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### Course: Precalculus > Unit 2

Lesson 2: Trigonometric identities on the unit circle# Tangent identities: periodicity

Sal solves a problem by considering the periodicity of the tangent function. Created by Sal Khan.

## Want to join the conversation?

- So this whole time, we've been describing the slopes of the rays of these points on the unit circle with tan(x)? So the slope of the ray through 60 degrees is rad3/2/.5?(16 votes)
- Or simplified further to just sqrt(3)...this is the article for the sides of special triangles for those who might be searching... https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles/trig-ratios-special-triangles/a/trig-ratios-of-special-triangles(3 votes)

- what exactly is periodicity?(5 votes)
- Periodicity is the quality of being periodic; that is, the tendency to recur at intervals.(10 votes)

- Since tan ø = opp/adj = y/x and this is a unit circle I get how the numerator in tan 0.46 = 1/2 can be 1 but how can the denominator be 2?(3 votes)
- Think of it this way.

The tangent is a ratio of 2 sides of the triangle ( opposite / adjacent ).

The 1 / 2 represents that ratio ,so, instead of representing actual lengths of the sides, it just indicates that the adjacent side is twice the length of the opposite side.

So, for instance, the adjacent side could be .6 long and the opposite side then would be .3.(8 votes)

- is the terminal ray is the tangent of the subtended angle?(3 votes)
- No, Sal's not talking about "tangent" as being a line (or ray), but about the trigonometric function (tan), which can be viewed as the slope of the terminal ray.

There is, however, a reason to why the tangent function is called tangent.

A ray, 𝑅, that emanates from the origin and forms the angle 𝜃 (0 < 𝜃 < 𝜋/2) with the 𝑥-axis intersects the unit circle in point 𝑇.

The straight line that touches the unit circle in 𝑇 is a tangent of the unit circle.

This line intersects the 𝑥-axis in a point 𝑋.

It is also perpendicular to 𝑅, which means that the length of the line segment between 𝑇 and 𝑋 will be tan 𝜃.(9 votes)

- Why is the tan of pi + 0.46 also a positive slope? If you start at the origin, isn't it then going downward since the arrow is going from right to left in a downward direction?(4 votes)
- Yes but as you cross the pi angle, you enter into the 3rd quadrant, where the tangent of an angle is always +ve, which you can understand by :

Recollecting that sine and cosine of an angle will always be of the same sign (-ve) in the third quadrant. or the first quadrant (+ve), or

Imagining it as such that the line whose angle with the positive x-axis is a reflex angle, in counterclockwise sense, will come back to the first and third quadrant, similar to the initial position, assuming you slowly turn the line about the origin in a counterclockwise fashion, hence making the tangent of the angle effectively same. Therefore

tan(pi+0.46)=tan(0.46)(6 votes)

- How do you do this without drawing a circle or without eyeballing it?(3 votes)
- By definition, an angle with the same slope is going to have the same tangent value. So, you don't have to visualize it on a unit circle in order to determine if two angles have the same tangent value.(6 votes)

- What is the differences in (2 pi + theta) and ( 2 pi - theta) on the unit circle? Thanks in advance.(2 votes)
- So they are both different angles assuming theta != 0. So they will just be two different points on the unit circle.(5 votes)

- my videos got stuck and wont let me finish or count the points for it. what should I do?(4 votes)
- You should go to the Help Center (a link is at the bottom of all KA screens).(3 votes)

- The tangent of angle pi/2+0.46 is 2/1 correct?

an answer would be great I feel like my entire understanding of this concept is balancing on the factuality of this statement.(4 votes)- You have the correct magnitude, but the sign is opposite.

When an angle crosses the y axis, x changes from + to -, so the tangent y/x does also, which makes tan(pi/2+.46) = -2. When the angle crosses the x axis, y changes sign, so the tangent changes sign every time it goes from one quadrant to the next.(2 votes)

- At2:58Sal mentions that with any degree, you can add pi and get the same tan 'function'. Will that work with radians since radians are another way to measure degrees?(3 votes)
- Sal says that if you have an
*angle*and add π to it, you get another angle with the same tangent. He is already talking about radians, since the problem at hand is in radians.

If you were measuring angles in degrees, then adding 180º to any angle would give you another angle with the same tangent.(4 votes)

## Video transcript

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.