In Exercises 1–12, solve each triangle. Round lengths to the near...

Question

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle.B = 37°, a = 12.4, b = 8.7

Accepted Answer

Hey, everyone in this problem, we're asked to identify whether the following measurements of two sides and an angle can produce one triangle, two triangles or no triangle. And to find the missing side links and angles of those triangles or of the triangle. Whereas to express the lengths and angles in one decimal place, we have angle B is 32 degrees. Side A is 15.7 and side B is 11.2. We're given four answer choices. We're gonna come back to these in just a minute. Option A and B have that we have one triangle. Option C is that we have two triangles. And option D is that we have no triangle and then A B and C give those additional um measurements as well. So what start now, we have angle B and side B and then side A. So by having a corresponding angle and a side, what we can do is use the law of sine and recall that the law of sines tells us that we can write sine of angle A divided by A is equal to sine of angle B divided by B. The ratio of sine of an angle to its corresponding side is equal for all three of the angle inside in our triangle, right. So we are given angle B side B side A. This sign law allows us to calculate angle A. So let's go ahead and do that. We'll substitute in our values. We get sine of angle A divided by 15.7 is going to be equal to sine of angle B 32 degrees divided by SB 11.2. If we multiply both sides by 15.7 we have that sine of angle A will be 15.7 multiplied by sine of 32 degrees divided by 11.2. We can simplify and take the inverse sign on both sides. And we're gonna get that angle A is equal to 47 0.97 degrees. Now we have to be careful here when we get this A value, there is actually another possible value for A as well because sign is positive in these two quadrants in a row. OK. So in quadrant two and quadrant three, we can also have angle A as 132.03 degrees, which still falls under that 180 degrees. It still makes it possible to be an angle in a triangle. And so in this case, we actually end up with two possible angles for A and that kind of gives us a hint that we're probably gonna end up with two triangles here. So we're gonna look at case one where we have A is equal to 47.97 degrees. And then we'll look at case two or we have A as 132.03 degrees. OK. So let's start with this case one. And again, we have A is equal to 47.97 degrees. Now, we have angle A and angle B, side A and side B. So we need to figure out angle C and side C in the simplest approach here, once we have two angles, we can figure out the third very easily. OK. So we know that the angles in a triangle add up to 180 degrees. So angle C must be equal to 180 degrees minus angle A minus angle B, it's gonna be 180 degrees minus 47.97 degrees minus 32 degrees. So angle C in this case will be equal to 100.03 degrees. Now that we have Anglesey, we can use that law of signs again in the exact same way we did before. But now we're gonna use it to find side C. What we're gonna do is relate side B to side C, we could do it with side A as well. We just have to remember that when we found angle A, there's gonna be a little bit of round off error from doing the calculation. So if we use side B and we were given both of those values in the question, we can avoid a little bit of that extra round off error. And again, either one will work, but we'll try to avoid the round off error as much as we can. So we get sign of angle B divided by side B is equal to sine of angle C divided by side C substituting in our values sine of 32 degrees divided by 11.2 is equal to sign of 100.03 degrees divided by side C. We're gonna multiply both sides by see we're gonna multiply both sides by 11.2. And then we're gonna divide by sine of 32 degrees. And what we are gonna get is that C is going to be equal to 11.2 multiplied by sign of 100.03 degrees divided by sign of 32 degrees which gives us the C value of 20.812. OK. So for case one, we have everything we need, we have angle A, we found angle C, we found side C. Those are the three unknowns. Now we can do the same for case two. And let's see if case two gives us another triangle. So for case two, we have that th uh angle A where A should be equal to 132 0.03 degrees. We're gonna do the exact same process. We can find Anglesey by taking 180 degrees and subtracting the other two angles. Angle A and angle B tells us angle C will be 180 minus 132.03 degrees minus 32 degrees. OK. So angle C will be 15.97 degrees. And so again, we have angle A, we now have angle C and so these, this triangle is going to work out. We have three angles that's gonna form our triangle. We just need to figure out this last side length. And again, we can do that using the law of science sine of angle B divided by B will be equal to sine of angle C divided by C sign of 32 degrees divided by 11.2 is equal to sine of 15.97 degrees divided by side C rearranging just like before side C will be 11.2 multiplied by sign of 15.97 degrees divided by a sign of 32 degrees. When we work this out on our calculator, we're gonna find that side C in this case is gonna be about 5.81 50, right. And that's it. Those are all of the missing values for both of those triangles that we can form. If we take a look at the answer choices we were given. OK, we can already narrow it down to being answer choice C just based solely off the fact that option A and option B only give us one triangle and option D gives us no triangles. But we found two. And we can look further to make sure that everything we've done is correct. And we're gonna round our answers and we can see that we have two triangles. Hey, the first, we have angle A with 48 degrees angle C 100 degrees and side C 20.8 units. Exactly what we found. The second triangle A is 132 degrees C is 16 degrees and side C, it's 5.8 units again, exactly what we found. And so option C is going to be the correct answer here. Thanks everyone for watching. I hope this video helped you in the next one.

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle.B = 37°, a = 12.4, b = 8.7

Key Concepts

Law of Sines

Ambiguous Case of the Law of Sines

Triangle Sum Theorem

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