Select the correct choice below and, if necessary, fill in the answer box.
A. [tex]$\tan ^{-1}\left(\frac{\sqrt{3}}{3}\right)=$\square[/tex]
(Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed)
Answer (2)
Recognize that tan − 1 ( 3 3 ) asks for the angle whose tangent is 3 3 .
Simplify 3 3 to 3 1 .
Recall the common trigonometric value tan ( 6 π ) = 3 1 .
Conclude that tan − 1 ( 3 3 ) = 6 π , so the answer is 6 π .
Explanation
Understanding the problem We are asked to find the value of tan − 1 ( 3 3 ) . This means we need to find the angle whose tangent is 3 3 .
Simplifying the expression Recall that 3 3 = 3 1 . So, we are looking for an angle θ such that tan ( θ ) = 3 1 .
Recalling trigonometric values We know that tan ( 6 π ) = c o s ( 6 π ) s i n ( 6 π ) = 2 3 2 1 = 3 1 = 3 3 .
Finding the angle Therefore, 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 want the ramp to have a slope of 3 3 , you need to determine the angle of elevation. Using the arctangent function, you can find that the angle is 6 π radians, or 30 degrees. This ensures the ramp is neither too steep nor too gentle.
The value of tan − 1 ( 3 3 ) is 6 π , as this is the angle whose tangent equals 3 3 . Thus, the answer is 6 π .
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