Question

The period of the function \( f(x) = \frac{2\sin\left(\frac{\pi x}{3}\right) \cos\left(\frac{2\pi x}{5}\right)}{3\tan\left(\frac{7\pi x}{2}\right) - 5\sec\left(\frac{5\pi x}{3}\right)} \) is

• A. 30
• B. 60
• C. 300
• D. 150

Hint:

When finding the LCM of fractions, convert all integers to fractions with denominator 1. The formula is LCM(Numerators) / GCD(Denominators).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:

The period of a function involving sums, differences, products, or quotients of trigonometric functions is the Least Common Multiple (LCM) of the individual periods of its components.
Step 2: Key Formula or Approach:

For \( \sin(ax), \cos(ax), \sec(ax) \), Period \( T = \frac{2\pi}{|a|} \). For \( \tan(ax) \), Period \( T = \frac{\pi}{|a|} \). \( \text{LCM}\left(\frac{a}{b}, \frac{c}{d}, \dots\right) = \frac{\text{LCM}(a, c, \dots)}{\text{GCD}(b, d, \dots)} \).
Step 3: Detailed Explanation:

Calculate periods of individual terms: 1. \( \sin\left(\frac{\pi x}{3}\right) \): \( T_1 = \frac{2\pi}{\pi/3} = 6 \). 2. \( \cos\left(\frac{2\pi x}{5}\right) \): \( T_2 = \frac{2\pi}{2\pi/5} = 5 \). 3. \( \tan\left(\frac{7\pi x}{2}\right) \): \( T_3 = \frac{\pi}{7\pi/2} = \frac{2}{7} \). 4. \( \sec\left(\frac{5\pi x}{3}\right) \): \( T_4 = \frac{2\pi}{5\pi/3} = \frac{6}{5} \). Find LCM of \( \{6, 5, \frac{2}{7}, \frac{6}{5}\} \): This is equivalent to LCM of fractions \( \frac{6}{1}, \frac{5}{1}, \frac{2}{7}, \frac{6}{5} \). \[ \text{Numerator LCM} = \text{LCM}(6, 5, 2, 6) = 30 \] \[ \text{Denominator GCD} = \text{GCD}(1, 1, 7, 5) = 1 \] \[ \text{Period} = \frac{30}{1} = 30 \]
Step 4: Final Answer:

The period is 30.