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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.
```
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