# A Better Law of Sines/Cosines Project

A while back I had students finding distances on our school’s campus using the Law of Sines and Cosines, and I made the point that I thought students appreciated finding things out that related to their lives a little more.  However, this activity still seemed superficial to me because of (1) the ease with which students could look online to find these distances (using Google maps and a latitude/longitude distance calculator can be surprisingly precise!–see below), (2) after finding 1 or 2 distances it got old fast, and (3) I’m pretty sure this is not how surveyors do it in the real world.

I was talking with the father of a student in my Precalculus class, who happened to be a farmer, and “finding the area of his land” came up somehow.  Now a good chunk of the students live on the reservation and so it is very, very rural.  One thing they have, however, is land, and a lot of it.  I realized that breaking polygons into triangles is a cool theorem, and one that has practicality in finding the area of a plot of land if, say, you wanted students to use the Law of Sines, Cosines, and the Trig Area formula for a triangle.

I’m going to start with the following warm-up and hope that students recognize the connection between the project and the warm-up (I may have to spell it out for some of them).

I decided to turn this into an individual project for the students and use it in place of a test, since this would probably interest them significantly more and would demonstrate a deeper understanding of how to use the equations than “apply it to this triangle”.  I’m going to hand them this sheet below, show them how to find distances on Google Maps via Wolfram Alpha, and then let them go.

I’m hoping that students will solve this problem in a number of different ways.  One way that came to my mind immediately was, after breaking the land into triangles, find all the distances using Google Maps, then use Law of Cosines to find an angle, and lastly use the trig area of a triangle formula which involves two sides and the included angle.  Of course, if students are familiar with Heron’s formula, they could jump to that, hence the requirement they use 2 of the 3 formulas (is that too false a requirement?).

Lastly, you’ll notice I’ve made accommodations for the “urban” and suburban students as well: they can measure the square footage of their house/apartment or even a room as long as it isn’t a rectangle.

Let me know what you think, and in return I hope to show off some students’ work!

Instructions for finding Distance using Google Maps and Wolfram Alpha

1) First you should find the place in Google Maps.  Right-click and select “What’s here?”  You should see a green arrow now on the location.  In the search bar it will leave some coordinates, which are the very precise latitude and longitude of the place.  Copy these coordinates.

2) Next, go to Wolfram Alpha, and paste the coordinates as follows (and shown in the picture below):

distance from (35.528149 N,108.654796 W) to (35.52855 N, 108.656035 W)

Make sure you change +/- to N/S or E/W otherwise Wolfram Alpha will simply plot them as points and find the distance.  You’ll have to go back to Google Maps to select the other point and find the other set of coordinates.

One cool thing that Wolfram Alpha does is gives the distance is a number of different measurements.  And I’m not just talking about m, km, feet, inches, etc., but it gives crazy ones like “times the traditional length of Noah’s Ark” or “about the height of the world’s tallest tree” or “Light travel time in a vacuum” or “Maximum distance visible from that height”.  Crazy cool, right?

Filed under Teaching

### 5 responses to “A Better Law of Sines/Cosines Project”

1. Rose

Really enjoyed this post – modifying it for my Triangle Trig unit in Pre-Calculus next week!

2. Jeanette

This is a great assignment. Did you have students simply use google maps to decide the shape of a plot of land? Or did they have to use a specified plot of land (where we are, they are mainly rectangular). We are looking to use it, but we are required to make our students take it further. Our idea was to have them also find the value of the property and then compare the values of the properties in different areas of the country. Debating if they should be required to create a plot of land that is not rectangular or use their plot of land (from zillow or trulia or something like that) and adjust it so it is slightly less rectangular.

Thanks for sharing this great project. Decent trigonometry problems are hard to come by. Would love any ideas you might have.

– Jeanette

• Yes, most plots of land around them in New Mexico are not square (too many hills/mountains). Perhaps you could look for famous landmarks (area of a nearby baseball stadium, for example?)

I like the “value of the property” idea!

I did make the students with supposedly rectangular plots of land to still use the law of cosines to calculate the angles–the students never put them *exactly* in the shape of a rectangle, so it forced them to use numbers like 89 or 91 degrees. Often even the rectangular-looking ones weren’t, but perhaps that’s a function of nothing being rectangular around the area.