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Chapter 4: Problem 18

Evaluate the expression without using a calculator. $$\tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right)$$

Short Answer

Expert verified
The solution to the exercise \(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\) is \(-30^{掳}\) or \(-蟺/6\) radians.

Step by step solution

01

Understanding the inverse tangent function

The inverse tangent function \(\tan^{-1}(x)\), also written as \(\arctan(x)\), returns the angle whose tangent is \(x\). This function is the opposite of the tangent function, which gives the ratio of the sides of a right triangle. The output of the inverse tangent function is usually in the interval \(-蟺/2\) to \(蟺/2\) (or \(-90掳\) to \(90掳\)).
02

Identifying the Ratio

The ratio \(-\sqrt{3}/3\) is a negative number. In order to find the angle associated with this ratio, one needs to identify that -sqrt(3)/3 is equivalent to the tangent of -30 degrees or -蟺/6 radians.
03

Final answer

From the knowledge and understanding that the inverse tangent of \(-蟺/6\) or \(-30^{掳}\) is equal to \(-\sqrt{3}/3\), it's concluded that the evaluation of the expression \(\tan^{-1}(-\sqrt{3}/3)\) gives \(-30^{掳}\) or \(-蟺/6\) radians as the final result.

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