Answered

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[tex]\[
\begin{array}{l}
\sec \theta=\frac{1}{\cos \theta}=\frac{1}{-\frac{5 \sqrt{26}}{26}}=\frac{-\sqrt{26}}{5} \\
\cot \theta=\frac{1}{\tan \theta}=\frac{1}{\frac{1}{5}}=5
\end{array}
\][/tex]

Which of the following explains whether all of Francesca's work is correct?

A. Each step is correct because she plotted the point, drew a line to the [tex]\( x \)[/tex]-axis to form a right triangle, used the Pythagorean theorem to find the hypotenuse, and finally wrote the correct ratios for all six functions.

B. She made her first error in step 1 because she should have drawn the line to the [tex]\( y \)[/tex]-axis to form the right triangle.

C. She made her first error in step 2 because she should have used a negative value for [tex]\( \cos \theta \)[/tex].

D. 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 cotangent functions.

Sagot :

GinnyAnswer
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.
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