Question

Find the exact value of the given expression. $$ \cot ^{-1}(-1) $$

Short answer

The exact value of the given expression is: \[ \cot^{-1}(-1) = 135^\circ \]

Step-by-step solution

Understanding cotangent

Cotangent of an angle in a right triangle is defined as the ratio of the adjacent side to the opposite side, which is the reciprocal of the tangent function: \(\cot(\theta) = \frac{\text{adjacent side}}{\text{opposite side}}\). In this exercise, we need to find the angle, \(\theta\), whose cotangent is -1.

Identifying the quadrant

Since the cotangent value of the angle is negative, we need to find in which quadrants the cotangent is negative. The cotangent function has the same sign pattern as the tangent function, which is positive in quadrant I and III, and negative in quadrant II and IV. Therefore, the angle we are looking for must be in either quadrant II or quadrant IV.

Finding the angle with cotangent = -1

Now we will find the angle whose cotangent is -1. We know that \(\cot(\theta) = -1\), so \(\tan(\theta) = -\frac{1}{1} = -1\). First, we will examine the angle in quadrant II: In quadrant II, the angle \(\theta\) is supplementary to an acute angle, say \(x\). So, \(\theta = 180^\circ - x\) and \(180^\circ > \theta > 90^\circ\). Now, \(\tan(\theta) = -1\) and \(\tan(180^\circ - x) = -\tan(x)\), thus we have: \[-\tan(x) = -1\] \[\tan(x) = 1\] Since \(\tan(45^\circ) = 1\), we have \(x = 45^\circ\) and \(\theta = 180^\circ - 45^\circ = 135^\circ\). Now, let's examine the angle in quadrant IV: In quadrant IV, the angle \(\theta\) is the exterior angle of an acute angle, say \(y\). So, \(\theta = 360^\circ - y\) and \(360^\circ > \theta > 270^\circ\). Now, \(\tan(\theta) = -1\) and \(\tan(360^\circ - y) = -\tan(y)\), thus we have: \[-\tan(y) = -1\] \[\tan(y) = 1\] Similarly, since \(\tan(45^\circ) = 1\), we have \(y = 45^\circ\) and \(\theta = 360^\circ - 45^\circ = 315^\circ\). Since principal values of inverse cotangent lie within the interval \((0, 180^\circ)\), the cotangent value of the angle in the fourth quadrant wouldn't be considered. Hence, the angle in quadrant II would be the required angle.

Final answer

We have found the angle whose cotangent value is -1, which is \(\theta = 135^\circ\). Thus, we can conclude: \[\cot ^{-1}(-1) = 135^\circ\]