Find the exact value of inverse tan of tan of four-thirds 𝜋 in radians in terms of 𝜋, which is located in the first quadrant.
There are two ways to solve this problem. The first is to evaluate tan of four-thirds 𝜋 and then find the inverse of that number. However, by considering what the inverse really means, we should be able to save ourselves some time.
Remember the inverse is the opposite of a function. The inverse of multiplying is dividing, the inverse of adding is subtracting, and the inverse of the tangent function is inverse tan.
For any trigonometric function, therefore, the inverse function of the function in terms of 𝜃 is simply 𝜃. In this case then, the inverse tan of tan of four-thirds 𝜋 is just four-thirds 𝜋.
However, the question wants the solution which is located in the first quadrant. Currently, four-thirds 𝜋 is located in the third quadrant. Remember the tan function is periodic with a period of 𝜋. This means it repeats every 𝜋 radians or 180 degrees.
To find the solution which lies in the first quadrant then, we’ll subtract 𝜋 from four-thirds 𝜋. The exact value which is located in the first quadrant is, therefore, a third 𝜋.