The Problem with Inverse Sine
When using the law of sines to solve for angles of a triangle, it is common to run into the issue of not having all of your resulting angles add up to 180 degrees. Why is this? The issue comes with the implementation of inverse sine.
Inverse sine is used to determine which angle gives us a specific value of sine. The problem is that for any value of inverse sine there are two possible values that correspond to it. This can be seen in the following sine graph:
Inverse sine is used to determine which angle gives us a specific value of sine. The problem is that for any value of inverse sine there are two possible values that correspond to it. This can be seen in the following sine graph:
Both B and B' correspond to same value of sine. When typing values of inverse sine into a calculator, the calculator will always pick the value that is farthest to the left. With this graph it would be point B. This means that sometimes you will get angles of a triangle that add up to less than 180 degrees.
To fix this problem, check the other possible value for sine. With this graph it would be B'. You can do this by evaluating the difference between your angle value and 90 degrees, and then adding that difference to 90 degrees.
To fix this problem, check the other possible value for sine. With this graph it would be B'. You can do this by evaluating the difference between your angle value and 90 degrees, and then adding that difference to 90 degrees.
Using the unit circle, we can see that this works because the two possible values of sine are equidistant from the coordinate pi/2, which equals 90 degrees. So by evaluating the difference between our angle and 90 degrees we can obtain the other possible value that has the same sine.