Hey, everyone. So up to this point, we have worked a lot with right triangles. We've discussed some special case right triangles, as well as how we can solve any kind of right triangle that has one angle and one side and nothing else. Well, in this video, we're going to be learning about some strategies to solve any kind of right triangle that has 2 or more sides given to you. So if you have 2 of the side lengths or more, you can actually solve for all other sides and angles in the right triangle. And it's very important to have this skill because you're going to see these types of problems show up a lot through this course. So without further ado, let's see how we can solve these types of problems.

Now here we have an example where we're asked to find the angles of the given triangle, and we'll start with situation A. In this situation, notice that we have two sides, and we have 2 missing angles. This is the 90 degree right angle right there, and we need to find the missing angles. Now our first step is going to be, if there are any missing sides to use the Pythagorean theorem to find that side. And I can see that we are missing the hypotenuse, and recall that the Pythagorean theorem looks something like this:

Now a and b are going to be the two sides of the triangle that are not the hypotenuse, so I'll say that this is a and that's b. These are interchangeable. They could be either way. And then c always has to be the hypotenuse or the longest side. So using this equation, we're going to have 12² plus 5² is equal to c². Now 12² is 144, and 5² is 25. And that's all going to be equal to c². Now 144 plus 25 comes out to 169. And if I go ahead and solve for c, I can take the square root on both sides of this equation, leaving me with c is equal to 169, which is 13. So that means that 13 is the missing side of this right triangle, so that's what we have for the hypotenuse.

So now that we found this missing side, our next step is going to be to for any of the non 90 degree angles to write a trig equation that involves the known sides. And to find a trig equation that would do this, we can use SOHCAHTOA, which is the memory tool for each trig function relating to the sides of the triangle. Now what I'm gonna do is take a look at angle x as our reference here, and I'm going to use the sine, which is opposite over hypotenuse. So we'll have that the sine of our angle x is equal to the opposite side of the triangle, which in this case would be 12 divided by the hypotenuse, which is the long side or 13. Now to go ahead and find x, which this is what we're setting up in step 2, what we need to do is take the inverse on both sides of the equation for the trig function to find our angle. So I'll take the inverse sine on the left side, and the inverse sine on the right side. That's leaving me with just our angle x being equal to:

Now from here, what you can do is take this value and plug it into a calculator. And if you plug this into a calculator, you should get an approximate value of 67.38 degrees. Now to simplify this, I'm going to round this to just 67. So we're gonna say that this missing angle x is 67 degrees. So now that we found this angle x, our final step is going to be to find this angle y. And we can do that by just recognizing that these two angles are going to be complementary, meaning that we can just take 90 degrees and subtract off the angle we just calculated. So what I can do is say that our angle y is going to be equal to 90 degrees minus this 67 degree angle. And 90 minus 67 turns out to be 23. So that means the missing angle y is 23 degrees.

And notice how we were able to use the SOHCAHTOA memory tool as well as this relationship with the Pythagorean theorem to solve for all of the missing angles in the right triangle when all we were given were 2 of the sides.

Now to make sure we're understanding this, let's go ahead and try one more example to really make sure we've got this down. So in this example, we're already given all the sides of the right triangle, so we don't really need to do this step where we use the Pythagorean theorem. So what I'm going to do is this next step where we need to find a trig equation that represents one of the angles. Now in this example, I'll focus on angle b here, and this time I'm going to use the cosine to find this missing angle. Since we have all the sides, you could actually use any one of these trig functions you wanted to, but just for the sake of practice, I'm gonna use the cosine. So we're gonna have that the cosine of our angle b is equal to adjacent over hypotenuse. So the adjacent side of this triangle is 8, and then the hypotenuse is the long side or 17. Now what I can do from here is take the inverse cosine on both sides of the equation to get the angle B by itself. That's going to get this cosine to cancel giving us that b is the inverse cosine of 8 over 17. Now what you can do is plug this value into a calculator, and you should get an approximate value of 61.92 degrees, but to simplify this, I'm going to round this to 62 degrees. So, we'll say that angle b is 62 degrees, and then all we need to do is find angle a. And we can do that by taking 90 degrees and subtracting it from the angle that we just calculated. So we're going to say that angle a is equal to 90 degrees minus what we just calculated for angle b, which is 62 degrees. And 90 degrees minus 62 degrees comes out to be 28 degrees, meaning 28 degrees is our missing angle. So that is how you can solve for all of the angles if you're only given 2 or more of the sides of a right triangle.

So I hope you found this video helpful. Thanks for watching, and let me know if you have any questions.