Find the unknown angles in triangle ABC for each triangle that...

Question

Find the unknown angles in triangle ABC for each triangle that exists.

B = 74.3°, a = 859 m, b = 783 m

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine the measure of the remaining angles of a triangle. ABC with the measurements angle B being equal to 81.8 degrees A equals 897 m and B is 824 m long. For our answer choices A says that angle A is 38.5 degrees and angle B is 59.7 degrees. B says angle A is 59.7 degrees and angle B is 38.5 degrees. C says angle A is 19.1 degrees and B equals 97.8 degrees. And D says that no triangle exists no first to get an idea of what's really going on here. Let's assume that a triangle exists. And so let's try to sketch a diagram of that triangle. So we're seeing here that for our triangle, OK. Let's let me dry the chang here a little bit looking like this. We're saying that it has an angle B of 81.8 degrees. OK? And it has other sides, other angles, sorry A and C and by how we write or rather the naming convention we follow for triangles is that the triangle with the same name or the side with the same name as an angle, but in common letters is opposite to that angle. So if angle A is here, then side A is going to be opposite to it. Side A is 897 m. And if angle B is 81.8 degrees, side B is opposite to it and equal to 824 m. And you know, we're trying to find the triangle, OK, are the remaining not the triangle, the remaining angles of that triangle. Now here, let's think about the information we have notice that we have a pair of angle and side that are opposite to each other. And for another angle, we don't know what the angle is, but we know the side that's opposite to it. So what do we know about sides and opposite angles in a triangle? Well, the love of science can help us recall that by the law of science, the Law of Science is that the ratio of the sign of an angle to its opposite side in a triangle is the same throughout the entire triangle. So in other words, the Law science tells us that the sine of A, the ratio of the sine of A to side A is the same as the ratio of the sine of B to side B, which is the same as the ratio of the sine of C to side C. Now, here we want to figure out the value of the anger A and we already, well, we don't know it. I should say we want to figure it out and we already know the length of side A, the value of sides angle B and the value of side B. So we can use that to help us to figure out our unknown angle A because now by the law of signs, this tells us that the sine of A to side A which is 897 m equals the sine of B which is 81.8 degrees to the value of side B which is 824 m. Now we can solve for A, if we multiply both sides by 897 then the sine of A equals 897 multiplied by the sine of 81.8 degrees divided by 824. Now, when we work this expression, not, we get the sign of A to be equal to, on your calculator, you should get 1.077 for 6271738. And for some of you, that value continues. So therefore, if the sign of A is equal to this value, that means A would be equal to the inverse sign of 1.07746271738. But here's the thing about that. When we put this into our calculator, we're going to get an error. And that's because the value of sign can't be more than one. So since the value of sign can't be more than one, there is no possible value for a, OK, there is no possible value for A and thus, OK. Thus, that tells us that the triangle is not possible, no triangle exists. Therefore, the correct answer is choice D thanks a lot for watching everyone. I hope this video helped.

Find the unknown angles in triangle ABC for each triangle that exists.

B = 74.3°, a = 859 m, b = 783 m

Key Concepts

Law of Sines

Triangle Angle Sum Property

Ambiguous Case of the Law of Sines (SSA Condition)

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