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### Course: Get ready for AP® Calculus > Unit 1

Lesson 17: Trigonometric identities on the unit circle# Tangent identities: symmetry

Sal finds several trigonometric identities for tangent by considering horizontal and vertical symmetries of the unit circle. Created by Sal Khan.

## Want to join the conversation?

- How does the pi in tan(theta + pi) get cancelled out?

because tan(theta + pi) = tan(theta)...it doesn't make sense!(28 votes)- Thinking about it as 'cancelling out' is the wrong way to look at it because, as you rightly say, you can't do that!

Instead, think that the tangent of an angle in the unit circle is the slope.

If you pick a point on the circle then the slope will be its y coordinate over its x coordinate, i.e. y/x. So at point (1, 0) at 0° then the tan = y/x = 0/1 = 0.

At 45° or pi/4, we are at an x, y of (√2/2, √2/2) and y / x for those weird numbers is 1 so tan 45° is 1.

Now you should be able to see that slope of the any ray taken from the centre point is going to be the same as another ray with which it forms a straight line, i.e. like the thick green line at2:17.

Those two rays are always pi apart. Therefore tan(x) = tan(x+pi*n) for any value of n because the rays of all of the lines will have the same slope.

And that means we can delete the pi in tan(theta + pi) knowing that we're right.

Does that make sense?(63 votes)

- Wouldn't it be possible to figure out all of this without the algebra? So what is the value of doing all of this work?(8 votes)
- The main reason for learning to do all of these proofs is to teach you how to think logically, to go step-by-step without making unfounded assumptions, to reach a conclusion you can prove. While you are unlikely to need to prove which triangles are similar in daily life, the ability to think through problems logically is extremely important. So, geometry class is an excellent tool for teaching you logic.(73 votes)

- Do I have to memorize all of these formulas? Is it important enough to know?(4 votes)
- Hello Chris,

I have a confession - most us us don't remember all of the formulas. Perhaps that is why the textbooks always put the trig identities on the back cover.

With that said, please do you level best to understand how these formula are derived and used. The work that you do now will be very useful as you continue through high school and move on to college.

Regards,

APD(38 votes)

- 2:20

How can something plus something equal the original something? pi /= 0. 0*should*be the only exception to this rule. Can someone explain?(7 votes)- He wrote that in a somewhat confusing way, here is what it means:

tan (θ) = tan (θ+π) ←**This is correct**

It is not:

tan (θ) = tan(θ) + π ←**This is wrong**(31 votes)

- am i suppose to memorize this(9 votes)
- Novices memorize, experts reconstruct.(22 votes)

- so is it reasonable to say that tangent will always be positive in the first and third quadrents , while tangent will always be negative in the second and fourth quadrents?(8 votes)
- Yes, the tangent is always
**positive**in the 1st and 3rd quadrant while**negative**in the 2nd and 4th quadrant.(14 votes)

- How does unit circle relate to finding out the angles of right triangles?(4 votes)
- if you know the x value and the y value, or one of them, you can use inverse trig functions.. the radius of the unit cirlce(hypothenuse) is 1, if your x value is 1/2 you have: sin(θ)=0.5/1=0.5, then arcsin(0.5) gives you the angle(sin(30°)=1/2, arcsin(1/2)=30°), when you have both x and y values you can use inverse tangent: tan(θ)=y/x, arctan(y/x)=θ, but as the video shows, keep in mind you can have the same result from trig functions with different angles, for example: cos(θ)=cos(-θ)(10 votes)

- As there are so many identities is there any quick method to memorize them or do we then have to memorize each and every by heart or know how to derive them?(4 votes)
- Well I don't know if you need to memorize all of them but (as far as I know) there is no device you can use remember them all. What I did when trying to remember these is to write them all down on one sheet of paper, not just the identities but all the really important stuff I tended to forget. When I did homework and studied I used the list like it was my bible. That's how I did it.(8 votes)

- Sometimes! I get confused by the wheel or the circle and the words?

so can someone just help me.(3 votes)- If you want any specific help, you'll need a more specific question, but my main suggestion is, if you're just having trouble understanding the unit circle, then watch the basic videos on it again. And again and again and again. And maybe search on YouTube for introduction videos on the unit circle, because maybe someone else will explain it in a way that makes more sense to you. Don't go past basic videos until you have a solid understanding.(10 votes)

- How are we able to find the tangent identity for pi + theta if it no longer forms a triangle?(5 votes)
- from what i understood, the tan is considered slope and you dont have to form a triangle to find the slope of a line you just need the rise (the length between the point and 0 on the y axis) over run (the length between the point and 0 on the x axis) where the rise is sin(pi+theta) over run which is cos(pi+theta) = -sin(theta) over -cos(theta) = tan(theta)(4 votes)

