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 = 5, b = 7, C = 42°

Accepted Answer

Hello everybody. I hope you're doing. All right. 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 length to one decimal place and the angle to the nearest degree. Now, the side lengths will be represented by a lower case letter and the angles will be represented by upper case letters. So the information that they give us is that side, P is nine units. Side Q is eight units and angle R is 72 degrees. Now the answer choices give us angles, P and Q and side length R with answer choice A saying that P equals 49 degrees, Q equals 59 degrees and R equals 10, B as for choice, B says that P equals 59 degrees, Q equals 49 degrees and R equals 10. As a choice C says that P equals 59 degrees, Q equals 10 degrees and R equals 49. And as choice D states that P equals 10 degrees, Q equals 49 degrees and R equals 59. Now, the first thing we'll do in this question is use these specific laws that we know and will be useful in this case. And that specific or those specific laws are called the cosine laws. So using the cosine laws, we will write three formulas with our variables. So equation number one will be side length Q squared is equal to side R squared plus side P squared minus two, multiplied by side R and side P multiplied by cosine of angle Q. Now the other two equations. So equation number two, now it'll be the same idea as equation number one, except the variables will be moved around a bit. So in this case, let's say we're solving for side length R squared that will be equal to side Q squared plus side P squared minus two multiplied by Q side Q multiplied by side P multiplied by cosine of angle R. And finally, for the third equation with the only side we're missing is sideline P squared, love equal to side R squared plus side Q squared minus two multiplied by sides R and Q multiplied by cosine of angle P. Now we can use these equations to go ahead and start solving. So for example, let's say we're trying to find side R. So to find side R, we just look at our equations and which one is solving for sin R that is equation number two. So we use equation number two where we have side R squared is equal to. And now we plug in numbers that we know, so that will be equal to eight squared plus nine squared minus two, multiplied by eight, multiplied by nine, multiplied by the cosine of 72 degrees. And we can start simplifying. So we'll have that R squared is equal to plus 81 minus multiplied by cosine of 72 degrees. And if we simplify further using our calculator, we'll have that R squared is around 100.501. Now to solve for R, we take the square root of both sides. So we have that R is equal to the square root of 100.501 which is around 10.025. And since they asked us to express the length to one decimal place, we will run this to R equals 0.0. So 10 and now we have solved for side R and we can move on and solve for, let's say angle P. So to find angle P and you can solve for either angle P or angle Q, I'm choosing to solve for angle P. In this case, no reason why just what I decided to do. Now, we look at our cosine laws, we see that equation number three has cosine of P in its equation. So if we rearrange it, we can set our equation equal to cosine of P. And after rearranging, we'll have cosine of P is equal to a fraction where in the numerator, we'll have Q squared plus R squared minus P squared. And that will all be divided by the denominator which will be two multiplied by Q multiplied by R and all those were sides. So we can plug some in. So we have cosine of angle P is equal to eight squared plus 10 squared minus nine squared divided by two multiplied by eight multiplied by 10, which will leave us with cosine of P is equal to divided by 160. Which if we simplify a bit, we'll have, we'll have that cosine of P is around 0.51875. Now we'll take the inverse cosine of our value to be able to solve for the angle. Now, inverse cosine is denoted by a negative one exponent. So we have cosine to the negative one exponent of 0.51875 and that will be equal to our angle peak. Now, after plugging that into our calculator, we'll have that angle P is around 58. which they asked us to round to the nearest degree. So it will round out to be around 59 degrees. And that will be our answer for angle P. And finally, we can move on to find our missing angle. In this case that will be angle Q if you took the other route and solve for Q first, now you would be solving triangle P but you would use the same way. So this will define the last angle. It's a bit easier because we know that the angles inside of a triangle will all add up to give us 100 and degrees. So what does this mean? Well, therefore, we have PP plus or angle P plus angle, Q plus angle R that will be equal to degrees. And we already have two of these angles so we can just solve for the last one. Now, we plug in our values, we'll have that 59 degrees plus angle Q plus 72 degrees is equal to 180 degrees, which means that Q is equal to or angle Q is equal to 180 degrees minus 72 degrees minus 59 degrees. Which once we plug into our calculator and simplify, we'll have that angle Q is equal to degrees and that will be our final answer. So now we saw for the missing side and the missing two angles and when we look at the answer choices, we see that answer choice B matches up with our answers. Angle P is 59 degrees. Angle, Q is 49 degrees and side R is 10. units. So this has shown us how important it is to be able to know how to write the cosine laws to manipulate them to solve for the different angles and different sides. Of a triangle. 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 = 5, b = 7, C = 42°

Key Concepts

Law of Sines

Triangle Sum Theorem

Side-Angle Relationships

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