Question

use a reference triangle in an appropriate quadrant to find the given angle.

$$\\cot^{-1}\\left(\\frac{1}{\\sqrt{3}}\ ight)$$

$$\\cot^{-1}\\left(\\frac{1}{\\sqrt{3}}\ ight)=\\square$$

(type an exact answer, using \\( \pi \\) as needed. use integers or fractions for any numbers in the expre

Explanation:

Step1: Recall the range of \(y = \cot^{-1}(x)\)

The range of \(y=\cot^{-1}(x)\) is \((0,\pi)\).

Step2: Use the relationship between cotangent and tangent

We know that \(\cot\theta=\frac{1}{\tan\theta}\). If \(\cot\theta = \frac{1}{\sqrt{3}}\), then \(\tan\theta=\sqrt{3}\).

Step3: Find the angle in the range \((0,\pi)\)

We know that \(\tan\frac{\pi}{3}=\sqrt{3}\) and \(\frac{\pi}{3}\in(0,\pi)\)

Answer:

\(\frac{\pi}{3}\)