Question
A student solved the equation Start Fraction sine 2 x over cosine x End Fraction equals 2, 0 less than or equals x less than or equals pi, and found an answer of Start Fraction pi over 2 End Fraction. Describe the student's error. no ai or big words!
Answers
GPT-4o mini
The student's error is in the equation \(\frac{\sin(2x)}{\cos(x)} = 2\).
First, let's rewrite it:
\[
\sin(2x) = 2\cos(x)
\]
The student found \(x = \frac{\pi}{2}\) as a solution. However, if we evaluate \(\sin(2x)\) and \(\cos(x)\) at \(x = \frac{\pi}{2}\):
1. \(\sin(2 \cdot \frac{\pi}{2}) = \sin(\pi) = 0\)
2. \(\cos(\frac{\pi}{2}) = 0\)
This makes the left side \(\frac{0}{0}\), which is undefined.
The student should have checked all possible values for \(x\) in the range \(0 \leq x \leq \pi\) to find valid solutions. They need to solve the original equation properly to find other solutions instead of just guessing.
First, let's rewrite it:
\[
\sin(2x) = 2\cos(x)
\]
The student found \(x = \frac{\pi}{2}\) as a solution. However, if we evaluate \(\sin(2x)\) and \(\cos(x)\) at \(x = \frac{\pi}{2}\):
1. \(\sin(2 \cdot \frac{\pi}{2}) = \sin(\pi) = 0\)
2. \(\cos(\frac{\pi}{2}) = 0\)
This makes the left side \(\frac{0}{0}\), which is undefined.
The student should have checked all possible values for \(x\) in the range \(0 \leq x \leq \pi\) to find valid solutions. They need to solve the original equation properly to find other solutions instead of just guessing.