Question:
The period of the function
\[
f(x)=\log(\cos x)+\cot^4\left(\frac{x}{2}\right)+\sin(3x+7)
\]
is
The period of the function
\[
f(x)=\log(\cos x)+\cot^4\left(\frac{x}{2}\right)+\sin(3x+7)
\]
is
Show Hint
For sums of periodic functions, first find the fundamental period of each term. Then determine the least positive common period, while also checking domain restrictions of functions such as logarithms.
Updated On: Jul 29, 2026
- \(4\pi\)
- \(\dfrac{2\pi}{3}\)
- \(2\pi\)
- \(\pi\)
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The Correct Option is C
Solution and Explanation
Concept:
The period of the sum of periodic functions is the least positive number that is a period of each constituent function and preserves the domain of the function.
Step 1: Find the period of \(\log(\cos x)\). For the logarithm to be defined, \[ \cos x\gt 0. \] Also, \[ \log(\cos(x+2\pi)) = \log(\cos x). \] Since \[ \log(\cos(x+\pi)) = \log(-\cos x) \] is not defined whenever \(\cos x\gt 0\), \(\pi\) is not a period. Hence, the period of \[ \log(\cos x) \] is \[ 2\pi. \]
Step 2: Find the period of \(\cot^4\left(\frac{x}{2}\right)\). Since \[ \cot\theta \] has period \(\pi\), \[ \cot\left(\frac{x}{2}\right) \] has period \[ 2\pi. \] Therefore, \[ \cot^4\left(\frac{x}{2}\right) \] also has period \[ 2\pi. \]
Step 3: Find the period of \(\sin(3x+7)\). For \[ \sin(ax+b), \] the period is \[ \frac{2\pi}{|a|}. \] Hence, \[ \text{Period of }\sin(3x+7) = \frac{2\pi}{3}. \]
Step 4: Find the least common period. The individual periods are \[ 2\pi,\qquad 2\pi,\qquad \frac{2\pi}{3}. \] The least positive common period is \[ 2\pi. \] Therefore, the period of \(f(x)\) is \[ \boxed{2\pi} \] \[ \boxed{\text{Answer = (C)}} \]
Step 1: Find the period of \(\log(\cos x)\). For the logarithm to be defined, \[ \cos x\gt 0. \] Also, \[ \log(\cos(x+2\pi)) = \log(\cos x). \] Since \[ \log(\cos(x+\pi)) = \log(-\cos x) \] is not defined whenever \(\cos x\gt 0\), \(\pi\) is not a period. Hence, the period of \[ \log(\cos x) \] is \[ 2\pi. \]
Step 2: Find the period of \(\cot^4\left(\frac{x}{2}\right)\). Since \[ \cot\theta \] has period \(\pi\), \[ \cot\left(\frac{x}{2}\right) \] has period \[ 2\pi. \] Therefore, \[ \cot^4\left(\frac{x}{2}\right) \] also has period \[ 2\pi. \]
Step 3: Find the period of \(\sin(3x+7)\). For \[ \sin(ax+b), \] the period is \[ \frac{2\pi}{|a|}. \] Hence, \[ \text{Period of }\sin(3x+7) = \frac{2\pi}{3}. \]
Step 4: Find the least common period. The individual periods are \[ 2\pi,\qquad 2\pi,\qquad \frac{2\pi}{3}. \] The least positive common period is \[ 2\pi. \] Therefore, the period of \(f(x)\) is \[ \boxed{2\pi} \] \[ \boxed{\text{Answer = (C)}} \]
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