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Chapter 7: Q. 11 (page 503)

In Problems 9鈥�23, find the exact value, if any, of each composite function. If there is no value, say it is 鈥渘ot defined.鈥� Do not use a calculator.

tan-1tan2蟺3

Short Answer

Expert verified

The exact value is-蟺3.

Step by step solution

01

Step 1. Given Information

We are given a composite function tan-1tan2蟺3.

We need to find its exact value.

02

Step 2. Concept Used

The tangent function and its inverse, form these properties

tan-1tanx=x where -蟺2<x<蟺2

and

tantan-1x=xwhere-鈭�<x<鈭�

03

Step 3. Find the exact value

The angle 2蟺3is not in the interval -蟺2,蟺2, so we cannot use the property tan-1tanx=x.

Angle 2蟺3is in the second quadrant so its tangent value is negative, and thus the corresponding reference angle is 2蟺3-蟺=-蟺3. So we get

tan2蟺3=tan-蟺3

Now, -蟺3lies in the interval -蟺2,蟺2. So using the property formed by tangent and its inverse function we have

tan-1tan2蟺3=tan-1tan-蟺3=-蟺3

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