Question

The period of the function \(f(x)=2\sin 4x+3\cos 2x\) is

• A. \(\frac{\pi}{2}\)
• B. \(\pi\)
• C. \(\frac{3\pi}{2}\)
• D. \(2\pi\)
• E. \(3\pi\)

Hint:

For sum of trigonometric functions, always find the LCM of individual periods. That gives the fundamental period.

Answer 1

The Correct Option is B

Step 1: Identify individual functions.
The function is: \[ f(x)=2\sin 4x+3\cos 2x \] It is a sum of two trigonometric functions: \(\sin 4x\) and \(\cos 2x\).

Step 2: Recall the period formula.
For \(\sin(ax)\) or \(\cos(ax)\), the period is: \[ \text{Period}=\frac{2\pi}{a} \]

Step 3: Find the period of each term.
For \(2\sin 4x\):
\[ \text{Period}=\frac{2\pi}{4}=\frac{\pi}{2} \] For \(3\cos 2x\):
\[ \text{Period}=\frac{2\pi}{2}=\pi \]

Step 4: Find the common period.
The period of the sum of two periodic functions is the least common multiple (LCM) of their individual periods.

Step 5: Compute LCM of \(\frac{\pi}{2}\) and \(\pi\).
\[ \frac{\pi}{2}=\frac{\pi}{2}, \quad \pi=\frac{2\pi}{2} \] So LCM is: \[ \pi \]

Step 6: Verify the result.
After interval \(\pi\), both functions complete an integer number of cycles, hence the sum repeats.

Step 7: Final answer.
Thus, the period is: \[ \boxed{\pi} \] which matches option \((2)\).