Answer :
To determine where Francesca made her first error, let's carefully analyze the problem step-by-step:
1. Evaluate [tex]\( \sec \theta = \frac{1}{\cos \theta} \)[/tex]:
- Given as [tex]\( \frac{1}{-\frac{5 \sqrt{26}}{26}} \)[/tex].
- Upon simplifying: [tex]\( \frac{1}{-\frac{5 \sqrt{26}}{26}} = \frac{26}{-5 \sqrt{26}} \)[/tex].
- Further simplification should yield: [tex]\( \frac{26}{-5 \sqrt{26}} = \frac{-26}{5 \sqrt{26}} = - \frac{\sqrt{26}}{5} \)[/tex], not [tex]\( \frac{-\sqrt{2b}}{5} \)[/tex].
2. Evaluate [tex]\( \cot \theta = \frac{1}{\tan \theta} \)[/tex]:
- Given as [tex]\( \frac{1}{\frac{1}{5}} = 5 \)[/tex], which is correctly simplified.
From the above steps, it's clear where the errors and incorrect simplifications occurred. Specifically:
- In Step 1, Francesca simplified [tex]\(\sec \theta\)[/tex] incorrectly, mixing up the expression and resulting in a wrong ratio.
Therefore, the correct explanation is:
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.
Thus, the correct answer is:
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.
1. Evaluate [tex]\( \sec \theta = \frac{1}{\cos \theta} \)[/tex]:
- Given as [tex]\( \frac{1}{-\frac{5 \sqrt{26}}{26}} \)[/tex].
- Upon simplifying: [tex]\( \frac{1}{-\frac{5 \sqrt{26}}{26}} = \frac{26}{-5 \sqrt{26}} \)[/tex].
- Further simplification should yield: [tex]\( \frac{26}{-5 \sqrt{26}} = \frac{-26}{5 \sqrt{26}} = - \frac{\sqrt{26}}{5} \)[/tex], not [tex]\( \frac{-\sqrt{2b}}{5} \)[/tex].
2. Evaluate [tex]\( \cot \theta = \frac{1}{\tan \theta} \)[/tex]:
- Given as [tex]\( \frac{1}{\frac{1}{5}} = 5 \)[/tex], which is correctly simplified.
From the above steps, it's clear where the errors and incorrect simplifications occurred. Specifically:
- In Step 1, Francesca simplified [tex]\(\sec \theta\)[/tex] incorrectly, mixing up the expression and resulting in a wrong ratio.
Therefore, the correct explanation is:
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.
Thus, the correct answer is:
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.