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Sine & cosine identities: symmetry

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

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  • leaf green style avatar for user sarahh
    Do I have to memorize all of these formulas? Is it important enough to know?
    (32 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      No, you can get through a lot of math without memorizing, but it just takes a lot longer to do the problems. Sometimes it is just plain easier to memorize a couple of formulas than to try to dig back to the basics and reconstruct the formulas.

      In the case of the symmetry relationships, it is a great time-saver to know these. There are ways of reconstructing the information if you forget. One way is to memorize the signs for the different trig functions in the four quadrants. The way I remind myself of these formulas is to think of a point in the first quadrant (both x and y will be positive, so all sine and cosine values will be positive, as will tangent).

      Then I think of a point in the second quadrant (x will be negative, since all the values for x will be less than zero, and y will be positive. As a result, sine will be positive, but cosine will be negative, and all tangent values will be negative.) In the third quadrant, all x and y values will be negative, so all sine and cosine values will be negative. Tangent will be positive because a negative divided by a negative is positive.) The final quadrant is the fourth quadrant, and there, all x values are positive, but all y values are negative, so sine will be negative, cosine will be positive and tangent values will be negative.

      So, you CAN recreate the information by logic. In the meantime, others can use the symmetries and be done with the problem and maybe with the next problem as well. Also, there are some ways that the questions can be asked that make it difficult to use this method, and if you are not very conversant with unit circle and translating points to sine and cosine, then you may have some tough slogging ahead. Knowing this set of symmetries becomes handy in Physics and many other applications.
      (130 votes)
  • starky ultimate style avatar for user armystorms
    What is a radian? I think I have some vague description of a ratio and something with PI but I don't honestly remember.
    (35 votes)
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  • male robot donald style avatar for user Neal
    I must say that was a lot to take in for one video, usually there's the odd "There's this reason" or "There's that" , not much made alot of sense to me with without examples or references .
    (18 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      I thought it was mostly review, but just pulled together as a different way of looking at the symmetries around a unit circle. So, what exactly was confusing? There are a number of more basic videos on trigonometry that may help, but I am not sure where to recommend that you start. Then after you are familiar with the basics, it would be easier to see how the concepts fit together. The key was showing how to name the reflected angles, and then how to construct the cosine and sine statements for each angle, and finally how to relate the pairs. And it all starts with the unit circle, so if you are hazy on that, it would be a great place to start your review.

      For example, let's say that we are looking at an angle of π/3 on the unit circle. The value of sin (π/3) is ½√3 while cos (π/3) has a value of ½
      The value of
      sin (-π/3) is -½√3 while cos (-π/3) has a value of ½
      Already we can see that cos theta = cos -theta with this example.
      And look at that: sin -theta = -sin theta just like Sal said.
      If we go through all the other reflected angles, with this specific example of an angle, we will get the same relationships that Sal just walked us through.
      One other example is π - theta
      This looks mysterious until you realize that in our situation, theta equals π/3
      and π - π/3 gives us an angle of ⅔π (In degrees, that is 120 degrees)
      cos ⅔π is going to be a negative value because it is in the second quadrant, and from working with unit circles, we should remember that it is evaluated as cos π/3 except for the negative. So, cos (π - π/3) = - cos π/3 and cos π/3 = - cos (π - π/3)
      Basically, if you have these symmetries, you have a multitude of sine and cosine values as long as you know what sine of theta is and cosine of theta is.

      It may help you to continue around the circle with common angles like π/6 and π/4 (not to mention the rest of the π/3 gang). That may help you to see how the symmetrical relationships are great time savers.
      (16 votes)
  • blobby green style avatar for user Leamsi Alconaba
    why sin(-theta)=-sin(theta)
    (8 votes)
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    • mr pants green style avatar for user Peta Coward
      Sin(theta) produces a positive value on the y axis.

      -Sin(theta) or, the negative of Sin(theta) takes that value, and multiplies it by -1, giving us a negative value on the y axis.

      That same negative value on the y axis, can be produced with Sin(-theta)
      (5 votes)
  • male robot donald style avatar for user Isaacskilling
    what is the diffrence between -data and +data
    (2 votes)
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  • marcimus pink style avatar for user Joy
    so, this is basic trig right? still amaze me with its complications! i still don't get it :( any websites that helps for all basic trig ?? and college math ... thanks!
    (0 votes)
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  • piceratops ultimate style avatar for user Unknown Incognito
    Why, why, why was this video placed AFTER solving sinusoidal equations?

    You literally CANNOT solve them without these identities, but they don't teach you said identities until after you spend an hour failing to solve those equations and moving on. Staff members, please consider repositioning this video in the curriculum.
    (5 votes)
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  • duskpin sapling style avatar for user DJ_SLIME
    I know this might be a dumb question but is Theta?
    (3 votes)
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  • aqualine ultimate style avatar for user Sameen Farooqui
    At , Sal wrote sin(- theta) = - sin(theta). When I tried to remake this circle, I wrote sin(theta) = - sin(- theta). Are both these expressions the same? If not, why?
    (1 vote)
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  • male robot johnny style avatar for user ctfitz123
    At Sal says that the angle in yellow is theta...bt isnt it vertically opposite to -theta and thus = -theta ?? if so, shouldnt the formula be pi-(-theta) = +theta??
    (4 votes)
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    • blobby green style avatar for user InnocentRealist
      It has magnitude theta. The actual angle of the yellow ray is the complement of the original theta, pi-theta. He uses "theta" to mean 1) the original theta in the 1st quadrant, or 2) an angle of the same magnitude. 3) any angle. It might be a little confusing, but when you get it I think it all makes sense.
      (1 vote)

