Question

The period of the function \(g(x)=5\cot(\frac{\pi}{3}x+\frac{\pi}{6})+2\) is equal to

• A. 2
• B. 3
• C. 4
• D. 5
• E. 6

Answer 1

The Correct Option is B

The period of the cotangent function \( \cot(x) \) is \( \pi \). For a function of the form \( \cot(kx + \phi) \), the period is given by: \[ \text{Period} = \frac{\pi}{|k|} \] In this case, the argument of the cotangent is \( \frac{\pi}{3} x + \frac{\pi}{6} \), where \( k = \frac{\pi}{3} \). Thus, the period of \( g(x) = 5 \cot \left( \frac{\pi}{3} x + \frac{\pi}{6} \right) + 2 \) is: \[ \text{Period} = \frac{\pi}{\left| \frac{\pi}{3} \right|} = 3 \]

The correct option is (B) : \(3\)

Answer 2

The general form of the cotangent function is \(f(x) = A\cot(Bx + C) + D\), where:

• \(A\) is the amplitude (vertical stretch)
• \(B\) affects the period
• \(C\) is the horizontal shift
• \(D\) is the vertical shift

The period of the standard cotangent function \(\cot(x)\) is \(\pi\).

For the given function \(g(x) = 5\cot\left(\frac{\pi}{3}x + \frac{\pi}{6}\right) + 2\), we have \(B = \frac{\pi}{3}\).

The period of \(g(x)\) is given by \(\frac{\pi}{|B|}\), so:

\(\text{Period} = \frac{\pi}{\left|\frac{\pi}{3}\right|} = \frac{\pi}{\frac{\pi}{3}} = \pi \cdot \frac{3}{\pi} = 3\)

Therefore, the period of the function \(g(x)\) is 3.