Let’s do an example that uses the definitions of sine and cosine, unit circle definitions. The problem says if theta is in quadrant 4 and cosine theta is 5/13, find sine theta.
The first thing I do is I draw a diagram showing theta in quadrant 4 and the cosine, remember corresponds with the x coordinate of point P is 5/13. The first thing you want to remember is the unit circle has equation x² plus y² equals 1. That’s true for every point on the unit circle including this one. When we apply it to this point we get that (5/13)² plus y² equals 1.
Now (5/13)² is 25 over 169. This equals 1. If I’m going to subtract 25 over 169 from sides, I’m going to want 1 in the form of a fraction with denominator of 169 so that’s 169/169. I want to subtract 25/169 from both sides, I get something over 169. 169 minus 25 which is 144 and I just have to take the square root. I get plus or minus the square root of 144 is 12, square root of 169 is 13. Y is going to be plus or minus 12/13.
Now when I look at my diagram here in quadrant 4, the y values are below the x axis that means that the y value has to be negative. I choose the negative value, y equals -12/13. And by the unit circle definition of sine, that means sine theta is -12/13.