How many solutions of the following equation exist on the interval $[0,2 \pi)$ ? $\sin (7 x)=0$
Answer
So, there are \(\boxed{14}\) solutions to the equation \(\sin(7x) = 0\) in the interval \([0, 2\pi)\).
Step 1 ：The solutions to the equation \(\sin(7x) = 0\) are given by \(7x = n\pi\), where \(n\) is an integer.
Step 2 ：To find the number of solutions in the interval \([0, 2\pi)\), we need to find the number of integers \(n\) such that \(0 \leq 7x < 14\pi\). This means \(0 \leq n < 14\).
Step 3 ：The solutions are \[0, 0.44879895, 0.8975979, 1.34639685, 1.7951958, 2.24399475, 2.6927937, 3.14159265, 3.5903916, 4.03919055, 4.48798951, 4.93678846, 5.38558741, 5.83438636\]
Step 4 ：So, there are \(\boxed{14}\) solutions to the equation \(\sin(7x) = 0\) in the interval \([0, 2\pi)\).
