Discovering right triangle side-angle relationships
This activity is designed to help you explore the right triangle relationships, discovered by geometers centuries ago, that led to the creation of the first trigonometric functions. Use the accompanying interactive tool to gather data and answer the questions below.
- Make three tables of six columns and six rows each (Excel or another spreadsheet program would work very well for this). In the top left cell of one table, write "AC/AB"; fill the top left cells of the other two tables with "BC/AB" and "AC/BC" respectively.
- Pick five different hypotenuse lengths between (and including) 1 and 20. Record these five lengths in the first column of EACH table.
- Pick five different angle measures between 0 and 90 degrees: two of the angle measures you pick must both be non-zero and add to 90 degrees (I suggest that you make 45 degrees one of your three remaining choices, but I'll leave that up to you). Record your five angle measures in the first row of EACH table.
- Using the sliders on the tool provided, fill in each table with the appropriate ratio (AC/AB, BC/AB, or AC/BC) for the 25 combinations of hypotenuse length and angle measure you chose.
- Pick one hypotenuse length: graph the data for AC/AB as a line (angle as x, ratio as y). Repeat this on the same set of axes (the same graph) for BC/AB and AC/BC as well.
- Answer the following questions:
- Describe the changes you see happening to the triangle as you manipulate the hypotenuse slider: what is changing, and how? What isn't changing?
- Describe the changes you see happening to the triangle as you manipulate the angle slider: what is changing, and how? What isn't changing?
- Looking at your data tables, what conclusions can you draw about the relationship between the side ratios, the hypotenuse length and angle B?
- Looking at your graphs, what patterns do you notice for each line? What relationships appear to exist between the three side ratios?
- Suppose you knew one side of a right triangle. Based on your discoveries today and any previous knowledge you might have, what more information might you need to solve the triangle (to solve a triangle means to determine all of its side lengths and angle measures). There is more than one correct answer: find as many as you can!