Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Let's analyze the given statement and the options one by one.
### Given:
1. [tex]\(\tan \theta = -\frac{12}{5}\)[/tex]
2. [tex]\(\csc \theta = -\frac{13}{5}\)[/tex]
3. The terminal point determined by [tex]\(\theta\)[/tex] is in quadrant 3.
### Options:
A. Cannot be true because [tex]\(\tan \theta\)[/tex] must be less than 1.
B. Cannot be true because if [tex]\(\tan \theta = -\frac{12}{5}\)[/tex], then [tex]\(\csc \theta = \pm \frac{13}{12}\)[/tex].
C. Cannot be true because [tex]\(\tan \theta\)[/tex] is greater than zero in quadrant 3.
D. Cannot be true because [tex]\(12^2 + 5^2 \neq 1\)[/tex].
### Analysis:
1. Option A:
- [tex]\(\tan \theta = -\frac{12}{5}\)[/tex]
- The value of [tex]\(\tan \theta\)[/tex] is [tex]\(-\frac{12}{5}\)[/tex], which is clearly less than 1. So, this option makes a false statement. Hence, this option cannot be true.
2. Option B:
- We are given [tex]\(\csc \theta = -\frac{13}{5}\)[/tex].
- The cosecant function, [tex]\(\csc \theta\)[/tex], is defined as [tex]\( \frac{1}{\sin \theta}\)[/tex].
- Since [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex] and [tex]\(\sin \theta = \frac{1}{\csc \theta}\)[/tex], it is evident that [tex]\(\csc \theta = -\frac{13}{5}\)[/tex] is consistent.
- If [tex]\(\tan \theta = -\frac{12}{5}\)[/tex], it would be incorrect to state that [tex]\(\csc \theta = \pm \frac{13}{12}\)[/tex]. Therefore, this option contains a false statement. Hence, this option cannot be true.
3. Option C:
- In quadrant 3, both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are negative. Therefore, [tex]\(\tan \theta\)[/tex], which is [tex]\(\frac{\sin \theta}{\cos \theta}\)[/tex], should be positive (a positive divided by a negative).
- However, [tex]\(\tan \theta = -\frac{12}{5}\)[/tex] is given as negative.
- A negative [tex]\(\tan \theta\)[/tex] is indeed possible in quadrant 3. So, stating that [tex]\(\tan \theta\)[/tex] greater than zero in quadrant 3 is incorrect.
- Therefore, this option contains a misleading statement. Hence, this option cannot be true.
4. Option D:
- They state [tex]\(12^2 + 5^2 \neq 1\)[/tex].
- Calculating it: [tex]\(12^2 = 144\)[/tex] and [tex]\(5^2 = 25\)[/tex], thus [tex]\(144 + 25 = 169\)[/tex], which is not equal to 1.
- This statement is true.
### Conclusion:
Hence, the only option that is indeed consistent or true based on the given values is:
A. Cannot be true because [tex]\(\tan \theta\)[/tex] must be less than 1.
Therefore, the correct option is:
```
The answer is Option A.
```
### Given:
1. [tex]\(\tan \theta = -\frac{12}{5}\)[/tex]
2. [tex]\(\csc \theta = -\frac{13}{5}\)[/tex]
3. The terminal point determined by [tex]\(\theta\)[/tex] is in quadrant 3.
### Options:
A. Cannot be true because [tex]\(\tan \theta\)[/tex] must be less than 1.
B. Cannot be true because if [tex]\(\tan \theta = -\frac{12}{5}\)[/tex], then [tex]\(\csc \theta = \pm \frac{13}{12}\)[/tex].
C. Cannot be true because [tex]\(\tan \theta\)[/tex] is greater than zero in quadrant 3.
D. Cannot be true because [tex]\(12^2 + 5^2 \neq 1\)[/tex].
### Analysis:
1. Option A:
- [tex]\(\tan \theta = -\frac{12}{5}\)[/tex]
- The value of [tex]\(\tan \theta\)[/tex] is [tex]\(-\frac{12}{5}\)[/tex], which is clearly less than 1. So, this option makes a false statement. Hence, this option cannot be true.
2. Option B:
- We are given [tex]\(\csc \theta = -\frac{13}{5}\)[/tex].
- The cosecant function, [tex]\(\csc \theta\)[/tex], is defined as [tex]\( \frac{1}{\sin \theta}\)[/tex].
- Since [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex] and [tex]\(\sin \theta = \frac{1}{\csc \theta}\)[/tex], it is evident that [tex]\(\csc \theta = -\frac{13}{5}\)[/tex] is consistent.
- If [tex]\(\tan \theta = -\frac{12}{5}\)[/tex], it would be incorrect to state that [tex]\(\csc \theta = \pm \frac{13}{12}\)[/tex]. Therefore, this option contains a false statement. Hence, this option cannot be true.
3. Option C:
- In quadrant 3, both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are negative. Therefore, [tex]\(\tan \theta\)[/tex], which is [tex]\(\frac{\sin \theta}{\cos \theta}\)[/tex], should be positive (a positive divided by a negative).
- However, [tex]\(\tan \theta = -\frac{12}{5}\)[/tex] is given as negative.
- A negative [tex]\(\tan \theta\)[/tex] is indeed possible in quadrant 3. So, stating that [tex]\(\tan \theta\)[/tex] greater than zero in quadrant 3 is incorrect.
- Therefore, this option contains a misleading statement. Hence, this option cannot be true.
4. Option D:
- They state [tex]\(12^2 + 5^2 \neq 1\)[/tex].
- Calculating it: [tex]\(12^2 = 144\)[/tex] and [tex]\(5^2 = 25\)[/tex], thus [tex]\(144 + 25 = 169\)[/tex], which is not equal to 1.
- This statement is true.
### Conclusion:
Hence, the only option that is indeed consistent or true based on the given values is:
A. Cannot be true because [tex]\(\tan \theta\)[/tex] must be less than 1.
Therefore, the correct option is:
```
The answer is Option A.
```
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
