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### Course: High school geometry>Unit 5

Lesson 8: Sine & cosine of complementary angles

# Trig word problem: complementary angles

Sal solves a problem about a submerged pyramid by finding congruent angles in the diagram and using the fact that the cosine of a given angle is always the same, no matter how large the triangle it appears in. Created by Sal Khan.

## Want to join the conversation?

• At ,how can we assume it is an isosceles pyramid? We don't know for sure,do we? That question might have been a trick question.
• I guess because we are using the actual Great Pyramid of Giza for the problem, and it is in real life isosceles, we can assume this.
• How would you find the sine/cosine/tangent of the right angle?
• If you look up the unit circle it will help explain it. You should find that:
sin90 = 1
cos90=0
tan90 = sin90/cos90 which is undefined
• Such a convoluted solution for something that could be done simply by 139(108/180). No trigonometry needed at all.
• The ratio 108/180 is the cosine of the angle theta. You are doing trigonometry whether you know it or not.
• I went the other way around sin^-1 (139/180) to get the bottom angle of 50.5539 to figure its complementary angle is 39.4461. Then using the Cos (39.4461) * 108 to get 83.4 as the height of the water. Do not know if it is easier, but this is the way I figured it out prior to Sal giving his explanation.
• The issue with this kinda approach is that we have to use a calculator for that which sometimes is not available.
• Did this scenario every happen in real life?
• Probably not, cause it would of absolutely demolished the entire Egyptian civilisation. Also the Nile 26 to 36 feet deep on average. The pyramid of Giza was 481 ft high at its creation. So it is absolutely impossible.

EDIT: and insane.
• You could also use similar triangles right?
• Yes you can set up a proportion with similar triangles, but x would be the part of the vertical distance above the water. 72/180 = x/139, so x = 55.6, then subtract 139 - 55.6 to get 83.4 which is the same answer of 83.4.
Or you could do 180-72=108, then 139/180 = x/108 to get x = 83.4.
• is there an easier way to do this and if there is what is it
• You didn't have to follow all the steps that Sal did in this video. Here are the basic steps for this problem:
1) Find two triangles with the same angles.
2) Determine which sides and angles you need for soa cah toa ratios.
3) He found the two lengths of the large triangle and one length from the smaller triangle with corresponding angles
4) Use the ratio of the sides adjacent / hypotenuse (cah) to get 139/180 = h/108
If you recognized the same angles right away, you can go right to the ratios to determine the height.
• couldnt we just say that angle x and angle theta are alternate interior? cuz the two lines (side of triangle and the line h) are parellel (both are 90 degrees)
• If sin90=1 then cos0=1 ,but 0 degrees isn't an angle!
• You will expand the trig ratios when you learn the unit circle
(1 vote)
• Here's my preamble before my argument that this video also needs a preamble. Bear with me: I love Sal Khan and I love everything he's created. I am re-discovering my love for math! And, I have watched a lot of his videos, but this is the first time I've had this though while watching: "Wow, there's a much easier to way to do this.(1)" I know these videos have been up for almost a decade and honestly that's a great thing about instructional math videos: the information being taught never changes or becomes obsolete from new information. That all being said, I really feel like this video needs some sort of preamble where he says something to the effect of like, "There are some much easier ways of doing this problem and getting the correct answer but I'm going to do it the hard way because we're learning trig ratios and when the math gets harder, there might not be an easier way to do it." It also would address the perhaps vague albeit real stereotype that "math teachers make it harder than it needs to be."

1. For reference, the much easier way of solving the problem is mentioned in comments below but to reiterate, 1) the big pyramid and small (al fresco) pyramid are similar. 2) 180/72=2.5 3)139-(139/2.5)=83.4