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## Sine & cosine of complementary angles

# Trig word problem: complementary angles

CCSS.Math:

## Video transcript

The Nile River has
overflowed and covered its entire surroundings,
except for the tip of the Great Pyramid
in Giza, Egypt. An expedition was sent to find
how high the water had risen. The people measured the
edge of the pyramid that's above the water and found
it was 72 meters long. So this distance right
over here is 72 meters. They knew that the
entire length of the edge is 180 meters, when
it's not flooded. So this entire
length is 180 meters. They also knew that the
vertical height of the pyramid is 139 meters. So this is 139 meters. What is the level of the
water above the ground? So the ground is right over
here at the base of the pyramid. And so they want the level of
the water above the ground. So that's this height, this
height right over here. So let's just call that h. We want to figure out what h is. Round your answer, if necessary,
to two decimal places. So what do we know, and
what do we not know? So they've labeled this
little angle here theta. And this, of course,
is a right angle. So this angle here at
the base of the pyramid, this is going to be the
complement of theta. It's going to be 90
degrees minus theta. And using that
information, we can also figure out that this angle up
here is also going to be theta. If that looks a little
bit strange to you, let me just draw it out here and
make it a little bit clearer. If we have a triangle,
a right triangle, where this angle right over
here is 90 minus theta, and we wanted to figure
out what this is up here, let's say this is x. Well, we could say x
plus 90 minus theta, 90 degrees minus
theta, plus 90 degrees is going to be equal
to-- well, the sum of the angles of a triangle
are going to be 180 degrees. Well, if we subtract 180 from
both sides, so that's 100, and that's 100. That's 180 from the
left, 180 from the right. We get x minus
theta is equal to 0. Or if you add theta to both
sides, x is equal to theta. So this thing up here is
going to be theta as well. So this is also
going to be theta. And what else do we know? Well, we know this is 72. We know that the
whole thing is 180. So this is 72, and the
whole thing is 180. The part of this edge that's
below the water, this distance right over here. Let me draw it without
cluttering the picture too much. I'll do it in that black color. This distance right over
here is going to be 108. 108 plus 72 is 180. So what does this do for us? We need to figure
out this height. We know that this right over
here is a right triangle. I could color this in just to
make it a little bit clearer. This thing in yellow, right
over here is a right triangle. If we look at that
right triangle, and if we wanted to solve for
h and solve for h using a trig ratio based on this angle
theta right over here, we know that relative
to this angle theta, this side of length h
is an adjacent side. And this length of
108 right over here along the edge,
that's the hypotenuse of this yellow triangle
that I just highlighted in. So which trig ratio involves an
adjacent side and a hypotenuse? Well, we just write SOHCAHTOA. Sine is opposite
over hypotenuse. That would be this distance
over the hypotenuse. Cosine is adjacent
over hypotenuse. So we get the cosine of theta is
going to be equal to the height that we care about. That's the adjacent side
of this right triangle over the length of the
hypotenuse, OVER 108. Well, that doesn't
help us yet because we don't know what the
cosine of theta is. But there's a clue here. Theta is also sitting up here. So maybe if we can figure
out what cosine of theta is based up here, then
we can solve for h. So if we look at this data,
what is the cosine of theta? And now we're looking at a
different right triangle. We're looking at this
entire right triangle now. Based on that entire
right triangle, what is cosine of theta? Well, cosine of
theta, once again, is equal to adjacent
over hypotenuse. The adjacent length is this
length right over here. We already know
that's 139 meters. So it's going to be
equal to 139 meters. And what's the length
of the hypotenuse? Well, the hypotenuse is
this length right over here. It's 72 plus 108. Oh, we already have
it labeled here. It's 180. We can assume that
this is an isosceles-- that this pyramid is
an isosceles triangle. So 180 on that side
and 180 on that side. So the cosine is
adjacent-- 139-- over the hypotenuse,
which is 180, over 180. And these data
are the same data. We just showed that. So now we have cosine
of theta is h/108. Cosine of theta is 139/180. Or we could say that h/108,
which is cosine of theta, also is equal to 139/180. Both of these things are
equal to cosine of theta. Now to solve for h, we just
multiply both sides by 108. So h is equal to
139 times 108/180. So let's get our calculator
out and calculate that. So that is going to be 139
times 108, divided by 180, gets us to 83.4 meters. So h is equal to 83.4 meters. The height of the water is 83.4.