Question

In Problems 1– 8, find the exact value of each expression. Do not use a calculator.

${\cot }^{-1}(-1)$

Short answer

The exact value is $\frac{3\mathrm{\pi }}{4}$.

Step-by-step solution

Step 1. Given Information

We are given an expression $co{t}^{-1}(-1)$.

We need to find its exact value.

Step 2. Find the exact value

The inverse cotangent function$y=co{t}^{-1}x$ means that $x=cot\left(y\right)$where $-\infty <x<\infty$and $0<y<\mathrm{\pi }$.

Let $\theta =co{t}^{-1}(-1)$, so using the definition of inverse cotangent function it can be written as $-1=cot\theta$where $0<\theta <\mathrm{\pi }$.

In this interval the angle for which the cotangent value is $-1$is $\frac{3\mathrm{\pi }}{4}$.

So the value of $co{t}^{-1}(-1)$is $\frac{3\mathrm{\pi }}{4}$.