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The Definitions of Sine and Cosine - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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