The word "trigonometry" is enough to strike fear into the hearts of many high school students. But it simply means "triangle measuring." A trigon is another name for a triangle -- think pentagon and hexagon, for example -- and "-metry" means "measuring," just as it does in "geometry," or "earth measuring." Trigonometry is about measuring similar right triangles.
The triangles below are all similar to the original triangle above. Find the length a for each triangle. Explain how you did it. Note 7
Set up a proportion between the corresponding sides of the two triangles, the original one and the new one with unknown sides. There are many different proportions you might set up depending on what you're looking for (e.g., a1/c1 = a2/c2, or, a1/a2 = c1/c2, etc.). In each proportion try to use three known side lengths, so the only unknown is the side you're looking for. Close Tip
For each triangle above, find the length b. Explain how you did it.
If you solved the problems above, you probably used the sine and cosine functions, even if you didn't know you were doing it. To find the missing value of a, you need to multiply the length of the hypotenuse by 3/5. To find the length b, you multiply the hypotenuse by 4/5.
Those ratios -- (side a)/(hypotenuse) = 3/5 and (side b)/(hypotenuse) = 4/5 -- will hold for any right triangle that has another angle the same as A. All such right triangles will be similar to each other, so the ratios must be the same as in our original triangle.