# Triangulation

Triangulation is a technique that works well over short distances, assuming you have the coordinates of two fixed locations and know the bearing (angle) from the line connecting the fixed locations and the point of interest: 1. You are given the following information about the image above: Point A is at (0, 4), Point B is at (12, 0), Angle A is 24.146 degrees, and Angle B is 35.609 degrees. What are the coordinates of the ship (Point C)? Make sure to show all work.
2. Problem 1 probably turned out to be a lot of work and it would be very annoying to complete that whole process every time you needed a new position. Mathematicians proposed a formula based on the picture below: Can you prove that this formula is correct?
3. Finding d is only part of the battle, can you develop a quick way to find the location of C once you know d?
4. The beauty of having a formula is the ability to put it into Desmos, Excel, etc. so that you can simply change the coordinates/bearings and the computer will immediately provide you the coordinates of the point of interest (Point C). Try this and explain your process. Do not just stop once you have something that works, but try to make your “code” as short and efficient as possible. The longer your “code,” the more the computer has to process.
5. Although original thought is often celebrated, another important skill is to be able to read the work of others and comprehend it. Brilliant discoveries are often the result of extensive reading and experience, not simply being a math person. Mathematicians, using the images below, created formulas to find the coordinates of an unknown point: You can find the proofs here (to be added). Work through the mathematics, highlight sections that do not initially make sense, investigate those further, and explain your process of making sense of the math.