Question

If \( n \in \mathbb{N} \), then \( |\sin nx| \) is

• A. \( \leq n |\sin x| \)
• B. \( \geq n |\sin x| \)
• C. \( = n |\sin x| \)
• D. None of these

Hint:

When dealing with multiple-angle sine functions, use the fact that \( |\sin nx| \) is always bounded by the maximum value of the sine function, scaled by \( n \).

Answer 1

The Correct Option is A

Step 1: Recognize the periodic nature of \( \sin nx \).
We are given that \( n \in \mathbb{N} \), meaning \( n \) is a positive integer. The function \( \sin nx \) is simply the sine function evaluated at a multiple of \( x \), and it has the same range as the basic sine function, i.e., \( [-1, 1] \).

Step 2: Use the properties of absolute value.
The absolute value of \( \sin nx \) will always lie between 0 and 1, just like the basic sine function: \[ |\sin nx| \leq 1 \]

Step 3: Analyze the expression \( |\sin nx| \).
The expression \( |\sin nx| \) is bounded by \( n |\sin x| \), because multiplying by \( n \) scales the sine function, but the absolute value ensures that the inequality holds true: \[ |\sin nx| \leq n |\sin x| \]

Step 4: Conclusion.
Thus, \( |\sin nx| \leq n |\sin x| \), corresponding to option (A).