Question
Sandra wants to evaluate 2 cotangent left parenthesis negative Start Fraction 9 pi over 2 End Fraction right parenthesis. Her work is shown below. What did she do wrong?
2 cotangent left parenthesis negative Start Fraction 9 pi over 2 End Fraction right parenthesis equals 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis equals 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis equals 2 left parenthesis 0 right parenthesis equals 0
(1 point)
Responses
Since cotangent is an odd function, 2 cotangent left parenthesis negative Start Fraction 9 pi over 2 End Fraction right parenthesis not equals 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis.
Since cotangent is an odd function, Image with alt text: 2 cotangent left parenthesis negative Start Fraction 9 pi over 2 End Fraction right parenthesis not equals 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis .
Since cotangent has a period of π, 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis not equals 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis.
Since cotangent has a period of π , Image with alt text: 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis not equals 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis .
Since cotangent evaluated at Start Fraction pi over 2 End Fraction is not 0, 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis not equals 2 left parenthesis 0 right parenthesis.
Since cotangent evaluated at Image with alt text: Start Fraction pi over 2 End Fraction is not 0, Image with alt text: 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis not equals 2 left parenthesis 0 right parenthesis .
Since cotangent cannot be multiplied by a factor, 2 left-parenthesis 0 right-parenthesis not-equal-to 0.
2 cotangent left parenthesis negative Start Fraction 9 pi over 2 End Fraction right parenthesis equals 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis equals 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis equals 2 left parenthesis 0 right parenthesis equals 0
(1 point)
Responses
Since cotangent is an odd function, 2 cotangent left parenthesis negative Start Fraction 9 pi over 2 End Fraction right parenthesis not equals 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis.
Since cotangent is an odd function, Image with alt text: 2 cotangent left parenthesis negative Start Fraction 9 pi over 2 End Fraction right parenthesis not equals 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis .
Since cotangent has a period of π, 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis not equals 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis.
Since cotangent has a period of π , Image with alt text: 2 cotangent left parenthesis Start Fraction 9 pi over 2 End Fraction right parenthesis not equals 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis .
Since cotangent evaluated at Start Fraction pi over 2 End Fraction is not 0, 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis not equals 2 left parenthesis 0 right parenthesis.
Since cotangent evaluated at Image with alt text: Start Fraction pi over 2 End Fraction is not 0, Image with alt text: 2 cotangent left parenthesis Start Fraction pi over 2 End Fraction right parenthesis not equals 2 left parenthesis 0 right parenthesis .
Since cotangent cannot be multiplied by a factor, 2 left-parenthesis 0 right-parenthesis not-equal-to 0.
Answers
GPT-4o mini
Let's review Sandra's work to identify where she went wrong:
1. She started with \(2 \cot\left(-\frac{9\pi}{2}\right)\).
2. She incorrectly equated this to \(2 \cot\left(\frac{9\pi}{2}\right)\) without considering that cotangent is an odd function. In fact, the correct relationship is:
\[
\cot(-x) = -\cot(x).
\]
This means:
\[
2 \cot\left(-\frac{9\pi}{2}\right) = -2 \cot\left(\frac{9\pi}{2}\right).
\]
3. Next, Sandra computed \( \cot\left(\frac{9\pi}{2}\right) \). Given the periodic nature of cotangent, with a period of \(\pi\), we can simplify:
\[
\frac{9\pi}{2} = 4\pi + \frac{\pi}{2}.
\]
Since cotangent is periodic with period \(\pi\), it follows that:
\[
\cot\left(\frac{9\pi}{2}\right) = \cot\left(\frac{\pi}{2}\right).
\]
Evaluating this, we find:
\[
\cot\left(\frac{\pi}{2}\right) = 0.
\]
4. Therefore, at this point, she correctly finds that:
\[
\cot\left(\frac{9\pi}{2}\right) = 0.
\]
However, based on her earlier mistake, she incorrectly wrote:
\[
2 \cot\left(-\frac{9\pi}{2}\right) = 2(0) = 0.
\]
Instead, she should have found:
\[
2 \cot\left(-\frac{9\pi}{2}\right) = -2(0) = 0.
\]
Considering these points, the correct response to what Sandra did wrong is:
**Since cotangent is an odd function, \(2 \cot\left(-\frac{9\pi}{2}\right) \neq 2 \cot\left(\frac{9\pi}{2}\right)\).**
This captures the core issue with her inaccurate handling of the function's properties when substituting negative angles.
1. She started with \(2 \cot\left(-\frac{9\pi}{2}\right)\).
2. She incorrectly equated this to \(2 \cot\left(\frac{9\pi}{2}\right)\) without considering that cotangent is an odd function. In fact, the correct relationship is:
\[
\cot(-x) = -\cot(x).
\]
This means:
\[
2 \cot\left(-\frac{9\pi}{2}\right) = -2 \cot\left(\frac{9\pi}{2}\right).
\]
3. Next, Sandra computed \( \cot\left(\frac{9\pi}{2}\right) \). Given the periodic nature of cotangent, with a period of \(\pi\), we can simplify:
\[
\frac{9\pi}{2} = 4\pi + \frac{\pi}{2}.
\]
Since cotangent is periodic with period \(\pi\), it follows that:
\[
\cot\left(\frac{9\pi}{2}\right) = \cot\left(\frac{\pi}{2}\right).
\]
Evaluating this, we find:
\[
\cot\left(\frac{\pi}{2}\right) = 0.
\]
4. Therefore, at this point, she correctly finds that:
\[
\cot\left(\frac{9\pi}{2}\right) = 0.
\]
However, based on her earlier mistake, she incorrectly wrote:
\[
2 \cot\left(-\frac{9\pi}{2}\right) = 2(0) = 0.
\]
Instead, she should have found:
\[
2 \cot\left(-\frac{9\pi}{2}\right) = -2(0) = 0.
\]
Considering these points, the correct response to what Sandra did wrong is:
**Since cotangent is an odd function, \(2 \cot\left(-\frac{9\pi}{2}\right) \neq 2 \cot\left(\frac{9\pi}{2}\right)\).**
This captures the core issue with her inaccurate handling of the function's properties when substituting negative angles.