Find the exact value of the following expression. [tex]$\tan ^{-1}\left(\frac{\sqrt{3}}{3}\right)$[/tex]
Select the correct choice below and, if necessary, fill in the answer box to
A. [tex]$\tan ^{-1}\left(\frac{\sqrt{3}}{3}\right)=[/tex] $\square$
(Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed.
Answer (2)
The exact value of tan − 1 ( 3 3 ) is 6 π . This is derived because tan ( 6 π ) equals 3 3 . Hence, the answer is 6 π .
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The problem asks to find the exact value of tan − 1 ( 3 3 ) .
Recall that tan ( 6 π ) = 3 3 .
Since 6 π is in the range of the inverse tangent function ( − 2 π , 2 π ) , we have tan − 1 ( 3 3 ) = 6 π .
The exact value of the expression is 6 π .
Explanation
Understanding the problem We are asked to find the exact value of tan − 1 ( 3 3 ) . This means we need to find an angle whose tangent is 3 3 .
Range of inverse tangent Recall that the range of the inverse tangent function is ( − 2 π , 2 π ) . So, we are looking for an angle θ in this interval such that tan ( θ ) = 3 3 .
Simplifying the expression We know that 3 3 = 3 1 . We need to find an angle whose tangent is 3 1 . Recall the values of the tangent function for common angles.
Finding the angle We know that tan ( 6 π ) = c o s ( 6 π ) s i n ( 6 π ) = 2 3 2 1 = 3 1 = 3 3 . Since 6 π is in the range of the inverse tangent function, we have tan − 1 ( 3 3 ) = 6 π .
Final Answer Therefore, the exact value of the expression is 6 π .
Examples
Imagine you're designing a ramp for a skateboard park. The angle of the ramp is crucial for the skaters' safety and performance. Using the inverse tangent function, you can determine the exact angle needed to achieve a specific ratio of height to horizontal distance. For instance, if you want the ramp to rise 3 meters for every 3 meters of horizontal distance, you can use tan − 1 ( 3 3 ) to find the precise angle, which is 6 π radians or 30 degrees. This ensures the ramp is neither too steep nor too shallow, providing an optimal experience for the skaters.
