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The Definitions of Sine and Cosine - Problem 3
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We’re talking about the unit circle definitions of sine and cosine and you may have noticed that now that we can define the sine and cosine of any angle it’s possible for sine and cosine to be positive or negative. I want to take a look at a demonstration that will allow us to figure out exactly when sine and cosine are positive and negative. Let’s take a look at geometry sketch pad.
I have the unit circle depicted here and what you see is the angle drawn in standard position, there is the vertex at the origin, there is the initial side of the angle on the positive x axis, and here’s the terminal side of the angle. The terminal side intersects the unit circle at point P whose coordinates are currently (0.5, 0.86). Remember the first coordinate give me the cosine of my angle, the second coordinate gives me the sine of my angle. And the angle is currently 1.043.
Well I can rotate the angle around. You can see how the cosine and sine values are going to change. Right now cosine’s negative and sine’s still positive. Now they’re both negative and now sine’s negative but cosine’s positive.
Let’s just recap. Where's sine positive? Since sine is the second coordinate on point P, it’s going to be positive whenever that point is above the x axis. That means quadrants 1 and 2. Those are the 2 quadrants that are above the x axis. Where is cosine positive? Cosine is positive to the right of the y axis. That’s where the first coordinate is going to be positive. Remember the first coordinate is cosine. So remember cosine if positive to the right of the y axis, quadrants 1 and 4. Sine is positive above the x axis, quadrants 1 and 2.
Let’s use what we’ve just learnt in an example. I have a problem. It says; for each angle, determines which quadrant its terminal side lies in. Are cosine and sine positive or negative?
We start with theta equals 3 pi over 4 and I want to get a sense for where that is. Remember I’m going to draw my angle in standard position, that means the initial side has to be to the positive x axis and the terminal side has got to cross the unit circle somewhere around this perimeter. If theta equals 3 pi over 4, that’s 3 times pi over 4. Pi over 4 is 45 degrees. 3 times 45 degrees is 135. It's going to be about here. Now all I want to finds out, I don’t want to find the values yet, we’ll do that later but all I want to find out is are cosine and sine positive or negative?
In this quadrant the x coordinate is going to be negative and the y coordinate is going to be positive, so cosine of 3 pi over 4 is negative and sine of 3 pi over 4 is positive. What about theta equals 11 pi over 6?
Well again I will start by drawing my initial side, 11 pi over 6, 2 pi is one complete revolution, one complete revolution around the circle. 11 pi over 6 is pi over 6 short of 2 pi so if I rotate all the way around but stop pi over 6 away which is about 30 degrees, this is what my angle is going to look like. What are the coordinates of this point on the terminal side? Or rather are the y negative or positive? Well the x coordinate is on the right side of the y axis so it’s going to be positive. That means cosine of 11 pi over 6 is positive. The y coordinate is below the x axis so it’s going to be negative. Sine of 11 pi over 6 is negative.
Remember with angles you can have positive or negative angles. Positive angles are always counter-clockwise, negative angles are clockwise. So we start with the initial side, now 2 pi over 3. Pi over 3 is 60 degrees so 2 pi over 3 is going to be 120 degrees and if I go clockwise 120 degrees I end up about there. So this is my theta.
This point will define my cosine and sine values. The x coordinate here is in the third quadrant is going to be negative. So cosine of -2 pi over 3 is negative and the y coordinate is also negative. Sine -2 pi over 3 is negative.
Finally theta equals 3, notice I don’t have a pi here. Let me start with my initial side, remember what pi is. approximately 3.14. 3 is just a little bit short of that and pi radians is the same angle as 180 degrees so this is going to be just a little short of 180 degrees. It is just a little short of a straight angle, something like this. So that puts me in the second quadrant, here is my point (x,y). The cosine of theta is going to be the x coordinate and that’s going to be negative, I’m sorry 3 and the sine of 3, it’s a little bit above the y axis so y is going to be positive. The sine of 3 is positive.
It’s important to know whether sine and cosine values are positive or negative when you’re evaluating them as we’ll see later on. So this is an important skill, knowing whether sine or cosine are positive or negative depending on the quadrant that the terminal side lies in is an important skill.
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