document.write( "Question 716976: find the exact value : arctan(tan(-5)) \n" ); document.write( "

Algebra.Com's Answer #440074 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
Since arctan and tan are inverses of each other, the temptation is to think that the answer would be -5. But it depends on whether the -5 is a number of degrees or a number of radians.

\n" ); document.write( "arctan only returns values between values between -90 and 90 degrees or between $\"-pi%2F2\"$ and $\"pi%2F2\"$ radians. So if the -5 is degrees then -5 is the answer because -5 degrees is between -90 and 90. But if the -5 is radians then the answer is not -5 because -5 radians is not between $\"-pi%2F2\"$ (approximately -1.57) and $\"pi%2F2\"$ (approximately 1.57) radians.

\n" ); document.write( "If the -5 is radians then what you must do is find an angle that is co-terminal with -5 radians that is between $\"-pi%2F2\"$ and $\"pi%2F2\"$ radians. (Co-terminal angles will always have the same tan's, sin's, etc.)

\n" ); document.write( "We can find this co-terminal angle by adding different multiples, possibly including negative multiples, of $\"2pi\"$ to -5 radians until we find one $\"-pi%2F2\"$ and $\"pi%2F2\"$ radians.

\n" ); document.write( "Since $\"2pi\"$ is approximately equal to 6.28, adding $\"2pi\"$ to -5 gets us a number approximately equal to 1.28 which is between $\"-pi%2F2\"$ and $\"pi%2F2\"$ radians.

\n" ); document.write( "Since we are asked for an exact answer we do not want to use the approximation of 1.27. We use $\"2pi+%2B+%28-5%29\"$ for the exact answer (if the -5 was a number of radians). \n" ); document.write( "

\n" );