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The Definitions of Sine and Cosine - Problem 4

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.

Hi, I’m talking about the unit circle definitions of sine and cosine and I want to do an exercise that will sort of help us get and idea for the relative size of sines and cosines of different values.

If you take a look at the problem it says fill in with less than, greater than or equal to and in parts a and b I’m dealing with 2 angles; 70 degrees and 60 degrees. In order to analyze these two I’m going to want to draw the two angles in standard position on the unit circle. Just to save time, I’m not going to draw the initial side. Usually the initial side goes across on positive x axis, I’m just going to draw the terminal side.

First, 60 degrees, I’m just approximating and then 70 degrees, about 10 degrees more. Now remember sine and cosine are defined by the coordinates of the point where the terminal side crosses the unit circle. For example, this point, the cosine of 60 degrees would be the x coordinate and the sine of 60 degrees would be the y coordinate. For this point, the cosine of 70 degrees would be the x coordinate and the sine of 70 degrees would be the y coordinate.

If you look at the relative position of these two points the x coordinate of these points to the left of this point, so this x coordinate is smaller. That means that the cosine of 70 degrees is smaller than the cosine of 60 degrees. This point’s higher than this point so the sine value here will be bigger than the sine value here. The sine value of 70 is bigger than the sine value of 60.

Let’s compare 165 degrees and 15 degrees. So going back over here, let me draw 15 degrees first. Now just to get an idea of what 15 degrees might be. If this is 90 degrees and you divided into thirds, each third would be 30 degrees and half of these could be, 15 so that’s about 15 degrees. And 165 degrees is 180 degrees minus 15 so I go all the way through 180 degrees and subtract 15. So this is 15 degrees, this is 165. And that means that what’s left over is also 15 degrees, so we have symmetry here and that means that when you compare the two, the coordinates of these points, the y coordinates are going to be the same, the x coordinates are going to be opposites.

So going back to our problem here, I’ll do the cosines first. Cosine of 165 was on the negative side so cosine of 165 is less than cosine of 15. The y values were exactly the same. So the sine of 165 is exactly equal to the sine of 15.

Pi over 9 and negative pi over 9. Let’s take a look at the unit circle. Pi over 9 is 20 degrees, I’ll just approximate that, negative pi over 9 is going to be the opposite of this, so just a reflection across the x axis. If you’re reflecting this terminal side across the x axis, you’re also reflecting this point across the x axis which means the coordinates of this point become (x,-y). This coordinate is going to be positive here and negative here. And so when we compare the cosines of the two angles they’ll be the same. But the sines will be opposite. The sine of pi over 9 is going to be the bigger one.

Let’s go back here the cosines are the same, the sines are opposite and remember the sine of pi over 9 was the positive one, so this is greater.

Finally we’re comparing 5 pi over 4 with -3 pi over 4. 5 pi over 4. Pi over 4 is 45 degrees so 5 pi over 4 is pi over 4 more than pi. 180 degrees plus 45 degrees. So here is 180 and then 45 more. So this angle here is 5 pi over 4.-3 pi over 4 is -135, it’s actually this angle’s the two angles arrive at the same terminal side so they’re called co-terminal angles. But since they’re co-terminal, they have exactly the same x and y coordinates at this point and that means that they have the same cosine and sine values exactly. So the cosines are the same and the sines are the same.

Co-terminal angles come in really handy when the you know some values of sine and cosine within the first zero to 2 pi you could just add 2 pi and find the sine and cosine values in the next period.

This is just an exercise that allows you to see the relative values of how they compare at different angles but remember that when you’re evaluating sine and cosine make use of the symmetries that you see in the unit circle. Symmetries across the x axis or the y axis and also co-terminal angles.

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