Hey, everyone. We now know all of the trig values of our 3 common angles in quadrant 1. And we also know that all of our other angles have a reference angle equal to one of those common angles. Now all that's left to find is the trig values of all of these angles. And learning that many more new trig values does not sound like fun. But luckily, you don't have to do that at all because the trig values of all of these angles are the exact same as those of their reference angle with just one tiny difference that I'm going to walk you through right now. So let's go ahead and get started.
Now I mentioned one tiny difference, and that difference is the sine. So the sine, cosine, and tangent of each of these angles are the exact same value as that of the reference angle, but just with a different sign depending on what quadrant our angle is located in. So let's go ahead and just come right down here and look at our unit circle. So in quadrant 1, we know all of these trig values, the cosine, sine, and tangent of each of these angles. And we see that being in quadrant 1, these are all positive values. But what about as we move over to quadrant 2? Let's specifically take a look at our 150-degree angle here. Now, remember that this has a reference angle equal to 30 degrees because that's the angle it makes with the nearest x-axis. So with that reference angle of 30 degrees, we can further visualize this by seeing that they form the exact same triangle just flipped. So it makes sense that the base and height of these triangles would be the exact same, which are cosine and sine values.
So we can take the cosine, sine, and tangent here, our trig values, and just copy them over here because they're the exact same value. Now doing that, I end up with 3/2 for that cosine or that x value, one-half for that y value, that sine, and 3/3 for tangent. Now the only thing that remains to consider here is the sign of each of these values because we're no longer located in quadrant 1. Now looking at our relationship to the x and y axes, I see that my x values over here should be negative, whereas my y values remain positive. So I want to reflect that in these trig values. So my x value is negative root 3 over 2. My y value is positive. Now when looking at the tangent, we just consider how we actually find the tangent. Right? So the tangent is the sine over the cosine or y over x. So knowing that our y value is positive and our x value is negative, if I were to divide a positive value by a negative value, that would give me a negative value. So here my tangent should also be negative.
Now looking at these values altogether, only one of them is positive. And this will be true for all of our trig values in quadrant 2. Only the sine value, our y value here, is going to be positive for all trig values in quadrant 2. Now let's move on to quadrant 3 and specifically 225 degrees. Now I know that the reference angle of 225 degrees is 45 degrees in quadrant 1, So I can take my trig values here and simply copy them over here for the trig values of 225 degrees. So here I can copy these in with the cosine and sine both being root 2 over 2, my x and y values, and then the tangent as 1. Now all that we need to consider here is remember the sign. So based on where we are in the x and y axis, we see that these are both these both should be negative values. My x and my y value are both negative. Now thinking about the tangent, in order to find the tangent, I would be dividing a negative by a negative, which would actually give me a positive-positive value. So my tangent here in quadrant 3 remains positive. Now looking at these values altogether, the only one that's positive is tangent, which will, of course, be true for all of my trig values in that quadrant 3. So here in quadrant 3, the tangent will always be positive, and it's the only positive one.
Now finally, let's take a look over here at quadrant 4. Now in quadrant 4, we're gonna look specifically at 300 degrees. Now we know that 300 degrees has a reference angle of 60, so I can go ahead and take my 60-degree trig values and copy them right down here for the trig values of 300 degrees. Now doing that, my cosine and my sine, I get one-half and root 3 over 2, literally just copying those values, then for the tangent, root 3. Now we need to consider, of course, the sign. Now the sign here, looking in this quadrant, quadrant 4, we know that we are in the negative y-values and positive x-values, so we need to reflect that on these trig values. So negative y-values, this negative root 3 over 2. And then for our tangent, we need to consider what we're doing. Of course, here we would be dividing a negative by a positive, which would give me a negative value. So our tangent here will also be negative. Now finally, looking at all 3 of these trig values as a whole, the only one that's positive is this cosine, this x-value, one-half. So here in quadrant 4, the cosine will always be positive.
Okay. We've looked at all of our quadrants here and we've seen the sign difference in each of them. All of our trig values are the same, just with a little variation in sign depending on our location on the unit circle. Now how can we remember this? We can remember this using a mnemonic device first letter of each function that's positive. So in quadrant 1, all of them are positive. Quadrant 2, the sine is positive. Quadrant 3, the tangent. And quadrant 4, the cosine, AST sign, ASTC. All students take calculus. Now this might not always be a true statement, but it's one that can help us remember which trig function is positive in our unit circle. So now we have all of the information we need in order to completely fill in an entirely blank unit circle. So let's continue getting some more practice together. Thanks for watching, and I'll see you in the next one.