find the exact value of \\( \\tan ^ { - 1 } \\left( \\frac { \\sqrt { 3 } } { 3 } \\right) \\).\n\n\\( \\tan…

find the exact value of \\( \\tan ^ { - 1 } \\left( \\frac { \\sqrt { 3 } } { 3 } \\right) \\).\n\n\\( \\tan ^ { - 1 } \\left( \\frac { \\sqrt { 3 } } { 3 } \\right) = \\square \\)\n(type your answer in radians.)

find the exact value of \\( \\tan ^ { - 1 } \\left( \\frac { \\sqrt { 3 } } { 3 } \\right) \\).\n\n\\( \\tan ^ { - 1 } \\left( \\frac { \\sqrt { 3 } } { 3 } \\right) = \\square \\)\n(type your answer in radians.)

Answer

Explanation:

Step1: Recall the range of the inverse tangent function

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

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

We know that (\tan\left(\frac{\pi}{6}\right)=\frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}), and (\frac{\pi}{6}\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right))

Answer:

(\frac{\pi}{6})