Let's review Sandra's work to identify where she went wrong:
-
She started with \(2 \cot\left(-\frac{9\pi}{2}\right)\).
-
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: \[ 2 \cot\left(-\frac{9\pi}{2}\right) = -2 \cot\left(\frac{9\pi}{2}\right). \]
-
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: \[ \cot\left(\frac{\pi}{2}\right) = 0. \]
-
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 answer