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# Tangent identities: periodicity

CCSS.Math:

## Video transcript

what angle whose tangent is is half is 0.46 radians so we're saying the D the tangent right over here is so the tangent so we're let me write this down so we're saying that the tangent 0.46 radians is equal to 1/2 and another way of thinking about the tangent of an angle is that's the slope of that angles terminal ray so it's the slope of this ray right over here then yeah that's that make sense that that slope is about 1/2 now what other angles have a have a tangent of 1/2 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 this one actually let me do it in a color you're more likely to see this one is going to look like this it's going to look like this where this is an angle of PI over 2 and just eyeballing 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 in fact 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 so they're not going to have the same Tanit they don't have the same slope now it's think about PI minus 0.46 so that's essentially pi is going along the positive x axis if you go all the way around or halfway around you're at PI radians then we're going to subtract 0.46 so it's going to look it's going to look something like this it's going to 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 array 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 our original in fact they look like they look like the negatives of each other so we could rule that one out as well 0.46 plus PI or PI plus 0.46 so that's going to take us that's going to take a few if you add PI to this you're essentially going halfway around the unit circle and you're getting to a point that is or you're forming array that is collinear with the original array 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 it's collinear it's going to have the exact same slope as the terminal ray for 0.46 radians 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 want to 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 - pi - is 0.46 so 2 pi if this if this is 0 degrees - pi gets you back to 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 not the same tangent now this one you're taking 0.46 and you're adding two pi so you're taking 0.46 0.46 and then you're adding two 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 tan Val tangent value going to 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 this is definitely also going to be true