SOLUTION: Find both real numbers x between 0 and 2pi radians, where tan(x) = -2.5. Put the smaller answer in the first answerbox, please, and round your answers to three decimal places.

Rules for doing all such problems asking for answers between 0 and  or
between 0° and 360°:

1. Find the reference angle by finding the inverse trig function of the
   ABSOLUTE VALUE of the given value, using a calculator.
2. Decide whether the answers are in the 1st, 2nd, 3rd, or 4th quadrants
   by:

sine and cosecant are positive in 1st and 2nd, negative in 3rd and 4th.
cosine and secant are positive in 1st and 4th, negative in 2nd and 3rd.
tangent and cotangent are positive in 1st and 3rd, negative in 2nd and 4th.

3. To find the answer in 1st quadrant, use the reference angle.
   To find the answer in 2nd quadrant, subtract the reference angle from ,
   or 180° if in degrees.
   To find the answer in 3rd quadrant, add the reference angle to , or to
   180° if in degrees.
   To find the answer in 4th quadrant, subtract the reference angle from ,
   or 360° if in degrees. 

I'll do it for tan(x) = -2.1, to force you to have to do it for 
tan(x) = -2.5 by yourself.  [Some students come to this site not to learn, 
but just so they can type in the correct answers for their homework and
get false credit.]

Make sure your calculator is in radian mode.

Now we find the inverse tangent of +2.1 on a calculator which will 
give us the reference angle.  

Reference angle = 1.126377117. 

tan(x) = -2.1, which is a negative number.  We know that the tangent 
is negative in the 2nd and 4th quadrants.

To get the 2nd quadrant answer, we subtract the reference angle from
:

3.1415926543 - 1.126377117 = 2.015215537 or to three decimals, 2.015.

To get the 4th quadrant answer, we subtract the reference angle from
:

2(3.1415926543) - 1.126377117 = 6.283185307 - 1.126377117 = 5.15680819,
or to three decimals, 5.157.

Answera: 2.015 and 5.157 

Now follow the same procedure to get your answers.

Edwin