which is an x - intercept of the graph of the function y = cot(3x)?\n(π/6, 0)\n(π/3, 0)\n(3π, 0)\n(6π, 0)

Answer:

$\left(\frac{\pi}{6},0\right)$

Explanation:

Step1: Recall the formula for x - intercept

For a function (y = f(x)), the x - intercepts are found by setting (y = 0). So, for (y=\cot(3x)), we set (\cot(3x)=0). Since (\cot\theta=\frac{\cos\theta}{\sin\theta}), (\cot\theta = 0) when (\cos\theta=0) and (\sin\theta\neq0). So, (\cos(3x) = 0) and (\sin(3x)\neq0). We know that (\cos\alpha=0) when (\alpha=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}). So, (3x=(2n + 1)\frac{\pi}{2}), then (x=(2n + 1)\frac{\pi}{6},n\in\mathbb{Z}).

Step2: Find a particular value of (n)

When (n = 0), (x=\frac{\pi}{6}). Substitute (x = \frac{\pi}{6}) into the function (y=\cot(3x)). We have (y=\cot\left(3\times\frac{\pi}{6}\right)=\cot\left(\frac{\pi}{2}\right)). Since (\cot\left(\frac{\pi}{2}\right)=\frac{\cos\left(\frac{\pi}{2}\right)}{\sin\left(\frac{\pi}{2}\right)}=\frac{0}{1} = 0).

For (x=\frac{\pi}{3}), (y=\cot\left(3\times\frac{\pi}{3}\right)=\cot(\pi)), and (\cot(\pi)=\frac{\cos(\pi)}{\sin(\pi)}), but (\sin(\pi)=0), so (\cot(\pi)) is undefined. For (x = 3\pi), (y=\cot(9\pi)), and (\cot(9\pi)=\frac{\cos(9\pi)}{\sin(9\pi)}), since (\sin(9\pi)=0), (\cot(9\pi)) is undefined. For (x = 6\pi), (y=\cot(18\pi)), and (\cot(18\pi)=\frac{\cos(18\pi)}{\sin(18\pi)}), since (\sin(18\pi)=0), (\cot(18\pi)) is undefined.