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Evaluating Sine and Cosine at Other Special Angles - Concept

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.

To find the value of sine and cosine at non-acute angles (from 90 to 360), first draw the angle on the unit circle and find the reference angle. A reference angle is formed by the terminal side and the x-axis and will therefore always be acute. When evaluating cosine and sine for the reference angle, determine if each value is positive or negative by identifying the quadrant the terminal side is in.

Okay I want to talk about how to find the sine and cosine of angles that aren't acute, so I'm going to start with an example finding the cosine and sine when theta equals 150 degrees. And I've drawn theta here in standard position so I need to find the coordinates of point q remember x is going to be the cosine of 150 degrees and y is the sine. And the trick to this is finding the reference angle. The reference angle is the angle made by the terminal side of your angle and the x axis and in this case it's 30 degrees and then you need to find the cosine and sine of 30 degrees which by now we found and the cosine of 30 degrees root 3 over 2, sine of 30 degrees is one half.
Now that does this have to do with the cosine and sine of 150 degrees? Well if you were to draw the angle 30 degrees it would be right here, this point would be the mirror image of point p across the y axis. Let's call it point q and because we know the cosine and sine values it's coordinates would be root 3 over 2 one half. And that means the coordinates of this point by symmetry across the y axis would be negative root 3 over 2 one half and that gives us the cosine and sine of 150. So how do we do this in general? Well find the reference angle, identify the cosine and sine. And the cosine and sine of the angles you're interested in are going to be plus or minus these values. So we can just write equals some space root 3 over 2 equals some space and one half, and just judge whether it is positive or negative based on the quadrate.
Remember in the second quadrate x coordinates are going to be negative, y coordinates are going to be positive. So here cosine value is negative and your sine value is positive and that's it. Reference angles and quadrant that's how to find the cosine and sine of an angle that's bigger than 90 degrees or smaller than 0.

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