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Chapter 5: Problem 3

Evaluate \(\tan ^{-1}(-1)\).

Short Answer

Expert verified
To evaluate \(\tan ^{-1}(-1)\), find the angle in the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) with a tangent value of -1. The reference angle with a tangent value of 1 is \(\frac{\pi}{4}\). Since the tangent is negative, the angle should be in the fourth quadrant. Therefore, \(\tan ^{-1}(-1) = -\frac{\pi}{4}\).

Step by step solution

01

Identify the function

We are asked to evaluate \(\tan ^{-1}(-1)\). It represents the angle whose tangent value is -1.
02

Find the reference angle

A reference angle is a positive acute angle that is associated with the angle in question. Since the tangent function has a period of \(\pi\), we can find a reference angle with tangent value 1 in the first or second quadrant. The angle of \(45^\circ\) or \(\frac{\pi}{4}\) radian has a tangent value of 1.
03

Determine the correct quadrant

Since the tangent of the angle is negative, we need to find an angle in a quadrant where the tangent function is negative. Tangent function is negative in the second (angles between \(90^\circ\) and \(180^\circ\) or \(\frac{\pi}{2}\) and \(\pi\) radians) and the fourth quadrants (angles between \(270^\circ\) and \(360^\circ\) or \(\frac{3\pi}{2}\) and \(2\pi\) radians). However, we are looking for an angle in the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), so it must be in the fourth quadrant.
04

Find the negative angle with the same reference angle

Since the reference angle is known (which is \(\frac{\pi}{4}\)), and it should be in the fourth quadrant with negative value, the angle whose tangent is -1 is \(-45^\circ\) or \(-\frac{\pi}{4}\) radians. Therefore, \(\tan ^{-1}(-1) = -\frac{\pi}{4}\).

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