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Trig word problem: complementary angles

Sal solves a problem about a submerged pyramid using the fact that the sine of an angle is equal to the cosine of its complementary angle. Created by Sal Khan.

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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.