One of the best ways to get comfortable with step-by-step processes is to see them in action. The first example problem we'll work through deals with calculating a specified angle of a 3-4-5 right triangle. This triangle is well understood and you may even know the answer we're solving for before we get started. A quick google search will tell you the answer. These types of well-known problems are great to try, because you can be sure that your answer is right.
Q: A right triangle has sides of lengths 3 and 4. Calculate the non-right angle closest to the side of length 4.
So, let's roll through the steps one by one and see how far we can get.
1) Identify the right angle in your triangle. This confirms that you will be able to use the sohcahtoa rule.
This will eventually become something that you can maybe even do in your head, but for now, let's be exhaustively descriptive. We've circled the right angle in red and added a little square, which is the common notation for a right angle. Check. Done. Next step.
2) Identify your angle of interest. This will change which sides are opposite and adjacent.
The identified angle is circled in red. Now we know which angle is the hypotenuse and can now identify the opposite and adjacent sides for this specific problem.
3) Identify your opposite and adjacent legs of the triangle.
Especially at the beginning stages, explicitly marking each of these angles is very important. It can lead to mistakes in the coming steps if you aren't tiresomely descriptive of what is what in this step.
4) Using the sohcahtoa rule, set up a relationship using sin/cos/tan, the angle, and two of the adjacent/opposite/hypotenuse sides.
The biggest trick here is to sort of "understand what you have". In this problem, we have the opposite and adjacent sides. We're looking for the angle, theta. The quickest way to our goal is to use toa, or tan[theta] = opp/adj. In this equation we have three variables, theta, adj, and opp. Two of them we already know. The reason we didn't choose sine or cosine is because those equations require knowing the hypotenuse, which isn't given in the problem. We could calculate it easily enough, but it's extra work which is not required of us. Good, concise mathematics is born out of a deep, internal laziness embedded in any good physicist or mathmetician.
5) Solve the equation for your desired variable.
At this point, we simply need to calculate the arctangent of our fraction, 3/4.
6) Box your answer, you're done!
At last, we arrive at our answer, 36.86 degrees. Make sure if your calculator gave you something like 0.64 that you understand that this is actually the right answer, but that the calculator is in radians mode, not degrees!
