Solving triangles means to find all of the missing side lengths and angles.  For now, we will limit our focus to right triangles, which allows us the use of quite a few facts.  Here’s what we have at our disposal.  Remember, “opp,” is an abbreviation for opposite, “adj,” is an abbreviation for adjacent, and “tan,” is an abbreviation for tangent.

We will not be running through an example of each one of these, but enough for you to get a sense of how you decide which tool to use.  You will most likely not be told to use a particular tool to find an answer.  One of the most difficult jobs is figuring out which tool is most useful.

With that in mind, the KISS principle should be employed.  KISS is an acronym, perhaps slightly rude, that stands for Keep It Simple Stupid.  In other words, if you can use what you know about complimentary angles to find a missing angle, do that.  It’s simple. 

Let’s see an example.

It is a matter of convention that lower case letters can represent an angle, while upper case would represent the opposite side to that angle.  So the angle “a” would have an opposite of “A,” and angle “b” would have an opposite side of “B,” and so on. 

In this diagram, angle b = 72°.

To solve this triangle, we need to find all missing sides and angles.  To keep out of trouble, we need to remember KISS.  That is, what is the simplest application of the information provided.  Let’s start with the sides A and B, though that’s arbitrary, we could start with angle a, just as easily. 

General Guide Lines:

1. KISS
2. Use provided information to find values whenever possible
   1. Opposed to using a value you discovered
      
      
3. Be neat, write your formula, then plug in values, then solve
   1. If using a calculator, do not do so until the last step. Do all of the manipulation on paper, first.
      
      
   2. Do not round any step, only round your final answer (unless directed otherwise for that problem).
      
      
4. If time is provided, verify your answers using a different method than what was used initially.

Side A:

1. (KISS): Side A is opposite of angle a.  We don’t know angle a, so that’s not going to help us much, yet.  However, side A is adjacent to the angle that’s 72°, and we know the hypotenuse.  Cosine is what we’ll use to find side A.  This is the simplest way.
   
   
2. We haven’t yet discovered new information, so this is not an issue.
   
   
3. This is typed, so it will be neat. Regardless, this is how to start.

Here’s what we’ve found, written on the diagram.  We could use arccosine of 9/2.78 to check our work, but let’s wait until we solve the entire triangle and use something other than cosine.  Since we used cosine to find the answer, and we made a mistake, we will likely make the same mistake again using arccosine, and not find our error.  However, if we later use sine and get a different answer, then we have a problem we can discover.

Now, let’s solve for side B.  Remember, KISS and use what’s provided, not discovered, whenever possible.  If we used the Pythagorean Theorem to solve for B here, and we made a mistake with side A, then we are guaranteed to be wrong with side B, also.  So, let’s use sine since side B is opposite of the angle given.

Now, the easiest thing to find in this problem was angle a.  Typically, if this were a test question, I’d advise you to find the simplest thing first.  But, it is important NOT to use discovered information to find further unknown information whenever possible.  It is just too easy to make a simple subtraction error and then have all of your work be off.

Since it is a right triangle, and triangles add to 180°, the angle marked 72° and angle a must add to 90° (they’re complimentary).  So, angle a = 18°.  Here’s our final, solved triangle.

This is a valid equality because we rounded our side lengths to three significant figures.

Let’s verify the angle 18° by using arctangent.  This will use all of our discovered answers together.  If arctangent of 2.78/8.56 is 18°, correct to three significant figures, I’d be 100% confident that this triangle is solved correctly.

If you made a simple subtraction error, a common one here would be saying the answer is 28°, then you’ll discover it with this verification.

This is 18.0, correct to three significant figures.  I’d say we’re rock solid!

Let’s see one more example.  Remember, the tricky part isn’t setting up and solving the equations.  The tricky part is deciding which tool is best to use.

Here we need to find both angles, and the hypotenuse.  A take a moment to recognize how the information provided is related to what is required and come up with an approach you’d use to solve this triangle.

To find the hypotenuse it is easiest to use the Pythagorean Theorem.  When you do so, you get 13.7, correct to three significant figures.

While you could use that answer with sine or cosine to find angles a and b, it would be best not to.  It would be best to use arctangent for both angles.

Summary

To solve a right triangle, the first thing you need to do is make sure you identify the relationship between the parts provided and the parts that are missing.  This will help you know which formula to use.  Without understanding this, Trigonometry will never make sense.  Trying to memorize arbitrary associations, like if the triangle looks like this then I use this set of steps, would be like trying to play a musical instrument while being deaf from birth.  It will make no sense.

Making sense of things is difficult, and often makes us feel insecure.  Nobody is born knowing this, everybody that learns it must make sense of it.  If you’re patient with yourself, willing to reflect on your decision making, this will be entirely approachable and eventually easy.

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