Question

The period of function f(x) = \(e^{log(sinx)}+(tanx)^3 - cosec(3x - 5)\)is

• A. π
• B. π/2
• C. 2π
• D. 2π/3

Answer 1

The Correct Option is C

To find the period of the function \( f(x) = e^{\log(\sin x)} + (\tan x)^3 - \csc(3x - 5) \), we need to determine the period of each individual component and then find their least common multiple (LCM).

1. The function \( e^{\log(\sin x)} \) simplifies to \( \sin x \). The basic period of \( \sin x \) is \( 2\pi \).

2. The function \( (\tan x)^3 \) is based on \( \tan x \), which has a fundamental period of \( \pi \). Therefore, the period of \( (\tan x)^3 \) is also \( \pi \).

3. The function \(\csc(3x - 5)\) has the same period as \(\sin(3x - 5)\) since \(\csc(x) = 1/\sin(x)\). The period of \(\sin(3x)\) is \(\frac{2\pi}{3}\). Hence, the period of \(\csc(3x - 5)\) is also \(\frac{2\pi}{3}\).

To find the overall period of \( f(x) \), we take the LCM of the individual periods:

\(\text{LCM}(2\pi, \pi, \frac{2\pi}{3})\).

Converting these to a common base:

• The period \( 2\pi \) is equivalent to \( \frac{6\pi}{3} \).
• The period \( \pi \) is equivalent to \( \frac{3\pi}{3} \).
• The period \( \frac{2\pi}{3} \) remains as is.

The LCM of \(\frac{6\pi}{3}\), \(\frac{3\pi}{3}\), and \(\frac{2\pi}{3}\) is \(2\pi\).

Thus, the period of the function \( f(x) \) is \( 2\pi \).

Conclusion: The correct answer is \( 2\pi \).