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. Write in simplified sentences

Answer 1

The student's error lies in misinterpreting the equation \(\frac{\sin(2x)}{\cos(x)} = 2\).

1. The equation can be rewritten using the double angle identity: \(\sin(2x) = 2\sin(x)\cos(x)\).
2. This gives us \(\frac{2\sin(x)\cos(x)}{\cos(x)} = 2\).
3. When simplifying, we need to consider that \(\cos(x) \neq 0\) to avoid division by zero. This leads to \(2\sin(x) = 2\).
4. Dividing both sides by 2, we have \(\sin(x) = 1\).
5. The solution to \(\sin(x) = 1\) in the interval \(0 \leq x \leq \pi\) is \(x = \frac{\pi}{2}\), which is correct, but the mistake lies in not considering the conditions under which we simplify the fraction.

To summarize, while the student found the solution \(x = \frac{\pi}{2}\) to be correct, they may have overlooked the importance of excluding values where \(\cos(x) = 0\) and should be clear about the principles used in simplifying the expression.