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# Sine & cosine identities: symmetry

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.