 Welcome back to our lecture series math 1060 trigonometry for students at Southern Intel University. As usual, I'll be your professor today, Dr. Angel Misaligned. Just as a reminder, in lecture four of our series, we've introduced properly the idea of right triangle trigonometry, and we talked about this idea of Sokotoa. That is to say that this is supposed to be a mnemonic device we can use to remember the three fundamental trigonometric ratios. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. What we want to do in lecture five is use the Sokotoa and the other trigonometric ratios to help us solve missing parts of right triangle diagrams, and then eventually apply those to various applications of right triangle trigonometry. So for example, look at the triangle you see on the screen here, ABC. What do we know about this triangle? We know that the angle A is measured at 40 degrees, and we know that the hypotenuse of the right triangle is given as a length of 12. And so we want to find the missing pieces of the triangle. I guess we also should mention that if 12 is the hypotenuse, that's because angle C is the right angle to 90 degree measure. So the things we don't know about this triangle is we don't know what little A is, where A, little A is opposite angle A. We don't know the length of side B, and we also don't know the measure of angle B. And so we want to figure out how can we fill in the rest of this information. And there's a couple of ways you could do it. So one thing to mention here is the Pythagorean equation comes into play that if you have A right triangle, then the sum of squares of the two legs will equal the square of the hypotenuse, so A squared plus B squared equals C squared. So if we knew two of the sides, like if we knew 12 and say A, we could solve for B, or if we knew 12 and B, we could solve for A. The Pythagorean equation is useful if you know two of the sides of our triangle, but you need to know these two of the sides. Also we should remember that the measure of the angles of a triangle, so the measure of A plus the measure of B plus the measure of C always adds up to be 180 degree for these triangles. And so from this, since we know A and we know C, we can very quickly find the measure of angle B. So notice that we have 180 degrees. Let's take away the measure of angle C since it's a right angle. We'll take away 90 degrees. Let's also take away the measure of angle A, which is 40 degrees. So 180 take away 90 degrees, of course, is 90. So we're really looking for the complement of 40 degrees, which would be 50 degrees. This is going to be the measure of angle B right here. So we see that B has a measure of angle 50, 50 degrees. And so one can very quickly find the missing angle of a right triangle. If you know two of the angles, then you know the third one by this principle right here. So how do we find the lengths of A and B? Well, this is where the Sokotoa relationship comes into play. With respect to angle A, right, we're going to just use angle A. Now that we know angle B, we actually could use angle B if we wanted to, but I'm going to stick with the angle that I was given just to avoid the possibility that maybe I computed B incorrectly. Since I know angle A is 40 degrees, look at side length A and the hypotenuse. What do we have here? With respect to angle A, I have the opposite side and I have the hypotenuse. And so this tells us I could determine little A using the sine ratio. So we get that sine of A is going to equal little A over 12, for which if we then fill in the information we know, we know that sine of 40 degrees is going to equal A, which I don't know over 12. Okay. So clearing the denominators here, you get that little A is equal to 12 times sine of 40 degrees. This right here is the exact answer. So if your question, you're like a homework question, you have our test question asked for the exact answer. This is what we're going to do right now. Now we can approximate the answer by consulting with our calculator. What is sine of 40 degrees? So this is going to be an irrational number. But this is something we just put into our calculator. Now be careful. Your calculator might, the syntax depends on the brand of calculator. If you want to do sine of 40 degrees, some calculators might require you type in the sine button. And then you follow that up with 40. Other options might actually have type in 40 first, and then you hit the sign button. So you have like a prefix versus a post fixed notation experiment with your calculator. One of the order that's wrong will probably give you an error. So check it out just to make sure what's going on here. Another thing that's important to note is that your calculator at this point needs to be in the degree mode. If you have a scientific calculator, or even a graphing calculator, there's typically angle mode where degree which might be abbreviated on the screen as DEG or just D for short. You want to do degrees and not radiance for this calculation. We'll do radiance of course later in this series. So for this question, though, it's degrees and many of your questions might be in degrees as well. Make sure you're in the correct mode. If you type in sine of 40 and your calculator is in radian mode, you'll get a very different calculation than degree mode and you'll get something that's incorrect. So make