Problem 8 - Calculus of a Single Variable

Question

Evaluate the expression without using a calculator. \(\operatorname{arccot}(-\sqrt{3})\)

Step-by-Step Solution

Answer

\(\operatorname{arccot}(-\sqrt{3})\) equals 150° or \(\frac{5\pi}{6}\) radians.

Step 1: Analyzing the input

It's known that \(\operatorname{arccot}\) function is the inverse of the cotangent function. It takes a number \(-\sqrt{3}\) as input and outputs an angle whose cotangent is that number.

Step 2: Connecting with the unit circle

The cotangent of an angle in the unit circle is defined as the ratio of the x-coordinate to the y-coordinate. It's noticed that the cotangent of the angle 150° or \(\frac{5\pi}{6}\) radians is \(-\sqrt{3}\), and the same for the angle 330° or \(\frac{11\pi}{6}\) radians. However, \(\operatorname{arccot}\) function returns the minor positive angle that cotangent equals to the input number, but as the input number is negative, the reference angle will be in the second quadrant.

Step 3: Evaluating the expression

Therefore, \(\operatorname{arccot}(-\sqrt{3})\) is equal to the angle 150° or \(\frac{5\pi}{6}\) radians.