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

Question

In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.

a = 63, b = 22, c = 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 angles of the triangle and express the angle to the nearest degree. Now, they tell us different facts about this triangle. For example, they tell us that side P is equal to 24 units. Side Q is equal to 75 units and side R is equal to 58 units. Now, the answer choices provided are a angle, P is 15 degrees angle, Q is 38 degrees and angle R is 100 and 27 degrees and I'll be referencing angles in all of our answer choices. Answer choice B says that P is 15 degrees, Q is 100 and 27 degrees and R is degrees. Answer choice C says that P is degrees, Q is 127 degrees and R is degrees. And finally answer choice D says that P is 38 degrees, Q is 15 degrees and R is degrees. Now, to solve this question, we're going to recall what we know is the cosine rule. And the cosine rule says that let's say you have cosine of an angle A that will be equal to a side B squared plus side C squared minus side A squared. And all of that divided by two multiplied by side B multiplied by side C. Now, we can give this a little bit more context. So let's just uh draw a random triangle of any shape and we can label the sides P Q and R and we can label the angles that correspond to each side. So opposite of side R, we can put angle R because it opens towards the side, R, opposite of angle Q or side Q will put angle Q and opposite of side P will put angle P. Now using the cosine rule that we just talked about and I'll scroll down a bit so we can have a bit of more room to work with. So using the cosine rule that we talked about, let's rewrite this. So let's first tackle angle P. So let's say we had cosine of P that will be equal to and side B and side C in the recall formula, those will be the two sides closest to our angle. So in this case, those will be Q and R. So we'll have cosine of P is equal to, in the numerator, we'll have Q squared plus R squared minus P squared because P is the remaining side and all that will be divided by two multiplied by Q multiplied by R. Now, we can plug in the values that we know since they told us what side P, side Q and side R are. So we can have cosine of P is equal to 75 squared plus squared minus 24 squared. And all of that will be divided by two multiplied by 75 multiplied by 58. And uh we simplify, we'll have that cosine of P is equal to 8413 divided by 8700 which means that cosine of P is around 0.9670. And now to obtain the angle, what we can do is take the inverse cosine of our value. So we'll have inverse cosine of 0.9670. That will be equal to the angle P. And once we plug that into our calculator, we'll have that P is around 14. degrees which rounded to the nearest degree is 15 degrees. And that will be our answer for angle P. And now we can move on and let's try angle R. So we'll have cosine of R is equal to. And once again, we use the side closest to the angle. So in the numerator of a fraction will have P squared plus Q squared minus R squared. And that will be divided by two multiplied by P multiplied by Q. And once we plug those in, we'll have cosine of R is equal to squared plus 75 squared minus 58 squared and all of that divided by two, multiplied by 24 multiplied by 75. Now, if we simplify all of this, we'll have a cosine of R is equal to 2837 divided by 3600. And once we plug that into our calculator, we'll have that cos sign of R is around 0.7880. And once again, we use the same technique to obtain the angle, we'll take the inverse cosine of our value. So in this case, inverse cosine of 0.7880, and that will be equal to R angle R, which means that angle R is around 37. 58 degrees which rounds out to degrees, the nearest whole degree. And that'll be our answer for R. Now, for our final angle. In this case, Q, we don't have to go through this entire process again because of one thing that we know and that is that a triangle is equal to 180 degrees. When you add up all the angles in a triangle, they will add up to 180 degrees. Therefore, we have that angle P plus angle, Q plus angle R is equal to degrees. And if we plug in what we know, we'll have that 15 degrees plus Q plus degrees is equal to 180 degrees, which means that angle Q is equal to 180 degrees minus 15 degrees minus 38 degrees. And once we calculate that we'll have that angle Q is equal to 127 degrees, which will be the answer for our final angle. And once we look at our answer choices, we'll see that answer, choice B matches up with the angles that we calculated. So as you can see by doing a simple recall of the cosine rule, we were able to use that to calculate all of the angles that we needed. 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 = 63, b = 22, c = 50

Key Concepts

Triangle Classification and Properties

Law of Cosines

Rounding and Angle Measurement

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