Question

find the exact value of the real number y if it exists. do not use a calculator.

( y = \tan ^ { - 1 } left( - \frac { sqrt { 3 } } { 3 }
ight) )

select the correct choice and fill in any answer boxes in your choice below.

a.
( y = \tan ^ { - 1 } left( - \frac { sqrt { 3 } } { 3 }
ight) = )
(simplify your answer. type an exact answer, using ( pi ) as needed. use integers or fractions

b.
( \tan ^ { - 1 } left( - \frac { sqrt { 3 } } { 3 }
ight) ) does not exist.

Explanation:

Step1: Recall the range of the inverse tangent function

The range of \(y = \tan^{-1}(x)\) is \((-\frac{\pi}{2},\frac{\pi}{2})\).

Step2: Find the angle whose tangent is \(-\frac{\sqrt{3}}{3}\)

We know that \(\tan(\frac{\pi}{6})=\frac{\sqrt{3}}{3}\). Since the tangent function is odd, \(\tan(-\theta)=-\tan(\theta)\). So \(\tan(-\frac{\pi}{6})=-\frac{\sqrt{3}}{3}\) and \(-\frac{\pi}{6}\in(-\frac{\pi}{2},\frac{\pi}{2})\).

Answer:

A. \(y = \tan^{-1}(-\frac{\sqrt{3}}{3})=-\frac{\pi}{6}\)