Video transcript

Voiceover:Let's explore the unit circle a little bit more in depth. Let's just start with some angle theta, and for the sake of this video, we'll assume everything is in radians. This angle right over here, we would call this theta. Now let's flip this, I guess we could say, the terminal ray of this angle. Let's flip it over the X and Y-axis. Let's just make sure we have labeled our axes. Let's flip it over the positive X-axis. If you flip it over the positive X-axis, you just go straight down, and then you go the same distance on the other side. You get to that point right over there, and so you would get this ray. You would get this ray that I'm attempting to draw in blue. You would get that ray right over there. Now what is the angle between this ray and the positive X-axis if you start at the positive X-axis? Well, just using our conventions that counterclockwise from the X-axis is a positive angle, this is clockwise. Instead of going theta above the X-axis, we're going theta below, so we would call this, by our convention, an angle of negative theta. Now let's flip our original green ray. Let's flip it over the positive Y-axis. If you flip it over the positive Y-axis, we're going to go from there all the way to right over there then we can draw ourselves a ray. My best attempt at that is right over there. What would be the measure of this angle right over here? What was the measure of that angle in radians? We know if we were to go all the way from the positive X-axis to the negative X-axis, that would be pi radians because that's halfway around the circle. This angle, since we know that that's theta, this is theta right over here, the angle that we want to figure out, this is going to be all the way around. It's going to be pi minus, it's going to be pi minus theta. Notice, pi minus theta plus theta, these two are supplementary, and they add up to pi radians or 180 degrees. Now let's flip this one over the negative X-axis. If we flip this one over the negative X-axis, you're going to get right over there, and so you're going to get an angle that looks like this, that looks like this. Now what is going to be the measure of this angle? If we go all the way around like that, what is the measure of that angle? To go this far is pi, and then you're going another theta. This angle right over here is theta, so you're going pi plus another theta. This whole angle right over here, this whole thing, this whole thing is pi plus theta radians. Pi plus theta, let me just write that down. This is pi plus theta. Now that we've figured out these have different symmetries about them, let's think about how the sines and cosines of these different angles relate to each other. We already know that this coordinate right over here, that is sine of theta, sorry, the X-coordinate is cosine of theta. The X-coordinate is cosine of theta, and the Y-coordinate is sine of theta. Or another way of thinking about it is this value on the X-axis is cosine of theta, and this value right over here on the Y-axis is sine of theta. Now let's think about this one down over here. By the same convention, this point, this is really the unit circle definition of our trig functions. This point, since our angle is negative theta now, this point would be cosine of negative theta, comma, sine of negative theta. And we can apply the same thing over here. This point right over here, the X-coordinate is cosine of pi minus theta. That's what this angle is when we go from the positive X-axis. This is cosine of pi minus theta. And the Y-coordinate is the sine of pi minus theta. Then we could go all the way around to this point. I think you see where this is going. This is cosine of, I guess we could say theta plus pi or pi plus theta. Let's write pi plus data and sine of pi plus theta. Now how do these all relate to each other? Notice, over here, out here on the right-hand side, our X-coordinates are the exact same value. It's this value right over here. So we know that cosine of theta must be equal to the cosine of negative theta. That's pretty interesting. Let's write that down. Cosine of theta is equal to ... let me do it in this blue color, is equal to the cosine of negative theta. That's a pretty interesting result. But what about their sines? Well, here, the sine of theta is this distance above the X-axis, and here, the sine of negative theta is the same distance below the X-axis, so they're going to be the negatives of each other. We could say that sine of negative theta, sine of negative theta is equal to, is equal to the negative sine of theta, equal to the negative sine of theta. It's the opposite. If you go the same amount above or below the X-axis, you're going to get the negative value for the sine. We could do the same thing over here. How does this one relate to that? These two are going to have the same sine values. The sine of this, the Y-coordinate, is the same as the sine of that. We see that this must be equal to that. Let's write that down. We get sine of theta is equal to sine of pi minus theta. Now let's think about how do the cosines relate. The same argument, they're going to be the opposites of each other, where the X-coordinates are the same distance but on opposite sides of the origin. We get cosine of theta is equal to the negative of the cosine of ... let me do that in same color. Actually, let me make sure my colors are right. We get cosine of theta is equal to the negative of the cosine of pi minus theta. Now finally, let's think about how this one relates. Here, our cosine value, our X-coordinate is the negative, and our sine value is also the negative. We've flipped over both axes. Let's write that down. Over here, we have sine of theta plus pi, which is the same thing as pi plus theta, is equal to the negative of the sine of theta, and we see that this is sine of theta, this is sine of pi plus theta, or sine of theta plus pi, and we get the cosine of theta plus pi. Cosine of theta plus pi is going to be the negative of cosine of theta, is equal to the negative of cosine of theta. Even here, and you could see, you could keep going. You could try to relate this one to that one or that one to that one. You can get all sorts of interesting results. I encourage you to really try to think this through on your own and think about how all of these are related to each other based on essentially symmetries or reflections around the X or Y-axis.