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 blank.
A. [tex]$\tan ^{-1}\left(\frac{\sqrt{3}}{3}\right)=[/tex]
(Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed.)
Answer (2)
We are looking for an angle θ such that tan ( θ ) = 3 3 .
Recall that tan ( θ ) = c o s ( θ ) s i n ( θ ) .
We know that sin ( 6 π ) = 2 1 and cos ( 6 π ) = 2 3 , so tan ( 6 π ) = 3 3 .
Therefore, tan − 1 ( 3 3 ) = 6 π .
Explanation
Understanding the Problem We are asked to find the exact value of the inverse tangent of 3 3 . The inverse tangent function, denoted as tan − 1 ( x ) or arctan ( x ) , gives the angle whose tangent is x . In other words, we need to find an angle θ such that tan ( θ ) = 3 3 .
Recalling Trigonometric Values Recall the unit circle and the values of trigonometric functions for common angles. We are looking for an angle θ in the range ( − 2 π , 2 π ) such that tan ( θ ) = 3 3 .
Finding the Angle Since tan ( θ ) = c o s ( θ ) s i n ( θ ) , we are looking for an angle where c o s ( θ ) s i n ( θ ) = 3 3 . We know that sin ( 6 π ) = 2 1 and cos ( 6 π ) = 2 3 . Therefore, tan ( 6 π ) = 2 3 2 1 = 3 1 = 3 3 .
Final Answer Thus, tan − 1 ( 3 3 ) = 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. If you know the ratio of the ramp's height to its horizontal length is 3 3 , you can use the arctangent function to find the exact angle of the ramp, which in this case would be 6 π radians or 30 degrees. This ensures the ramp is neither too steep nor too shallow, providing an optimal experience for the skaters.
The exact value of tan − 1 ( 3 3 ) is 6 π . This indicates that the angle whose tangent equals 3 3 is 3 0 ∘ or 6 π radians. Thus, the answer is tan − 1 ( 3 3 ) = 6 π .
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