3. What is the maximum value of \( |3 \sin (100 \theta)-1| \) ? Hint: consider the range of the sine function.

To find the maximum value of \( |3 \sin (100 \theta)-1| \), first recognize that the sine function, \( \sin(100 \theta) \), oscillates between -1 and 1. Thus, \( 3 \sin(100 \theta) \) oscillates between -3 and 3. Next, we can rewrite the expression: \[ |3 \sin (100 \theta) - 1| \] When \( 3 \sin(100 \theta) = 3 \), we get: \[ |3 - 1| = 2 \] When \( 3 \sin(100 \theta) = -3 \), we have: \[ |-3 - 1| = |-4| = 4 \] To find the maximum value, we need to evaluate the expression at the extremes of \( 3 \sin(100 \theta) \): 1. For \( 3 \sin(100 \theta) = 3 \): \[ |3 - 1| = 2 \] 2. For \( 3 \sin(100 \theta) = -3 \): \[ |-3 - 1| = 4 \] Thus, the maximum value of \( |3 \sin (100 \theta)-1| \) is: \[ \boxed{4} \]