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Tangent identities: symmetry

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

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  • aqualine seed style avatar for user micahskan
    How does the pi in tan(theta + pi) get cancelled out?
    because tan(theta + pi) = tan(theta)...it doesn't make sense!
    (24 votes)
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    • blobby green style avatar for user Keith Whittingham
      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 at .

      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?
      (49 votes)
  • male robot hal style avatar for user Logan LaFollette
    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?
    (7 votes)
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    • piceratops ultimate style avatar for user Just Keith
      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.
      (58 votes)
  • hopper cool style avatar for user BenR

    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)
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  • starky ultimate style avatar for user June
    Do I have to memorize all of these formulas? Is it important enough to know?
    (2 votes)
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    • purple pi purple style avatar for user APDahlen
      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
      (24 votes)
  • blobby green style avatar for user Luis
    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?
    (3 votes)
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  • aqualine ultimate style avatar for user Destiny
    Sometimes! I get confused by the wheel or the circle and the words?
    so can someone just help me.
    (2 votes)
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    • leafers sapling style avatar for user CaptainMurg
      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.
      (8 votes)
  • sneak peak green style avatar for user Cybernetic Organism
    How does unit circle relate to finding out the angles of right triangles?
    (3 votes)
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    • leaf grey style avatar for user Lucas Vasconcelos
      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)
      (5 votes)
  • aqualine ultimate style avatar for user Z
    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?
    (3 votes)
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    • primosaur tree style avatar for user Bradley Reynolds
      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.
      (6 votes)
  • starky tree style avatar for user sixhundredandsixtysix
    In this video, we derive trig identities based on the quadrant the angle is in. So, for example, we see that cos(-theta) = cos(theta), because the X-value is symmetric for quadrants I and IV. But what if theta > 90 degrees? Does that not mean the quadrants change, making at least some of the trig identities invalid?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      Actually, it still applies because in the II and III quadrant, the cosine is negative. So the identity still is valid. Same thing happens for sine where it is positive in the I and II quadrant and negative in the III and IV quadrant. It can also be seen on the sine and cosine function (wave) where any horizontal line will cross the function at two points in any one period.
      (3 votes)
  • blobby green style avatar for user Brian Hatzelhoffer
    What is the relation of the reciprocals for this subject? I found a web page that allows you to visualize sin cos and tan angles by hovering your mouse over it. https://www.mathsisfun.com/sine-cosine-tangent.html Anyone know if there is something like this with sec csc and cot? Is putting the negative in front of sin cos or tan the same as saying the reciprocal? On a calculator there is an option fo put the power of negative 1 on them. Is that the same thing? I have been watching all these trig videos and still don't seem to understand it fully.
    (2 votes)
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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.