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: Understanding the Concept:
The period of a function is the smallest positive value T for which \(f(x+T) = f(x)\) for all x. For a function that is a sum of two or more periodic functions, the period is the least common multiple (LCM) of the individual periods.
Step 2: Key Formula or Approach:
1. The period of \(a\sin(bx+c)\) and \(a\cos(bx+c)\) is \(T = \frac{2\pi}{|b|}\).
2. The period of \(f(x) = g(x) + h(x)\), where \(g(x)\) has period \(T_1\) and \(h(x)\) has period \(T_2\), is given by LCM(\(T_1, T_2\)).
3. For rational periods \(T_1 = \frac{p_1}{q_1}\) and \(T_2 = \frac{p_2}{q_2}\), LCM(\(T_1, T_2\)) = \(\frac{\text{LCM}(p_1, p_2)}{\text{HCF}(q_1, q_2)}\).
Step 3: Detailed Explanation:
The function is \(f(x) = 2\sin(4x) + 3\cos(2x)\).
Let's find the period of each term separately.
For the first term, \(g(x) = 2\sin(4x)\):
Here, \(b=4\). The period is \(T_1 = \frac{2\pi}{|4|} = \frac{\pi}{2}\).
For the second term, \(h(x) = 3\cos(2x)\):
Here, \(b=2\). The period is \(T_2 = \frac{2\pi}{|2|} = \pi\).
Find the period of f(x):
The period of \(f(x)\) is the LCM of \(T_1\) and \(T_2\).
Period = LCM(\(\frac{\pi}{2}, \pi\)).
To find the LCM, we can write the periods as fractions with a common base: \(T_1 = \frac{1}{2}\pi\) and \(T_2 = \frac{1}{1}\pi\).
Using the formula for LCM of fractions: LCM(\(\frac{a}{b}, \frac{c}{d}\)) = \(\frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}\).
Here, our fractions are \(\frac{1}{2}\) and \(\frac{1}{1}\) (ignoring \(\pi\) for a moment).
LCM(\(\frac{1}{2}, 1\)) = \(\frac{\text{LCM}(1, 1)}{\text{HCF}(2, 1)} = \frac{1}{1} = 1\).
So, the period is \(1 \times \pi = \pi\).
Alternatively, we can find the smallest number which is an integer multiple of both \(\pi/2\) and \(\pi\).
Multiples of \(\pi/2\): \(\pi/2, \pi, 3\pi/2, 2\pi, ...\)
Multiples of \(\pi\): \(\pi, 2\pi, 3\pi, ...\)
The least common multiple is \(\pi\).
Step 4: Final Answer:
The period of the function \(f(x)\) is \(\pi\). Therefore, option (B) is correct.