Originally Posted by

**theanonomous** Although the formula $\displaystyle \phi = sin^{-1} \left ( \frac{b}{\sqrt{a^2 + b^2}} \right )$ 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 $\displaystyle tan^{\phi}=\frac{b}{a} $ and $\displaystyle

\phi= cos^{-1} \left ( \frac{a}{\sqrt{a^2 + b^2}} \right )

$ and the quadrant $\displaystyle \phi $ is in is determined by the sine (+ or -) of a and b. The actual value of $\displaystyle \phi $ might be different from what your calculator tells you if $\displaystyle \phi $ is not in the quadrant your calculator is expecting it to be in. For example, if your a and b were negative your $\displaystyle \phi $ 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 $\displaystyle \phi $ is none of these. Therefore, the formula given in the previous would be more correctly written as $\displaystyle \phi = sin^{-1} \left ( \frac{b}{\sqrt{a^2 + b^2}} \right )+ \frac{n\pi}{2}$ 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 $\displaystyle \phi $ 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.