Multi-Step Trig Problems

Since they're really acing the introductory trig material, I am ramping my honors kids up to laws of sines and cosines. To do that, I am going to make them divide* scalene triangles into smaller right triangles, in order to really understand side and angle relationships within general scalene triangles. The longer-term vision is to build them up to Kristen Fouss's wonderful collection of complex trig problems. (Not to be overly confident in their abilities, but I really think that my honors 9th-graders may be able to handle dividing up quadrilaterals after a few days of practice dividing up triangles into smaller right triangles. --In groups, of course! Kristen's problems are just difficult enough where they will be forced to work together in order to get through them -- which would be absolutely perfect for these mathematically fearless warriors.)

Anyway, before we build up to all this fancy-schmancy laws and quadrilateral stuff, I thought it'd be good to just take a day and work out all of the kinks in their basic trigonometry application. Make sure they can fling sine, cosine, tangent at people in a hurry, that sort of thing. To that end, my Holt Geometry textbook (which I have a love-hate relationship with) has a nice collection of multi-step word problems. My honors kids were engrossed in them today, and feeling really good once they got through them. Check them out!

Intermediate problems (the book gave them the diagram for the first two of these... but I would expect the kids to be able to draw their own diagrams regardless):

From the top of a canyon, the angle of depression to the far side of the river is 58 degrees, and the angle of depression to the near side of the river is 74 degrees. The depth of the canyon is 191 m. What is the width of the river at the bottom of the canyon? Round to the nearest tenth of a meter.

Marion is observing the launch of a space shuttle from the command center. When she first sees the shuttle, the angle of elevation to it is 16 degrees. Later, the angle of elevation is 74 degrees. If the command center is 1 mile from the launch pad, how far did the shuttle travel while Marion was watching? Round to the nearest tenth of a mile.

Katie and Kim are attending a theater performance. Katie's seat is at floor level. She looks down at an angle of 18 degrees to see the orchestra pit. Kim's seat is in the balcony directly above Katie. Kim looks down at an angle of 42 degrees to see the pit. The horizontal distance from Katie's seat to the pit is 46 ft. What is the vertical distance between Katie's seat and Kim's seat? Round to the nearest inch.

Hard-ish:

A plane is flying at a constant altitude of 14,000 ft and a constant speed of 500 mi/h. The angle of depression from the plane to a lake is 6 degrees. To the nearest minute, how much time will pass before the plane is directly over the lake?

Really hard (differentiation for my super smarties):

Susan and Jorge stand 38 m apart, both to the west of Big Ben. From Susan's position, the angle of elevation to the top of Big Ben is 65 degrees. From Jorge's position, the angle of elevation to the top of Big Ben is 49.5 degrees. To the nearest meter, how tall is Big Ben?

A skyscraper stands between two school buildings. The two schools are 10 mi apart. From school A, the angle of elevation to the top of the skyscraper is 5 degrees. From school B, the angle of elevation is 2 degrees. What is the height of the skyscraper to the nearest foot?

I LOVED these! You have to assume for these problems that the heights of the observers are negligible, I guess. (Which makes it seem silly to round to the nearest foot or inch.) But, otherwise, I love these!! I went around and guided their thinking through a rigorous process of picking out what you're given, eliminating what you don't care about (ie. usually the hypotenuse, therefore we discard the cosine and sine), and then figuring out what to do with the rest of the info. After the first problem, they were all on their own and doing amazingly!!

*Would it be helpful to cut up the scalene triangles physically using scissors? I think so. I think I've found a hands-on Do Now task! :)

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On a different note, I was trying to explain to my regular kiddies today why sin(x) = cos(90-x), and even though I drew diagrams on the board and showed them that they come from the same 2 sides in the same right triangle, I still wasn't convinced that they understood. So, I drew an analogy! (I am reading a book about how to make ideas "stick", and one of the tips they give is that analogies help people connect to ideas in a concrete/almost visceral way.) I think it worked well. What I said was, "Let's say I say that Lourdes is sitting across from me, and Sofi disagrees, 'No! She's in front of me.' But in reality, we're both correct, because we're referring to the same position, just using different descriptions because Sofi and I are looking at her from different perspectives." After that, EVERY SINGLE KID understood! Wow to the power of analogies.

(Obvious, I know. But I think I need to consciously draw more analogies on abstract concepts, so for me it was a really good reminder of a little trick I can use to make ideas immediately more accessible to more kids.)