In Exercises 41–42, find a to the nearest tenth.

Question

Accepted Answer

Hey, everyone in this problem, we're asked to determine the value of A in the following triangle. And so write our answer to the nearest 10th. So we're giving this triangle P QR, OK? And within that triangle, we have this separate triangle, it can be broken into two triangles with this extra corner called S OK. So we have four answer choices. Option A 87.3 option B 86.2 option C 83.7 and option D 85.1. So looking at our triangle, we have side A that we're looking for and for the triangle, PSR that side A is a part of, we have two of the angles 98 degrees and 32 degrees, but no sidelines. Now we have two angles. That means we can find the third one. OK. So we have all three angles if we need, but we don't have a site. So what we need in order to solve for A is another side. So let's go ahead, we're gonna call this middle line that separates the two triangles P. If we can find side P, then we can use our law of signs to find side A because we'll know a side link and the corresponding angle. All right. Now for side P, let's rewrite this triangle. OK, we're gonna rewrite just this bottom triangle. So we have our us, this is a 90 degree angle. We have an eight degree angle. OK. This is a right triangle with side P on the hypos. Now because this is a right angle triangle, we can use our trigonometric ratios. And so we want to relate side P to the side that we know which is this bottom side. Now our eight degree angle, the angle we know is adjacent with the side that we know. So we wanna use co sign. Now we call that cosine of an angle theta it's gonna be the adjacent side invited by the hippos. So in this case, we have cosine of eight degrees is equal to 125 divided by P. This tells us that P is gonna be equal to divided by cosine of eight degrees. And we're gonna leave it in that form to try to avoid as much round off error as possible. OK. So we're gonna leave it in this exact form without ramp. But now we know side P so we know we know all the angles of our triangle or we can find them and we need to find side A. So let's recall the law of science, the law of signs tells us that the ratio of a side length to the sign of its corresponding angle is consistent for all three sides of a triangle. OK. So a divided by sine alpha is equal to B divided by sine beta is equal to C divided by sine of gamma. Now, in this case, we are going to use A and its corresponding angle of 32 degrees. Then we have side P that we know but we need to use the corresponding angle P. Now we have two of the other angles we've said we know how to calculate this angle. So recall that the sum of the angles of a triangle. So angle P plus angle S plus angle are add up to degrees. That tells us that angle P plus 98 degrees plus 32 degrees is equal to degrees. And if we subtract our 98 degrees plus 32 degrees from both sides, then we can find that angle P is going to be equal to 50 degrees. OK. So let's add that way up here to our diagram. So we don't lose track of it 50 degrees. And now what we have is we're looking for a, we know the corresponding angle. If we say that B is equal to P. In this case, then we know side P, we now know angle P and so we can use our law of science and let's do exactly that. So we have side A divided by sine of its corresponding angle which is 32 degrees is going to be equal to side P which we found to be 125 divided by cosine of eight degrees divided by sign of its corresponding angle which we just found to be 50 degrees. Now, you can see we have everything in this equation A is the un only unknown. So we're gonna multiply both sides by sine of 32 degrees. And we get that A is equal to 125 multiplied by sine of 32 degrees, the by cosine of eight degrees multiplied by sine of 50 degrees. And if we work all of this out on our calculator, we get that side A is about 87.32. All right. So we did a lot of things there, let's just kind of refresh. So to find side A, we wanted to use the sign law based on the information we had, but we needed to find one of the side links first. So we used this second triangle using our trigonometric ratio because it was a right angle triangle to find side length P, we found side link or sorry angle P using the fact that the sum of the angles in a triangle add up to 180 degrees. And then we could finally use our sign lot to calculate A which we found to be about 87.3 units which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 41–42, find a to the nearest tenth.
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Key Concepts

Understanding the Variable 'a' in Trigonometric Context

Application of Trigonometric Ratios and Laws

Rounding and Precision in Final Answers

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