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 = 3, b = 9, c = 8

Accepted Answer

Hello everybody. I hope you are 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 angles to the nearest degree. Now, they tell us that they're denoting the sides by a lower case letter and they're denoting the angles with a upper case letter. Now they tell us that the sides, side, P equals four, side, Q equals seven and side R equals 10. Now, for the answer choices, they are only referencing the angles and we have that angle P is equal to degrees. Angle, Q is equal to 129 degrees and angle R is equal to degrees for answer choice, A as a choice, B says that P equals 19 degrees, Q is 129 degrees and R equals degrees. As a choice C says that P is 33 degrees, Q is 100 29 degrees and R is A is degrees. N D says that P is 18 degrees, Q is 33 degrees and R is equal to degrees. Now, I'm going to scroll down a bit just to have a little more room to work with for this question. And the first thing that we do is do a simple recall of a rule that we can use to solve this. And that is the cosine rule which says that let's say you have cosine of an angle A that will be equal to a fraction that has in the numerator side B squared plus side C squared minus side A squared. And that'll be divided by the denominator which will be two multiplied by side B multiplied by side C. Now sides B and C will be the sides closest to the angle in this. In, in the, in this case, in the recall formula angle A and this will be a little more clear as we go along. And let's say, for example, for our question, let's just draw a random triangle in general shape and let's assign the sides and the angles. Now let's just say side P, side Q and side R. And to assign the angles, we just look at the opposite angle to that side. So which angle is opening up to side R, we label that angle R opposite of side Q, we have angle Q and opposite of side P, we have angle P. No, we can use our recall formula to, let's say soft wrangle P instead of cosine up A, we'll have cosine of P now. And that will be equal to the fraction. And as we said, B and C in the recall formula represent the sides closest to the angle we're solving for. In this case, those sides are Q and R. So we have that co sign of P is equal to, in the numerator, we'll have Q squared plus R squared minus P squared. And that'll be divided by the denominator which will be two multiplied by Q multiplied by R. Now, they told us the values of the sides so we can plug those in and we'll have that cosine of P is equal to seven squared plus 10 squared minus four squared. And all of that divided by two multiplied by seven multiplied by 10. And once we plug that into our calculator, we'll have a fraction that says cosine of P is equal to 133 divided by 140. Now, we simplify that we'll have that co sign of P is around 0.95. And now to obtain the angle, we do what we call what we refer to as inverse cosine. So have inverse cosine of our value. 0.95 will be equal to our angle P. Now, after plugging that into our calculator, we'll have that P is around 18 point 195 degrees which rounded to the nearest degree is 18 degrees. And that will be our first answer answer to for answer for angle P in this case. And we can move on to, let's say angle R, we'll have cosine of R is equal to our fraction. And the closest sides to angle R are side P and side Q. So we'll have that close set of R is equal to our fraction with the numerator having P squared plus Q squared minus R squared. And that will be divided by two multiplied by P multiplied by Q. Now, once we plug in the values that we know we'll have cosine of R is equal to four squared plus seven squared minus squared. And all of that divided by two multiplied by four, multiplied by seven. And once we simplify that we'll have cosine of R is equal to negative 35 divided by 56. And once again to solve for the angle, we do what we call inverse cosine. So we have inverse cosine of our fraction which once it's simplified it is negative 0.625. And that will be equal to the angle R. And once we plug that into our calculator, we'll have angle R is around 128.68 degrees which rounds off to 100 and degrees. And that will be our answer for angle R. Now, for the very last angle, we don't have to go through the cosine rule again because there's another rule that we know that says the A triangle is equal to degrees So all the angles within the triangle will add up to 180 degrees. Therefore, we'll have that angle P plus A plus angle Q plus angle R is equal to 180 degrees. Or that 18 degrees plus Q plus 100 degrees will be equal to 180 degrees. Which means that angle Q is equal to 180 degrees minus 18 degrees minus 129 degrees. And once we simplify that we'll finally have that angle Q is equal to 33 degrees. And that will be our final answer for the angles that we are looking for. And when we compare this to the answer choices, we see that answer choice D matches up with what we solved for. So we can see how important it is to be able to recall the cosine formula to software different angles as well as knowing how a triangle works and how the angles add up to 1 80. 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 = 3, b = 9, c = 8

Key Concepts

Law of Cosines

Law of Sines

Triangle Classification

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