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# Tangent identities: symmetry

CCSS.Math:

## Video transcript

in the previous video we explored how the cosine and sines of angles relate when we essentially take the terminal ray of the angle and we reflect it about the X or the y axis or 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 if you we remind ourselves slope 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 we're 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 terminal terminal on the stripe terminal for the terminal ray so the tangent of theta is a sine of theta over cosine theta or you could view it as the slope of this ray right over here so let's think about what other what other angles are going to have the exact same tangent of theta well this ray is collinear with this ray right over here in fact if you kind of 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 I'm just obviously you could write theta plus PI instead of PI plus theta this should be just based on this kind of slope argument this should be equal to the tangent of theta this should be equal to the tangent of theta let's 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 the slope of the ternal ray of course the other side of the angle is going to be the positive x-axis based on the conventions that we set up well let's look let's think about what it is what the tangent of thought of theta plus pi is in terms of sine and cosine so the tangent let me write this down in a pink color the tangent that's not pink the tangent of PI plus theta that's going to be equal to let's parentheses to avoid ambiguity that's equal to the sine of pi plus theta of theta plus pi over the cosine of theta plus pi and in the previous video we established 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 negative cosine of theta well you 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 could feel pretty good about that now what about what about this that what about the points or the terminal rays right over here right over here so let's think about this point what is the tangent what is the tangent of negative theta going to be well 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 we already established the sine of negative theta that's negative sine theta that's negative sine theta we see that right of your sine of negative theta that's the negative it'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 which is equal to negative tangent theta and so we see here if you make the angle 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 change changes signs but the denominator does the denominator does not so tangent of negative theta is the same thing as negative tangent of theta now what about 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 so 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 signs so this is equal to sine of theta while cosine of PI minus theta well it's it's the opposite of cosine of theta it's a negative of cosine of theta so this is negative cosine 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 as this ray right over here and we see that slope we could view this as negative tangent negative tangent of theta and we see just looking at we just see looking at these two I guess if you if you take if you combine the Rays that these two intersecting lines have the negative slope of each other there are mirror images across the x-axis and so we've just seen if you take an angle and you add PI to the angle your 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 in the opposite direction but the slope of your array 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 take you're gonna get the negative of the your tangent or or if you were to go go over here and if you were to take pi minus your angle if you were take pi minus your angle then you're also going to get the negative of your tangent so hopefully this makes you a little bit if this is very useful when you're trying to kind of work through trigonometric problems or try to find relationships or even when we're trying to use our identities or prove identities and essentially what we've done here is we have proven some identities but it's really helpful to think about these symmetries that we have within the unit circle