In Exercises 9–24, solve each triangle. Round lengths to the near...

Question

In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 6, c = 5, B = 50°

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're going to be looking at this question that states find the missing side length and angles of the triangle and express the lengths to one decimal place and the angle to the nearest degree. Now sides are denoted by lower case letters while angles are denoted by uppercase letters. So they tell us that side P is equal to seven, side, R is equal to three and angle Q is equal to 40 degrees. Now they tell us that the S choices are A P equals 118 degrees or angle, P side, Q equals 5.1 units and angle R equals 22 degrees. Now, for the answer choices, P and R will be angles and Q will be the missing side. So answer choice B says that P equals 23 degrees, Q equals 5.1 units and R equals 117 degrees as a choice C states that P equals 23 degrees, Q is 5.1 units and R equals 17 degrees. And as choice D says that P equals 117 degrees, Q equals 5.1 units and R equals 23 degrees. Now, the first thing we will do that we'll do here and we'll use a specific technique here is we're going to be using the cosine laws. So using the cosine laws, we can write three different expressions where we'll have one expression will be Q squared equals R squared plus P squared minus two R P multiplied by the cosine of Q. The second equation will be R squared equals Q squared plus P squared minus two Q P cosine R. And our third equation will be P squared is equal to R squared plus Q squared minus two R Q cosine of P angle P. Now, these are three equations that we are able to write by just recalling our cosine laws, we use our variables to write them and now we can use them. The first thing we're using for is to find side Q. Now, for this one, specifically, we'll use equation number one. So we'll have Q squared equals and now we can plug in the values that we know which will be three squared plus seven squared minus two, multiplied by three, multiplied by seven, multiplied by cosine of angle Q which was 40 degrees. Now, we'll have Q squared is equal to nine plus minus 42 multiplied by cosine of 40 degrees. And we can simplify this to have Q squared equals 25.826, which means that Q is equal to the square root of 25.826, which runs off to be around five point oh eight, which means that Q or side Q is equal to 5.1 once we put it to one decimal place, and now we can move on to find an angle. So I put to find angle P. Once again, we'll use the cosine laws, it'll be a bit rewritten. So for this case, we'll use equation number three where we were, we had cosine of angle P and we'll rewrite our equation to have cosine of P is equal to a fraction where we have in the numerator, Q squared plus R squared minus P squared. And that will be divided by two. QR and we can plug in the values that we know to have that cosine of P is equal to five points one squared plus three squared minus seven squared. And all of that divided by two multiplied by 5.1 multiplied by three. And now we'll have cosine of angle P is equal to negative 13.99 divided by 30.6, which ones we simplify will have cosine of P equals negative 0.4648. And now to obtain the angle, we do what we refer to as inverse cosine, which is noted by an exponent of negative one. So we'll have in inverse cosine of the answer we just obtained. So negative 0. and this will be equal to our angle P. Now, once we plug that into our calculator, we'll have that P is around 117.70 which runs off to 118 degrees. And that will be our answer to four angle peak. Now, lastly, to find angle R, it will be a bit easier since we already have two of our angles. And we know that a triangle is equal to 180 degrees. Meaning that once you add up all the angles, they will add up to 180. Therefore, angle P plus angle, Q plus angle R equals 180 degrees, which means that 118 degrees plus 40 degrees plus R equals 180 degrees, which means that R is equal to 180 degrees minus 100 degrees minus 40 degrees. And once we simplify that we'll be left with R equals degrees, which will be our final answer for the missing angle. Now, when we compare this to the answer, choices, we see that answer choice a matches with what we obtained. And we see how important it is to be able to use the cosine laws to solve for different sides as well as different angles. So I hope that was helpful. And until next time.

In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 6, c = 5, B = 50°

Key Concepts

Law of Sines

Triangle Angle Sum Theorem

Side-Angle Relationships

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