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Sagot :
Let's analyze each of the given explanations to determine the correctness of Francesca's work step by step.
### Step-by-Step Solution:
1. Identification of Trigonometric Values:
- Given:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{5\sqrt{26}}{26}} = \frac{-\sqrt{26}}{5} \][/tex]
and
[tex]\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{1}{5}} = 5 \][/tex]
2. Analyze the First Possible Explanation:
- Explanation: "Each step is correct because she plotted the point, drew a line to the x-axis to form a right triangle, used the Pythagorean theorem to find the hypotenuse, and finally wrote the correct ratios for all six functions."
- Evaluation: If Francesca drew the right triangle correctly and used the Pythagorean theorem accurately to determine the hypotenuse, she should then correctly compute the trigonometric ratios. This answer suggests no errors were made, but this needs verification.
3. Analyze the Second Possible Explanation:
- Explanation: "She made her first error in step 1 because she should have drawn the line to the y-axis to form the right triangle."
- Evaluation: Drawing to the y-axis instead of x-axis doesn't align with classic trigonometric problem methods, particularly when the x-axis usually serves for computation for triangles. This explanation seems unlikely.
4. Analyze the Third Possible Explanation:
- Explanation: "She made her first error in step 2 because she should have used a negative value for [tex]\( r \)[/tex]."
- Evaluation: The radius [tex]\( r \)[/tex] in trigonometry is generally taken as positive. The secant function already accounts for the sign based on the chosen angle quadrant. Therefore, using a negative [tex]\( r \)[/tex] can mislead results in trigonometric ratios. This suggests misunderstanding but isn't the likely source of error, assuming positive standard.
5. Analyze the Fourth Possible Explanation:
- Explanation: "She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions."
- Evaluation: If sine, cosine, and tangent calculations had errors then consequently other related trigonometric functions (secant, cosecant, cotangent) should reflect these inaccuracies.
Conclusion:
Given evaluations of potential errors and approaches, Francesca's miscalculations or misunderstandings for the fundamental trigonometric ratios indicate her first error was within the trigonometric ratios themselves.
Thus, the correct explanation for Francesca's error should be:
```
She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions.
```
Hence, the correct answer is:
```
4
```
### Step-by-Step Solution:
1. Identification of Trigonometric Values:
- Given:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{5\sqrt{26}}{26}} = \frac{-\sqrt{26}}{5} \][/tex]
and
[tex]\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{1}{5}} = 5 \][/tex]
2. Analyze the First Possible Explanation:
- Explanation: "Each step is correct because she plotted the point, drew a line to the x-axis to form a right triangle, used the Pythagorean theorem to find the hypotenuse, and finally wrote the correct ratios for all six functions."
- Evaluation: If Francesca drew the right triangle correctly and used the Pythagorean theorem accurately to determine the hypotenuse, she should then correctly compute the trigonometric ratios. This answer suggests no errors were made, but this needs verification.
3. Analyze the Second Possible Explanation:
- Explanation: "She made her first error in step 1 because she should have drawn the line to the y-axis to form the right triangle."
- Evaluation: Drawing to the y-axis instead of x-axis doesn't align with classic trigonometric problem methods, particularly when the x-axis usually serves for computation for triangles. This explanation seems unlikely.
4. Analyze the Third Possible Explanation:
- Explanation: "She made her first error in step 2 because she should have used a negative value for [tex]\( r \)[/tex]."
- Evaluation: The radius [tex]\( r \)[/tex] in trigonometry is generally taken as positive. The secant function already accounts for the sign based on the chosen angle quadrant. Therefore, using a negative [tex]\( r \)[/tex] can mislead results in trigonometric ratios. This suggests misunderstanding but isn't the likely source of error, assuming positive standard.
5. Analyze the Fourth Possible Explanation:
- Explanation: "She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions."
- Evaluation: If sine, cosine, and tangent calculations had errors then consequently other related trigonometric functions (secant, cosecant, cotangent) should reflect these inaccuracies.
Conclusion:
Given evaluations of potential errors and approaches, Francesca's miscalculations or misunderstandings for the fundamental trigonometric ratios indicate her first error was within the trigonometric ratios themselves.
Thus, the correct explanation for Francesca's error should be:
```
She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions.
```
Hence, the correct answer is:
```
4
```
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