 In number 17, we have another example that you may recognize once you write out your trigonometric ratio. Angle A, we have the adjacent leg, which is 8, and the hypotenuse, which is 16. So we are looking at cosine of angle A equals adjacent over hypotenuse, 8 over 16. Now, if you reduce 8 over 16, that's going to equal 1 half. Now, what kind of triangle do we know has a ratio of 1 to 2? Hopefully your answer is the 30-60-90 triangle. You would be correct. And if we have this side is 1 and this side is 2, that means this is root 3. And so what angle is always opposite of root 3? That would be the 60 degree angle. Now, again, if you enter this in your calculator, inverse cosine of 1 divided by 2, you will get 60 degrees. Number 18 is a little trickier. We want to figure out the measure of angle A, but that means that we need to know what this side is, what BD is. You have a couple of ways to figure that out, one of which would be Pythagorean theorem because you know that DC is 12 and BC is 12 root 2. And you know this is a right triangle, so you could use Pythagorean theorem, but much quicker would be to recognize that this is a special right triangle. If you look at triangle BDC, if you recognize that DC is 12 and BC is 12 root 2, this has to be a 4590 triangle. So if DC is 12, that means that BD also has to be 12. Again, you should get the same answer if you do Pythagorean theorem. It just takes you a little bit longer. Okay, so now that we know BD is 12, now I can set up to solve for angle A. I have the opposite leg which is 12 and the hypotenuse which is 13. And so that's going to be sine, sine of A equals 12 over 13. And if I go to my calculator and enter inverse sine of 12 divided by 13, I will find out that the measure of angle A is 67.4 degrees. So number 18 is kind of a two-part problem. You first have to figure out using this triangle, you have to figure out BD. And then to find angle A, you're going to use this triangle, ABD. Number 19 is a similar problem to number 18. What we're going to start off with is looking at this triangle BDC. Right away we see it is a 4590 triangle. So that means if CD is 4 root 2, then BC is also 4 root 2. And the hypotenuse BD, if you use the special right triangle rules, if this is 4 root 2 and this is 4 root 2, how do we figure out the hypotenuse? Well remember that this is x, x and x root 2. Well, so that means that x is equal to 4 root 2. So BD is going to be 4 root 2 times root 2. That's going to equal 4 root 4. And because the square root of 4 is 2, 4 times 2 is 8. So that means that BD is 8. Now I have the information I need to be able to find angle A. And so now I'm going to focus on triangle ABD. And because I have opposite and hypotenuse, that's going to be sine of angle A equals the opposite, which is 8 over hypotenuse, which is 10. And so if I evaluate inverse sine of 8 over 10, we find out that angle A should be about 53 degrees. So angle A is 53.1 degrees.