In Exercises 54–57, solve the right triangle shown in the figure....

Question

In Exercises 54–57, solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree.

A = 22.3°, c = 10

Accepted Answer

Hello, everyone. We are asked to find the missing side lengths and angles of the right triangle shown. We want to express the lengths in two decimal places and the angles in one decimal place. We are told that the angle at M measures 34.2 degrees inside N lowercase N equals 14. We have a right triangle onm where the angles are labeled with capital letters and N is the right angle. Our sides are lower case N which is our hypotenuse and measures 14 M lowercase M which is across from angle M and lowercase O which is across from angle O. We have four answer choices with slightly varying numbers for the angle O and the sides M and O. The first thing I'm going to do is label what I am given on my figure. So we are given lower case N is 14. So I'm labeling that as my hypotenuse and that the angle at M is 34.2 degrees. I personally see things better when I have it all together. The first thing I'm going to find is the measure of the angle at O. And I recall that using the triangle sum theorem, I know the angles of a triangle, the interior angles of a triangle total to 100 80 degrees. So I can find the angle at O by doing 180 degrees minus our right triangle or our right angle, I'm sorry, of 90 degrees minus M which is 34.2 degrees. And when we subtract all of that, we get that the angle at O must be 55. degrees. So that is part of our answer. The O is 55.8 degrees. I'm gonna label that on our diagram at the angle that matches O and now I'm gonna work on the sides. I'm gonna begin with side M, I notice that it is a cross or opposite angle M and I know I have an a, a hypotenuse. So opposite and hypotenuse this means that I'm gonna be working with sign. So the sign of angle M equals the side opposite it. So side M divided by the hypotenuse which is lowercase N. So this would be the sign of 34. degrees has to equal lowercase M divided by 14. I'm going to use a calculator. If you are also make sure your calculator is in degree mode. And I get that the sign of 34.2 degrees is approximately 0. and that equals M divided by 14. You need to multiply both sides by 14 to cancel it out and get the M by itself. And I find that lowercase M when rounded to two decimal places would be 7.87. So that's gonna be my length for side M, it's approximately 7.87. So we have one angle and one side done. We need to do one more side. We need to find the value of side. Oh, I'm going to still work in relation to angle M O is adjacent to angle M and we know the hypotenuse is 14. So we're gonna work with adjacent and hypotenuse which is cosine. So the cosine of angle M will be side O divided by side N and be very careful that you don't think the lower case O is a zero. If you need to substitute in a different value like an X or A Y, please do so. So we're gonna do the cosine of 34.2 degrees equals lower case O divided by 14. The cosine of 34.2 degrees is approximately 0.8271. And that equals lower case O divided by 14. I'm gonna multiply both sides by 14 to cancel out my division. And when I do lower case O when rounded is approximately 11.58. So I'm gonna label that on my diagram. Also that side O has a measure of 11.58. So I have both my sides and my angle. So I'm looking in my answer choices for one that has the angle at O being 55.8 degrees. So that's either answer choice C or D. Um The measure of side M needs to be 7.87. So that looks like it should be answer choice C and yep, zero equals 11.58 in answer choice C also, so answer choice C is our answer. Have a nice day.

In Exercises 54–57, solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. A = 22.3°, c = 10

Key Concepts

Right Triangle Properties

Trigonometric Ratios

Angle and Side Calculations

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