sure your calculator is in degrees and make sure you understand the syntax of your calculator before you go on from this from this point right here. If you type it incorrectly, sine of 40 degrees will be approximately 0.6428. If you times that by 12, then you'll get 7.7136 as this is an estimate accurate to four decimal places for this number A. And it's best to do all of the approximation in the calculator. Try not to do too much on paper once you get to the approximation part because we want as many decimals in this expansion as possible to guarantee our last answer is accurate to whatever the prescribed level is. So that's going to give us A, right? If we want to find B, we could use the Pythagorean equation because B will just equal the square root of C squared minus A squared. I'm not going to advise doing it that way because as we found A as an approximation, if A is an approximation, that is there's some error to it, maybe it's rounded that rounding that error will compound when we calculate B. And so it could be that maybe A is within the margin of error, but B will fall out. So it's best to try to use as much as possible the original measurements which we can assume are accurate to find the value. So that's why I avoided using angle B in my calculation. So can I figure out what little B is with respect to the hypotenuse and the angle A that's given to us? Absolutely. This is the adjacent side with respect to angle A. This is still the hypotenuse. So the cosine relationship comes into play here. We see that cosine of A is going to equal B over 12. In other words, B is equal to 12 times cosine of 40 degrees. Again, I just cleared the denominators right there. If we approximate cosine of 40 degrees following the same principles we talked about before to make sure this is an accurate approximation, we're going to get a cosine of 40 degrees is 0.7660. And when you times that by 12, you're going to get approximately 9. Let me write that nine again. You get 9.192 as our estimate. If you want an exact answer, it's 12 times cosine of 40. An estimate would be 9.192. And like I said, once you start seeing these approximation symbols on the screen, this means I'm doing everything in my calculator. In my calculator, I type in cosine of 40 degrees, I hit enter, and then I times that by 12 to get my final value right there. So we can use our Socatola relationships, the Pythagorean equation, maybe also the angle sum, the triangle sum theorem there to find out all the missing parts of a right triangle. Let's do one more example of this. This time we know two of the sides, we know side B, we know side A, we don't know side C, and we don't know angles A or B. Okay, so how can we proceed forward from this? Again, we want to stay as close to the original given information as possible. We should also mention that we know that angle C is the right angle. So that means that little C is the hypotenuse. And so this time, because I know two of the sides, this time I am going to use the Pythagorean equation, we know that a squared plus B squared is equal to C squared. That means that C is going to equal the square root of a squared plus B squared, which we have that information. So A, we can assume that this measurement is accurate to however many decimal places, how many significant digits, we're not going to worry about that in this video. But those are things that should be concerning as we do these calculations in practice. So C is going to equal the square root of 3.41 squared plus 2.73 squared. And again, feel free to use a calculator to help you out with the arithmetic here. 2.73 squared, excuse me, we'll do the 3.41 first. 3.41 squared is going to give us 11.6281. If we do 2.73 squared, that gives us 7.4529, all inside of the square root. Again, add those things together. That is the two numbers together, you got 19.081 inside the square root. And so that gives us an approximation of C, the square root of 19.081 is going to be approximately 4.37. Again, rounding to two decimal places, mostly to respect the original format. These things are two decimal places. So I'm going to do that here as well. If the instructions give you something different, then follow those instructions, of course. So we can use the Pythagorean equation to find the missing side. How do we find the angles A and B? Okay, well, we can still use a trigonometric ratio. So whenever the angle is involved, you need to use a trig ratio. Now, with respect to angle A, which we don't know, we do know the opposite side and we know the adjacent side. So the tangent ratio comes into play here. We get that tangent of A is equal to the opposite side 2.73 over 3.41. So tangent of A is equal to this ratio here. This is again where we're going to consult our calculator. We can compute this angle, of course, excuse me, this ratio. This ratio, so the tangent of A will be approximately when you throw that ratio 2.73 over 3.41 into your calculator, you get 0.8006. But keep as many decimal places in your calculator as much as possible. To get the measure of angle A, what we're going to do next is we're going to apply the arc tangent function on this number right here. And so we take arc tangent of 0.8006. So we're going to talk about what this symbol means a lot more in the future of this lecture series. So what we're writing right here is the inverse