period of [tex]\( \frac{\pi}{2} \)[/tex]. Hence, the statement that the period is [tex]\( \pi \)[/tex] is False.
- Answer: [tex]\( F \)[/tex]
### Statement 3: [tex]\(\tan(x)\)[/tex] and [tex]\(\cot(x)\)[/tex] are odd functions.
A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex]. For [tex]\(\tan(x)\)[/tex],
[tex]\[ \tan(-x) = -\tan(x) \][/tex]
and for [tex]\(\cot(x)\)[/tex],
[tex]\[ \cot(-x) = -\cot(x) \][/tex]
These properties confirm that both [tex]\(\tan(x)\)[/tex] and [tex]\(\cot(x)\)[/tex] are odd functions. Thus, the statement is True.
- Answer: [tex]\( T \)[/tex]
### Statement 4: [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex] are undefined for the same values of [tex]\(x\)[/tex].
The secant function [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex] is undefined where [tex]\(\cos(x) = 0\)[/tex]. Similarly, as previously discussed, [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex] is also undefined where [tex]\(\cos(x) = 0\)[/tex]. These points are [tex]\(x = \frac{(2n+1)\pi}{2}\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex] are indeed undefined for the same values of [tex]\(x\)[/tex]. Thus, the statement is True.
- Answer: [tex]\( T \)[/tex]
Therefore, the final answers to the statements are:
1. [tex]\( F \)[/tex]
2. [tex]\( F \)[/tex]
3. [tex]\( T \)[/tex]
4. [tex]\( T \)[/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
