Question

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

Short answer

\(\operatorname{arccot}(-\sqrt{3}) = -\frac{\pi}{3}\).

Step-by-step solution

Understanding Cotangent and Arccotangent

Consequently, the cotangent of an angle in a right triangle is the ratio of the adjacent side to the opposite side. It's also the reciprocal of tangent. Arccotangent, denoted as \(\operatorname{arccot}\), is the inverse cotangent function. That means if \(y = \operatorname{cot}(x)\), then \(\operatorname{arccot}(y) = x\).

Mapping the value

Next, try to map the given value to one of the known values in the unit circle. Though considering the negative value seems tricky, the factor that it is the square root of 3 makes it easier to be mapped to the 60° angle. So, we can say \(\operatorname{cot}(60°) = \frac{1}{\sqrt{3}}\). Therefore, \(\operatorname{cot}(-60°) = -\frac{1}{\sqrt{3}} = -\sqrt{3}\). Turn it around we get \(\operatorname{arccot}(-\sqrt{3}) = -60°\).

Converting angle measure to radians

However, usually, the convention is to express the angle measure in radians, especially when it's returned from the \(\operatorname{arccot}\) function. To convert degrees into radians use the formula: \[radians = degrees * \frac{\pi}{180}\]\ So, \[\operatorname{arccot}(-\sqrt{3}) = -60° * \frac{\pi}{180} = -\frac{\pi}{3}\]