function of tangent sometimes called arc tangent or tangent inverse. You see this negative one superscript right here. That doesn't mean an exponent. That just means the inverse function. Again, a topic will talk about a lot more in the future. For our purposes right now in our lecture series, I mostly just want you to be familiar with how to do the mechanics of this in your calculator. So when it comes to typing an inverse trig function, the format you have to worry about is similar to if you want to do like sine or tangent. Do you plug the number first then the button or do you do the button then the number that depends on the calculator experiment that to make sure you're familiar with that to find the tangent inverse button, it'll probably say something like that. It'll probably say like tangent inverse. You'll see that negative one as a superscript. And to access this, most likely if you're using a scientific calculator, there's a tangent button. There's a tangent button. And before you hit the tangent button, you're going to put some type of like secondary button, or there's a button that might change. Because when you look at the tangent button, you might see next to the tangent button like this, tangent inverse, but it's not actually a button that's just print on your keyboard. Sometimes there's like, there's a secondary button, or there might be like an auxiliary button that transforms the tangent button into tangent inverse, or it might be called shift. It's kind of like, it's kind of like you use a regular keyboard, if you push shift, and then you hit like s, you'll get a capital S scientific characters, graphic characters often have the same features. So look for whatever the shift or secondary auxiliary button is that typically will turn the tangent into a tangent inverse. And so then you click that button with the point zero, the point, excuse me, 0.8006. And doing that will give you the estimate of angle a, which you would get a 38.7 degrees right here. Now just like with the sine cosine and tangent buttons, when you're working with arc tangent, tangent inverse that is, if you do inverse sine or inverse cosine, same problems right there, you need to make sure your calculator is in degree mode, because the output here needs to be in degrees not in radians. So watch out for that as well. So we get 38.7 as the approximation for the measure of angle a. So to find the measure of angle B, there's basically two ways we could do it. With angle B, we could do a co tangent ratio because we have of course, the adjacent over hypotenuse. When it comes to solving these right triangle diagrams, for the most part, we don't want to use secant, cosecant or cotangent. And this mostly comes from the fact that our calculators come equipped with a sine cosine and tangent button. And likewise, an inverse sine inverse cosine inverse tangent button, but the keyboards rarely will have secant, cosecant or cotangent. So if we have a choice, it's probably better to use a tangent ratio over a cotangent ratio. Same thing, better to use cosine over secant, better to use sine over cosecant if it's somehow avoidable. So with respect to angle B, you could do opposite over adjacent, like so. And so you could do a tangent ratio, you get that tangent of B is equal to 3.41 over 2.73. In which case, then, in this situation, we could then solve it, we could find angle B using the inverse tangent function again, just like we did with tangent of A over here. I want to provide one alternative approach just to get as a, as a contrast. Because what we can do is since we know angle A, and we know angle C, of course, is 90 degrees, since the angle sum will out to be 180 degrees, this means that angle A and angle B are complements of each other. In which case, then, if I just take my measurement of angle A, and I subtract it from 90 degrees, that would give me angle B. Now, you might be wondering, well, didn't you say earlier that when it comes to computing these things, you want to use the original values, not these ones we computed. Generally speaking, that is very true. And the reason is we don't want to compound error. We don't want the error of one approximation to affect the error of another. The good news is as we just have to subtract the measure of angle A, which remember was 38.7. So since we just have to subtract that from 90 degrees, the error of B will be no worse than the error of A. So there's no compounding of error here. And so it's, it's okay to do. So the measure of angle B is going to equal 90 degrees, minus the measure of angle A, which is 38, 38.7 degrees, for which then we can easily see that's going to give us that B is approximately 51.3 degrees. And therefore, we finished the missing part of these triangles. And so when it comes to solving for the missing parts of our triangle, it's always these principles here. We can find a missing side by using either the Pythagorean equation, if we know two of the sides or trigonometric ratio, like sine cosine tangent, if we know an angle in one of the sides. If we're missing angles, we can find the third angle by subtracting the two known angles from 180. Or if we only know the right angle, we can use inverse sine inverse cosine inverse tangent to solve those. And so this video gives us a general strategy for finding the missing parts of a right triangle.