In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Video transcript

Hey, everyone in this problem, we're asked to find the missing sideline and angles of the triangle given here and to express the length to one decimal place and angles to the nearest degree. So we have triangle ABC given in this diagram, we're told that angle A is 76 degrees side B is five and side C is three. And so there are three things we need to find. We need to find side A, we need to find angle B and we need to find Anglesey and we have four answer choices in this problem A through D and we're gonna come back to those as we work through this problem. Now, we have two side links and the enclosed angle. So what we can do is use our log cosines to find that third side. OK. Now log cosines says that the side we are looking for which is A squared is gonna be equal to the sum of the other two sides squared B squared or C squared minus two multiplied by the product of those two other sites. So B multiplied by C multiplied by cosine of the enclosed angle A. OK. So we have A squared is equal to B squared plus C squared minus two BC cosine of A. And we have all of these values except for a little A. So we are gonna substitute in everything we know and solve for that value of A. We get that A squared is equal to five squared. Lost the squared minus two, multiplied by five, multiplied by three, multiplied by cosine of 76 degrees. OK. Simplifying a little bit, we have a squared is equal to 25 plus nine minus 30 sine of 76 degrees. And if we use our calculator and we simplify on the right hand side and take the square root, we're gonna take the positive route because we're talking about a length here and we get that A is equal to approximately 5. units. OK? So that's the missing side length we were looking for. If we compare this to our answer choices, we round to one decimal place, we can see that we have either option C or D, OK. Option A has that the side length A is step and D that's not what we found. So we can eliminate that. And option B has that, that length is 2.5. Again, not what we found, we can eliminate that. So we're looking at answers C or D. Now let's move on to finding these missing angles. So the first angle we're gonna find would start with B. Now we know sign A and angle A now and we also know side B, OK. With those three things, we can use our lost signs, OK? Looking at the ratio of the sign of an angle to its sideline in our triangle, right? So recall that the law of science tells us that sign of A divided by A is equal to sin of B divided by B. And again, we know angle A guide A and side B. The only thing left in this equation to find is angle B. So let's substitute in our values to get that signed of 76 degrees divided by 5.17 is equal to sign ob divided by five, multiplying both sides by five, gives us a sign of B is equal to five fine of 76 degrees divided by 5.17. And if we work this out on our calculator, OK. Taking the inverse sign of both sides, we get that B is equal to 69.78 degrees. Now again, rounding to the nearest degree, we have that B is gonna be 70 degrees. And we're left with finding our last angle which is going to be C. Now, there are two kind of ways we can go about this. We can use our law of signs again to find Anglesey or we can use something else. That's a little bit simpler recall that the sum of the angles in a triangle is 180 degrees. So we know that angle A what angle be? Fourth angle C must equal 180 degrees. We know that angle A is 76 degrees. We have that angle beat is 69. degrees. We're left with finding angle C OK. So 76 degrees plus 69. degrees plus angle C is gonna be equal to 180 degrees. Subtracting 76 degrees and 69.78 degrees from the left hand side to move it over to the right hand side. We get that angle C is going to be equal to 34 0. degrees. Yeah. All right. So that's it. We found our missing side length, we found our two missing angles and we found that our side length was equal to approximately 5.2 which narrowed us down to option C or D. Then we found that angle B was equal to about 70 degrees with angle C being equal to about 34 degrees. This corresponds with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.