Which of the following trigonometric functions has an amplitude of 3 ?

Options

A. `f(x)=1/3sinx`
B. `f(x)=cos 3x `
C. `f(x)=sin (1/3 x)`
D. f(x)=3 cos x

Answer

To determine which of the given trigonometric functions has an amplitude of 3, we will analyze each option step by step.

### Step 1: Understand the concept of amplitude
The amplitude of a trigonometric function is defined as half the distance between its maximum and minimum values. For sine and cosine functions, the general form is \( A \sin(Bx) \) or \( A \cos(Bx) \), where \( A \) represents the amplitude.

### Step 2: Analyze each option

**Option 1: \( f(x) = \frac{1}{3} \sin x \)**
- The amplitude is \( \frac{1}{3} \) (since \( A = \frac{1}{3} \)).
- Therefore, the range of this function is from \( -\frac{1}{3} \) to \( \frac{1}{3} \).
- **Conclusion**: This option does not have an amplitude of 3.

**Option 2: \( f(x) = \cos(3x) \)**
- The amplitude is \( 1 \) (since \( A = 1 \)).
- The range of this function is from \( -1 \) to \( 1 \).
- **Conclusion**: This option does not have an amplitude of 3.

**Option 3: \( f(x) = \sin\left(\frac{1}{3} x\right) \)**
- The amplitude is \( 1 \) (since \( A = 1 \)).
- The range of this function is from \( -1 \) to \( 1 \).
- **Conclusion**: This option does not have an amplitude of 3.

**Option 4: \( f(x) = 3 \cos x \)**
- The amplitude is \( 3 \) (since \( A = 3 \)).
- The range of this function is from \( -3 \) to \( 3 \).
- **Conclusion**: This option has an amplitude of 3.

### Final Conclusion
The function that has an amplitude of 3 is **Option 4: \( f(x) = 3 \cos x \)**.

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