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Evaluating Sine and Cosine at Special Acute Angles - Problem 3

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.

Pi over 3 is another special angle. I want to find the cosine and sine of Pi over 3. Now I’ve drawn the angle Pi over 3 on the unit circle point P is the point on the terminal side of Pi over 3 and I need to find the coordinates of Pi over3 in order to find the sine and cosine of the angle.

But I already know the coordinates of point Q which lies on the angle Pi over 6. The sine of Pi over 6 is ½ and the cosine of Pi over 6 is root 3 over 2 and that gives me the x and y coordinates. This is the x coordinate., this is the y coordinate of point Q.

Point Q is a refection of point P along the line y equals, y equals x. you can kind of see that because Pi over 3 is 60 degrees, Pi over 6 is 30 degrees. This angle is 30 degrees so there is symmetry across the diagonal. That means that to get the coordinates of point P all I need to do is interchange the coordinates of point Q. So point P has coordinates ½ root 3 over 2 and that means the cosine of Pi over 3 is ½ and the sine of 1/3 is root 3 over 2.

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