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

C = 41° 20', b = 25.9 m, c = 38.4 m

Video transcript

Welcome back. Everyone in this problem, we want to determine the measure of the remaining angles of a triangle. ABC with the measurements that says angle C equals 32 degrees and 35 minutes. Side B is 17.8 m long and side C is 25.6 m long. For our answer choices. A says that Anger A equals 125 degrees 26 minutes and B equals 21 degrees. 59 minutes. B says A is 21 degrees 26 minutes and B is 125 degrees. 59 minutes. C says that A is 21.59 degrees and angle B is 125 degrees 26 minutes. And D says that no triangle exists. Now, if we're going to determine the measures of the remaining angles, let's try to sketch what's really going on here. So what we're saying is that we have a triangle, OK? We have a triangle, we're for our triangle, we have an angle C being 32 degrees and 35 minutes I think. Let me try to draw it bigger just to make sure I have enough space here. So we're saying it's 32 degrees and 35 minutes. OK? We have the side B so that's Anger C let's label Anger A and Anger B OK? And we know that based on how we labor triangles, if angle C is right here, then the side sea is opposite to angle sea. And we know the side sea is 25.6 m. We also know that if, if angle B is here, then side B is opposite to that. And side B is 17.8 m as given here. And we're trying to find out the values of A and B. Now, if we're going to figure out the values of angles A and B first, let's think about what we have here. So we're talking about a triangle. OK? And we have a known pair of side with opposite angle. Now, in our second scenario, if we're solving for angle B, let's say here, we know the side, but we don't know the angle. So if we have a known pair of angles and sides, how could we use that to figure out an unknown angle? Well, we can use this law of science because recall that by the law of science, the law of science tells us that the ratio of the sine of an angle to its opposite side in a triangle is the same or throat. In other words, the sine of A to side A is equal to the sine of angle B to its opposite side B which is equal to the sine of angle C to its opposite side C. Now, here we have, well, we have angle C, we know angle C, we know side C, we know side B but we don't know angle B. So we can use this idea here to help us to solve for angle B because by the law of signs, it tells us then that the sign of anger B two, it's opposite side. 17.8 m is going to be equal to the sine of angle C which we know is 32 degrees and 35 minutes two. It's opposite side C which is 25.6 m. Now here, if we cross multiply or if we, well, we don't even have to cross multiply multiply, we can multiply both sides by 17.8 which now tells us then that the sine of B is going to be equal to 17.8 multiplied by the sine of 32 degrees and 35 minutes. And all of that is going to be divided by 25.6. Now, if the sine of an angle gives us that expression to figure out what that angle B is, we can find the inverse side of our express sine of our expression. So B is going to be equal to the inverse sine of our expression here, it's a lot to say. So I'm just going to write it. But it's the same expression that we've been working with. And the reason why we say it like this is or why we keep it like this is so that we can find the exact value when we try to solve it with our calculator. So now let's go ahead and put this expression into our calculator to see what value we get for our b keeping in mind that our angles in our answer have been expressed in terms of degrees and minutes. Now, when we put this expression into our calculator, you should find that angle B, OK. Angle B is equal to 21 degrees and 59 minutes. So that's the value of Anger B. Now that we have angle B, OK? Now that B is known here, so let's say put it here, we can use that idea to help us solve for a. Now remember we're working with a triangle here and for this triangle so far, we have two known angles. What do we know about angles in a triangle? Angles in a triangle sum to 180 degrees. So using that property, OK. So let's go a bit to the right, using the ango some property of triangles, then we know that all three angles will soar to 180 degrees, which means then that angle A plus 21.5 21 degrees and 59 minutes. OK? Plus 32 degrees. II I should have put this in red and 35 minutes equals 180 degrees. Therefore, angle A, let me get my red here. Angle A is going to be equal to 180 degrees minus the sum of those angles 21 degrees, 59 minutes and 32 degrees. 35 minutes. Now, when we try to solve that here, let me put that into our calculator. We should find that Anger A equals 125 degrees and 26 minutes. So those are two possible values for Angus A and B. No, the thing is just to make sure we want to check if there are any other possible triangles and any other possible values for A. Now, if another value exists, if we subtract or sorry, if another value for B exists, if we subtract it from 180 degrees and add it to C, then it would be less than 180 degrees. Because remember the sum of angles in a triangle has to be equal to 180 degrees. So we couldn't add two angles for it to be more than 180 degrees. So let's try to find that other value for B and check if it is less than 180 degrees, check if that sum is less than 180 degrees. No, this tells us that here the other value for A B is going to be equal to 180 degrees, ok? Minus 21 degrees and 59 minutes, which equals 158 degrees and one minute. Therefore, that means Anger B plus Anger C is going to be equal to 158 degrees and one minute plus angle sea which is 32 degrees and 35 minutes, which would be equal to 190 degrees and 36 minutes. Now, here, the sum of those two angles are greater than 180. And if they are greater than 180 that means if we add another angle to it, it would also be more than 180 it couldn't be equal to 180. So this tells us then that 190 degrees, sorry, uh 158 degrees and one minute could not be a valid measure for a bee. Therefore, the only valid measures for B and A are 21 degrees, 59 minutes and 125 degrees 26 minutes. If we look back on our answer choices, that would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.