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Studying for a test? Prepare with these 9 lessons on Trigonometry with right triangles.

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# Trig challenge problem: trig values & side ratios

Video transcript

Sort the expressions
according to their values. You can put any number
of cards in a category or leave a category empty. And so we have this
diagram right over here, then we have these cards
that have these expressions. And we're supposed to sort
these into different buckets. So we're trying
to say, well what is the length of segment AC
over the length of segment BC equal to? Which of these expressions
is it equal to? And then we should drag it
into the appropriate buckets. So to figure these
out, I've actually already redrawn this
problem on my little I guess you call it scratch
pad or blackboard, whatever you want to call it. This right over here
is that same diagram blown up a little bit. Here are the expressions that
we need to drag into things. And here are the
buckets that we need to see which of these
expressions are equal to which of these expressions. So let's first look at this. The length of segment AC over
the length of segment BC. So let's think about what AC is. The length of segment AC. AC is this right here. So it's this length right
over here in purple over the length of segment BC, over
this length right over here. So it's the ratio of
the lengths of two sides of a right triangle. This is clearly a right
triangle, triangle ABC. And I could color
that in just so you know what triangle
I'm talking about. Triangle ABC is
this entire triangle that we could focus on. So you could imagine
that it's reasonable that the ratio of two
sides of a right triangle are going to be the sine
of one of its angles. And they give us one of
the angles right over here. They give us this
angle right over here. You say, well all they
did is mark that angle. But notice, one arc is
here, one arc is here. So anywhere we see only one arc,
that's going to be 30 degrees. So this is 30 degrees as well. You have two arcs here,
that's 41 degrees. Two arcs here, this is going
to be congruent to that. This over here is
going to be 41 degrees. This is three arcs. They don't tell us how
many degrees that is, but this angle
with the three arcs is congruent to this
angle with the three arcs right over there. So anyway, this yellow
triangle, triangle ABC, we know the measure of
this angle is 30 degrees. And then they give
us these two sides. So how do these sides relate
to this 30 degree angle? Well, side AC is adjacent to it. It's literally, it's one of the
sides of the angle that is not the hypotenuse, so let
me write that down. This is adjacent. And what is BC? Well, BC is the hypotenuse
of this right triangle. It's the side opposite
the 90 degrees. So this is the hypotenuse. So what trig function,
when applied to 30 degrees, is equal to the adjacent
side over the hypotenuse? Let's write down sohcahtoa
just to remind ourselves. So soh cah toa. Sine of an angle is
opposite over hypotenuse, cosine of an angle is
adjacent over hypotenuse. So cosine, let's write this
down, cosine of 30 degrees is going to be equal to the
length of the adjacent side. So that is AC, over the length
of the hypotenuse, which is equal to BC. So this right over
here is the same thing as the cosine of 30 degrees. So let's drag it in there. This is equal to the
cosine of 30 degrees. Now let's look at the next one. Cosine of angle DEC. Where is DEC? So DEC-- D, E, C. So that's
this angle right over here. I'll put four arcs here so
we don't get it confused. So this is angle DEC. So what is the cosine of DEC? Well once again, cosine is
adjacent over hypotenuse. So cosine of angle DEC,
the adjacent side to this, well that's this
right over here. You might say, well
isn't this side adjacent? Well that side, side DE, that
is the actual hypotenuse. So that's not going to
be the adjacent side. So the adjacent side
is, I could call it EC. It's the length of segment EC. And then the hypotenuse
is this right over here. The length of the hypotenuse--
the hypotenuse is side DE, or ED, however you
want to call it. And so the length of it is,
we could just write it as DE. Now what is this also equal to? We don't see this
choice over here. We don't have the ratio EC
over DE as one of these choices here. But what we do have is one of--
we do get one of the angles here. They give us this 41 degrees. And the ratio of this green
side over-- the length of this green side over this
orange side, what would that be in terms of, if
we wanted to apply a trig function to this angle? Well, relative to this
angle, the green side is the opposite side,
and the orange side is still the hypotenuse. So relative to 41
degrees-- so let's write this down--
relative to 41 degrees, this ratio is the opposite
over the hypotenuse. It's the cosine of
this angle, but it's the sine of this
angle right over here. Sine is opposite
over hypotenuse. So this is equal to the sine
of this angle right over here. It's equal to the
sine of 41 degrees. So that is this one
right over here. The sine of 41 degrees. So let's drag that in to
the appropriate bucket. Sine of 41 degrees
is the same thing as the cosine of angle DEC. Only have two left. So now we have to figure out
what the sine of angle CDA is. So let's see, where is CDA? CDA is this entire angle. It's this entire
angle right over here. So I could put a bunch of
arcs here if I want just to show that it's different
than all the other ones. So that's that angle
right over there. So now we're really dealing
with this larger right triangle. Let me highlight it
in this pink color. So we're now dealing with
this larger right triangle right over here. We care about the sine
of this whole thing. Remember, sine is
opposite over hypotenuse. So the opposite side
is going to be side CA. So this is going to be
equal to the length of CA over the hypotenuse,
which is AD. So that is going to be over AD. And once again, we don't
see that as a choice here. But maybe we can express
this ratio-- maybe this ratio is a trig
function applied to one of the other angles. And they give us
one of the angles. They give us this
angle right over here. I guess we could
call this angle DAC. This is 30 degrees. So relative to this
angle, what two sides are we taking the ratio of? We're taking, now, the ratio
of, relative to this angle, the adjacent side
over the hypotenuse. So this is the adjacent
side over the hypotenuse. What deals with adjacent
over hypotenuse? Well, cosine. So this is equal to the
cosine of this angle. So this is equal to
cosine of 30 degrees. Sine of CDA is
equal to the cosine of this angle right over here. So this one is equal to
this right over here. So let me drag that in. So this one is equal to--
so you can see that I just dragged it in-- equal to that. Now we have one left. We have one left. Home stretch, we should
be getting excited. AE over EB. AE, let me use this color. Length of segment AE. That's this length
right over here. Let me make that stand out more. Let me do it in this red. This color right over here,
that's length of segment AE, over length of segment EB. This is EB right over here. This is EB. So now we are focused on this
right triangle right over here. Well, we know the measure
of this angle over here. We have double arcs
right over here, and they say this is 41 degrees. So we have double
marks over here, and this is also going
to be 41 degrees. So relative to this
angle, what ratio is this? This is the opposite
over the hypotenuse. Opposite over the hypotenuse,
this right over here is going to be sine of that
angle, sine of 41 degrees. So it's equal to this
first one right over there. So let's drag it. So this is going to be
equal to sine of 41 degrees. So none of the ones
actually ended up being equal to the
tangent of 41 degrees. Now let's see if we
actually got this right. I hope I did. We did.