Once again, in tabular form, to solve:tan θ=aIfaisandθis inthenθequalspositivefirst quadrantarctananegativesecond quadrant(arctana) +πpositivethird quadrant(arctana) +πnegativefourth quadrantarctanaor(arctana) + 2πThus, it appears that one can not uniquely solve tanθ=aunless it is known which quadrantθis in.This is correct.Fortunately, in applications when the equation needs to be solved,information will always be available to determine which possible angle is correct.How should one remember these results? Rather simple: ifθis in the second or third quadrants,just addπradians to the computer’s evaluation of arctana.While this alone is sufficient to get a correct answer, there are two simplifications sometimesused.First of all, the function arctan is anoddfunction, i.e., arctan(-b) =-arctan(b) for anyreal numberb. This comes into play especially in the second and fourth quadrants, where thetangent is negative. So, for example, if tanθ=-1.5, thenθcan be written asθ=-arctan(1.5)if it is in the fourth quadrant, orθ=π-arctan(1.5) if is is in the second quadrant. Noticethat we have followed exactly the rules above, except we have pulled the minus sign out of thearctangent function.The second simplification refers only to fourth quadrant angles. Although the computer willgive fourth quadrant angles as negative angles when evaluating the arctangent function, insome applications it will be desirable to haveθalways between 0 and 2πradians. In this case,only for fourth quadrant angles, one will need to add 2πto obtain the same angle writtenas a positive angle.If you have understood all of this, you will realize that the solution oftanθ=-1.2, whenθis known to be in the fourth quadrant, can be written three ways: (i)θ= arctan(-1.2); (ii)θ=-arctan(1.2) , which give exactly the sameθas the first way; (iii)θ= 2π+ arctan(-1.2) = 2π-arctan(1.2) , which givesθas a positive angle.IdentitiesThe three most important trig identities (and, we believe, the ones every scientist and engineershould know) are the following.(1)sin2x+ cos2x= 1 (True for anyx; for example, sin2et+ cos2et= 1)