## Working Backwards – Finding the Angle using trial and error

Sometimes you’ll get given a diagram of a triangle with two of the side lengths labelled, but not the angle.  The question will ask you to find the angle, usually to a certain degree of accuracy.  In these cases, start with a guess at the angle, and see if it works using the appropriate cos, sin or tan ratio.  If it doesn’t, try increasing the angle.  If this gets you closer keep going, otherwise try decreasing the angle.

 Sponsored Links Find θ to the nearest degree: Solution If the diagram is roughly to scale, we can make an initial guess at what θ is – let’s say it’s about 40°.  Now, to check whether that’s a good guess, we can use a trigonometric ratio.  But which one to use?  Well, relative to the θ angle, the ‘5’ side is the adjacent side, and the ‘6’ side is the hypotenuse.  So we want the cos ratio:                                                    So what we are trying to do is find a value of θ that will make this equation true.  Now, we’ve guessed that θ is 40°.  We can use our calculator to work out what cos of 40° is:                                                      So our guess of 40° gives us an answer of 0.7660, which is smaller than 0.8333.  Let’s try increasing our guess for θ to 41°:                                                      Uh-oh!  0.7547 is even further away from 0.8333, so perhaps we should be reducing our guess for θ, down to 39°:                                                      Aha!  We’re getting closer to 0.8333 now, so we should continue decreasing our angle.  Now, once you get used to this sort of thing, you may be able to jump around a little with your guesses.  For instance, going from cos 40° to cos 39°, the result only changed from 0.7660 to 0.7771, and we want to get all the way to 0.8333.  So let’s try skipping a few degrees, perhaps go all the way down to 35°:                                                      That’s a lot closer to 0.8333, but we’re still a bit short.  So let’s drop another 2 degrees:                                                      Now, 0.8387 is larger than 0.8333, so we’ve probably gone a bit too far.  Let’s go back to 34°:                                                      So the cosine of 33° is 0.8387, which is larger than 0.8333, and the cosine of 34° is 0.8290, which is smaller than 0.8333.  So the actual angle we’re looking for must be somewhere between 33° and 34°. The question asked for the angle to the nearest degree, so we’re going to have to pick one of them.  One approximate way to pick the closer angle is to see which of their cosines is closer to 0.8333: cos 33°:                                    cos 34°:                                    So the result of cos 34° is a smaller distance from what we want – 0.8333, so we’ll use it as our answer – θ = 34°. There is a much faster way to do this question using the inverse cosine function on your calculator, which is covered later. However, it’s good to have an idea of how to approach a problem like this using trial and error.

One of the common questions you get in trigonometry is one about a ship, person or plane etc… travelling a certain distance at a certain angle.  You’re then usually asked to find out how far in a certain direction, north for instance, the person is of where they started.  There are a lot of variations on this type of question, but the general procedure for solving them is the same:

-       Draw a diagram with as much information labelled on it as possible

-       Work out what you’re trying to find, and label it on the diagram

-       Work out whether you have enough information to finish the problem straightaway, or whether you’re going to need to find out some other values first before you can find the final answer

-       Work through the problem.  Each time you find out some more information, draw another diagram if possible and label it on the new diagram, or add it to the original diagram.