A
lthough the formula is correct, it might not give you the right answer when you enter this into your calculator; it is important to remember that this formula comes from the angle that a point (x=b, y=a) makes with the origin. This leads to the above formula and the equivalent formulas
and
and the quadrant
is in is determined by the sine (+ or -) of a and b. The actual value of might be different from what your calculator tells you if is not in the quadrant your calculator is expecting it to be in. For example, if your a and b were negative your would be based on a point in the third quadrant and would therefore be between pi and 3pi/2 (in degrees, between 180 and 270). However, if you used the tangent formula and entered a negative a divided by a negative b the calculator would read this as a positive and number and give an answer in the 1s quadrant (between zero and pi/2, in degrees between 0 and 90). If you used the sin function your answer would be in the fourth quadrant, probably written as a negative value (between zero and -pi/2, in degrees between 0 and -90). Using the cos formula would give an answer in the 2nd quadrant (between pi/2 and pi, in degrees between 90 and 180). However, the actual value of is none of these. Therefore, the formula given in the previous would be more correctly written as where n=1, 2, or 3 and is determined by the quadrant that the point (b,a) lies in.
Basically, when solving a problem like the one presented, chose one of the three formulas to find a value for using your calculator, and check to make sure this answer is in the same quadrant as (b,a). If it isn't, add ether pi/2, pi, or 3pi/2 to your answer so that it is in the correct quadrant.