Solve the right triangle shown in the figure. Round lengths to...

Question

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 = 52.6°, c = 54

Accepted Answer

Hello, everyone. We are asked to find the solution for the right triangle. In the given figure, expressing the angles to the nearest 10th of a degree and rounding the lengths to two decimal places. We have a right triangle, QR P with the right angle at R and lowercase letters PQ and R marking the sides with lowercase R as our hypotenuse. We are given that the angle at P measures 55. degrees. And that side R which is our hypotenuse measures 73. We have four answer choices with varying measures for both the angle Q insides P and Q. I'm gonna begin by finding the measure of the angle at Q. So I know that the sum of the interior angles of a triangle total 100 80 degrees. So we can work backwards from there and do 180 degrees minus our right angle of 90 degrees minus the angle at P which we were given is 55.7 degrees. And that will tell us what the measure of angle Q will be. So when we work backwards, we find that the measure of the angle at Q is 34. degrees. So I'm gonna label that on our figure 34. degrees. I'm just gonna label everything else we were given. So the angle at P is 55.7 degrees and side R is 73 as a measure. So we need to find other items. We need to find sides P and Q. So I'm gonna find side P next and using my right angle trig formulas, I know that I can use the sign of P to find side P because sine is opposite over hypotenuse or opposite, divided by hypotenuse. So if the sign of angle P is the opposite side divided by the hypotenuse, we can say that this is the sign of 57. degrees equals our unknown side, which is P divided by our hypotenuse, which is 73. So now I'm gonna put the sign of 55.7 degrees in a calculator, make sure your calculator is in degree mode. And I get that, that will equal 0. which will equal P divided by 73 next to get P by itself. I'm gonna multiply both sides by 73. And when I do so I get that P is 60.3053. Our direction is asked to have lengths rounded to two decimal places. So side P would be 60.31. So we now know the angle Q and we know the side at P. So I'm gonna put that on our diagram also that it is 0.31. So everything's in one place. And then I need to find side QE I could either use the Pythagorean theorem because I know one side and the hypotenuse I could solve for the missing side or I could use another trig identity. So I'm gonna use trig to find side Q because I know that is the adjacent side to P. And I know that cosine of P would be the adjacent side divided by the hypotenuse. So this seems just as easy as the Pythagorean theorem for me. So I'm gonna find the cosine of 55. degrees. It's gonna equal P, I'm sorry, not P. We just did PQ divided by the hypotenuse which is 73. So finding out what the cosine of 55.7 degrees is using a calculator, we find that it is 0. and that will equal Q divided by 73. We're gonna multiply both sides by 73 again, and when I do Q will equal 41.1355. And again, we want our sides rounded to two decimal places. So Q would be 41.14 for length. So we will highlight that. And then I'm gonna put that on my diagram also just so all my information is in one place. And then let's see which answer choice this matches with though, starting with angle Q being 34.3 degrees. So that's gonna be, could be answer choice B through C or sorry BC and D. So let's see what comes next. Uh The side P is 60.31. So that limits us down to answer choice B or D. And if side Q is 41.14 that means it's answer choice B. So angle Q is 34.3 degrees. Side P is 60.31 and side Q is 41.14. Have a nice day.

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 = 52.6°, c = 54

Key Concepts

Right Triangle Properties

Trigonometric Ratios (Sine, Cosine, Tangent)

Angle and Side Calculation Using Trigonometry

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