Answer :
Answer:
Step-by-step explanation:
To solve the equation \( \sin(2x) = \sin(x) \), we can use trigonometric identities.
Recall the double-angle identity for sine:
\[ \sin(2x) = 2\sin(x)\cos(x) \]
Now, we can rewrite the equation as:
\[ 2\sin(x)\cos(x) = \sin(x) \]
We have sin(x) on both sides of the equation, so we can divide both sides by sin(x) (assuming sin(x) ≠ 0):
\[ 2\cos(x) = 1 \]
Now, we solve for cos(x):
\[ \cos(x) = \frac{1}{2} \]
To find the solutions for \( x \) within the range of \( 0 \leq x \leq 2\pi \), we examine where the cosine function equals \( \frac{1}{2} \). This occurs at two places in the unit circle: \( \frac{\pi}{3} \) and \( \frac{5\pi}{3} \).
So, the solutions for \( x \) are \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \).
