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Sagot :
To determine the validity of the statement given [tex]$\cot \theta = \frac{12}{5}$[/tex] and [tex]$\sec \theta = -\frac{13}{5}$[/tex] with the terminal point in quadrant 2, let's carefully analyze each option step-by-step.
### Step-by-Step Analysis
Given:
- [tex]\(\cot \theta = \frac{12}{5}\)[/tex]
- [tex]\(\sec \theta = -\frac{13}{5}\)[/tex]
- [tex]\(\theta\)[/tex] is in quadrant 2
1. Option A:
- This option claims that [tex]\(\cot \theta\)[/tex] must be less than 1.
- However, [tex]\(\cot \theta = \frac{12}{5}\)[/tex] means [tex]\(\cot \theta = 2.4\)[/tex], which is indeed greater than 1.
- This option is therefore not true. Hence, A is incorrect.
2. Option B:
- This option suggests that [tex]\(12^2 + 5^2 \neq 1\)[/tex].
- True, [tex]\(12^2 + 5^2 = 144 + 25 = 169\)[/tex], which does not equal 1.
- However, this reasoning doesn't relate to an error in trigonometric identities or placement in the correct quadrant. Rather, it addresses the Pythagorean identity involving [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex].
- Therefore, B is not relevant to the validity of whether [tex]\(\cot \theta\)[/tex] and [tex]\(\sec \theta\)[/tex] fall into the described conditions properly. So, B isn't incorrect due to non-violation of the condition being checked.
3. Option C:
- In quadrant 2, [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex].
- Since in quadrant 2, [tex]\(\cos \theta\)[/tex] is negative and [tex]\(\sin \theta\)[/tex] is positive, [tex]\(\cot \theta\)[/tex] should be negative.
- Here, [tex]\(\cot \theta = \frac{12}{5}\)[/tex], which is positive.
- Thus, this condition is violated because [tex]\(\cot \theta\)[/tex] must indeed be negative in quadrant 2. Hence, C is true.
4. Option D:
- This option is checking values directly based on a different [tex]\(\cot\)[/tex] condition - if [tex]\(\cot \theta = -\frac{12}{5}\)[/tex] this does not align directly if juxtaposed directly as dependent on different trigonometric identity applied here.
- Given [tex]\(\sec\theta= \pm \frac{13}{5}\)[/tex] directly isn’t used reasonably so option checking if number alignment isn't deterministic.
- This cannot support or confirm trigonometrically where contradicts assigned cot-defined originally. Remains proposed other casting is seeing log denied.
### Conclusion:
Therefore, C correctly notes [tex]\(\cot \theta\)[/tex] should differ mistaken positive vice its justifying trigonometric possibilities.
Correct Answer:
[tex]\(C. \text{cannot be true because } \cot \theta \text{ is less than zero in quadrant 2}\)[/tex]
### Step-by-Step Analysis
Given:
- [tex]\(\cot \theta = \frac{12}{5}\)[/tex]
- [tex]\(\sec \theta = -\frac{13}{5}\)[/tex]
- [tex]\(\theta\)[/tex] is in quadrant 2
1. Option A:
- This option claims that [tex]\(\cot \theta\)[/tex] must be less than 1.
- However, [tex]\(\cot \theta = \frac{12}{5}\)[/tex] means [tex]\(\cot \theta = 2.4\)[/tex], which is indeed greater than 1.
- This option is therefore not true. Hence, A is incorrect.
2. Option B:
- This option suggests that [tex]\(12^2 + 5^2 \neq 1\)[/tex].
- True, [tex]\(12^2 + 5^2 = 144 + 25 = 169\)[/tex], which does not equal 1.
- However, this reasoning doesn't relate to an error in trigonometric identities or placement in the correct quadrant. Rather, it addresses the Pythagorean identity involving [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex].
- Therefore, B is not relevant to the validity of whether [tex]\(\cot \theta\)[/tex] and [tex]\(\sec \theta\)[/tex] fall into the described conditions properly. So, B isn't incorrect due to non-violation of the condition being checked.
3. Option C:
- In quadrant 2, [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex].
- Since in quadrant 2, [tex]\(\cos \theta\)[/tex] is negative and [tex]\(\sin \theta\)[/tex] is positive, [tex]\(\cot \theta\)[/tex] should be negative.
- Here, [tex]\(\cot \theta = \frac{12}{5}\)[/tex], which is positive.
- Thus, this condition is violated because [tex]\(\cot \theta\)[/tex] must indeed be negative in quadrant 2. Hence, C is true.
4. Option D:
- This option is checking values directly based on a different [tex]\(\cot\)[/tex] condition - if [tex]\(\cot \theta = -\frac{12}{5}\)[/tex] this does not align directly if juxtaposed directly as dependent on different trigonometric identity applied here.
- Given [tex]\(\sec\theta= \pm \frac{13}{5}\)[/tex] directly isn’t used reasonably so option checking if number alignment isn't deterministic.
- This cannot support or confirm trigonometrically where contradicts assigned cot-defined originally. Remains proposed other casting is seeing log denied.
### Conclusion:
Therefore, C correctly notes [tex]\(\cot \theta\)[/tex] should differ mistaken positive vice its justifying trigonometric possibilities.
Correct Answer:
[tex]\(C. \text{cannot be true because } \cot \theta \text{ is less than zero in quadrant 2}\)[/tex]
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