Tangent identities: periodicity

Voiecover:One angle whose tangent is half is 0.46 radians. So we're saying that the tangent right over here is... So the tangent... So we're gonna write this down. So we're saying that the tangent of 0.46 radians is equal to half. And another way of thinking about the tangent of an angle is that's the slope of that angle's terminal ray. So it's the slope of this ray right over here. Yeah that makes sense that that slope is about half. Now what other angles have a tangent of 1 half? So let's look at these choices. So this is our original angle, 0.46 radians, plus pi over 2. If you think in degrees, pi is 180. pi over 2 is 90 degrees. So this one... Actually let me do in a color you're more likely to see. This one is gonna look like this. Where this is an angle of pi over 2. And just eyeballing it, you immediately see that the slope of this ray is very different than the slope of this ray right over here. In fact they look like they are. They are perpendicular because they have an angle of pi over 2 between them. But they're definitely not going to have the same tangent. They don't have the same slope. Let's think about pi minus 0.46. So that's essentially pi is going along the positive x axis. You go all the way around. Or half way around to your pi radians. But then we're gonna subtract 0.46. So it's gonna look something like this. It's gonna look something like that where this is 0.46 that we have subtracted. Another way to think about it, if we take our original terminal ray and we flip it over the y axis, we get to this terminal ray right over here. And you could immediately see that the slope of the terminal ray is not the same as the slope of this one, of our first one, of our original, in fact they look like the negatives of each other. So we can rule that one out as well. 0.46 plus pi or pi plus 0.46. So that's going to take us... If you add pi to this you're essentially going half way around the unit circle and you're getting to a point that is... Or you're forming a ray that is collinear with the original ray. So that's that angle right over here. So pi plus 0.46 is this entire angle right over there. And when you just look at this ray, you see its collinear is going to have the exact same slope as the terminal ray for the 0.46 radion. So just that tells you that the tangent is going to be the same. So I could check that there. And in previous videos when we explore the symmetries of the tangent function, we in fact saw that. That if you took an angle and you add pi to it, you're going to have the same tangent. And if you wanna dig a little bit deeper, I encourage you to look at that video on the symmetries of unit circle symmetries for the tangent function. So let's look at these other choices. 2 pi minus 0.46. So 2 pi... If this is 0 degrees, 2 pi gets you back to the positive x axis and then you're going to subtract 0.46. So that's going to be this angle right over here. And that looks like it has the negative slope of this original ray right up here. So these aren't going to have the same tangent. Now this one, you're taking 0.46 and you're adding 2 pi. So you're taking 0.46 and then you're adding 2 pi which essentially is just going around the unit circle once and you get to the exact same point. So you add 2 pi to any angle measure, you're going to not only have the same tangent value, you're gonna have the same sine value, cosine value because you're essentially going back around to the exact same angle when you add 2 pi. So this is definitely also going to be true.