## Video transcript

Voiceover:The previous video we explored how the cosine and sines of angles relate. We essentially take the
terminal ray of the angle and we reflect it about the X
or the Y axis, or both axes. What I want to do in this
video is think a little bit about the tangent of
these different angles. So just as a little bit of a reminder, we know that the tangent
of theta is equal to the sine of an angle over
the cosine of an angle, and by the Unit Circle Definition, it's essentially saying,
"What is the slope "of the terminal ray right over here?" We remind ourselves
slope is rise over run. It is our change in the vertical axis over our change in the horizontal axis. If we're starting at the
origin, what is our change in the vertical axis if we
go from zero to sine theta? Well, our change in the
vertical axis is sine theta. What is our change in the horizontal axis? It's cosine of theta. So this is change in Y over change in X for the terminal ray. So the tangent of theta
is the sine of theta over cosine of theta, or you could view it as the slope of this ray right over here. Lets think about what other angles are going to have the exact
same tangent of theta? This ray is collinear with
this ray right over here. In fact if you put them
together you get a line. So the tangent of this
angle right over here, this pink angle going all the way around, the tangent of pi plus theta, or the tangent of theta plus pi, obviously you could write theta plus pi instead of pi plus theta. This should be, just based
on this slope argument, this should be equal to
the tangent of theta. Lets see if this actually is the case. So these two things should be equal if we agree that the
tangent of an angle is equal to the slope of the terminal ray. Of course the other side of the angle is going to be the positive X axis based on the conventions that we've set up. Lets think about what it
is when the tangent of theta plus pi is in
terms of sine and cosine. Let me write this down in the pink color. The tangent. That's not pink. The tangent of pi plus theta,
that's going to be equal to, put the parentheses to avoid ambiguity, that's equal to the sine of pi plus theta, or theta plus pi, over the
cosine of theta plus pi. And in the previous video we established that the sine of theta plus pi, that's the same thing
as negative sine theta. So this is equal to negative sine theta. And the cosine of theta plus pi, we already established that's the same thing as negative cosine of theta. We have a negative divided by a negative that you could say the
negatives cancel out, and we're left with sine
theta over cosine theta, which is indeed tangent of theta, so we can feel pretty good about that. Now what about the points, or the terminal rays right over here? Lets think about this point. What is the tangent of
negative theta going to be? We know that the tangent of negative theta is the same thing as the
sine of negative theta over the cosine of negative theta, and we already established
the sine of negative theta, that's negative sine theta. We see that right over here, sine of negative theta. That's the negative, that's the opposite, of the sine of theta,
so we have that there, but the cosine of negative theta is the same thing as the cosine of theta, so these things are the same. So we're left with
negative sine theta over cosine of theta, which is the same thing, equal to, negative tangent theta. So we see here if you take
the negative of the angle, you're going to get the
negative of the tangent, and that's because the
sine, the numerator, in our definition of tangent, changes signs, but the
denominator does not. So the tangent of negative theta is the same thing as
negative tangent of theta. Now, what about this
point right over here? Well over here, relative to theta, when we're looking at pi minus theta, so when we're looking at
tangent of pi minus theta, that's sine of pi minus theta
over cosine of pi minus theta. and we already established
in the previous video, that sine of pi minus theta
is equal to sine of theta, and we see that right over here, they have the exact same sines, so this is equal to sine of theta, while cosine of pi minus theta, well, it's the opposite
of cosine of theta, it's the negative of cosine of theta. And so this once again is going to be equal to the
negative sine over cosine, or the negative tangent of
theta, which makes sense. This ray should have the same slope as this ray right over here. And we see that slope, we could view this as negative tangent of theta. And we see just looking at these two, if you combine the rays, that these two intersecting lines have the negative slope of each other, they're mirror
images across the X axis. And so, we've just seen,
if you take an angle and you add pi to the
angle, you're tangent won't change because you're going to essentially be sitting on the same line. pi, everything in degrees,
you're going 180 degrees around. You're going the opposite direction but the slope of your ray has not changed. So tangent of theta is the same thing as the tangent of theta plus pi, but if you take the
negative of your angle, then you're going to get
the negative of your tangent or, if you were to go
over here, and if you were to take pi minus your
angle, then you're also going to get the negative of your tangent. Hopefully this makes you a little bit, this is very useful when you're trying to work though trigonometric problems or try to find relationships
or even when we're trying to use our identities
or prove our identities, and essentially what we've done here is we have proven some identities, but it's really helpful
to think about these symmetrys that we have
within the unit circle.