Solve each triangle ABC that exists. B = 34°36' a = 38.5 b =...

Solve each triangle ABC that exists. B = 34°36' a = 38.5 b = 30.9 Select the correct choice below and, if necessary, fill in the answer boxes within the choice. A. There is only one possible solution for the triangle. The measurements for the remaining angles A and C and side c are as follows. A = ° ' C = ° ' c = (Simplify your answer. Round to the nearest degree as needed. Round to the nearest minute as needed.) (Round to the nearest tenth as needed.) B. There are two possible solutions for the triangle. The measurements for the solution with the longer side c are as follows. A1 = ° ' C1 = ° ' c1 = (Simplify your answer. Round to the nearest degree as needed. Round to the nearest minute as needed.) (Round to the nearest tenth as needed.) The measurements for the solution with the shorter side c are as follows. A2 = ° ' C2 = ° ' c2 = (Simplify your answer. Round to the nearest degree as needed. Round to the nearest minute as needed.) (Round to the nearest tenth as needed.) C. There are no possible solutions for this triangle.

Transcript

00:01 We are given a triangle we see with angle b and side a and side b and we need to find all the other measurements and we need to tell whether this one such triangle exists two such triangle exists or whether no possible solution to the triangle exists.

00:18 So it's always a great idea to make a rough sketch of a triangle and indicating what is given and what is not so angle b is 34 degrees 36 minutes side b is 30.

00:32 Side a is 38 .5 and side c is unknown so i think it's a good idea if we can use law of signs because one opposite pair is given to us and another opposite pair we can use this one and get the value of angelaide so using law of signs we write uh 30 .9 over sign of 34 degrees 36 minutes is equal to 38 .5 over sign of angle a.

01:05 So if we cross multiply, sign of angle a is going to be 38 .5 times sign of 34 degree 36 minutes over 30 .9.

01:17 So from here, sign of a comes out as let me just grab my calculator, 38 .5 over 30 .9 times.

01:27 Okay, so if we have a problem converting the finding this since there is involvement of minutes so let me just help you out there because in order to convert minutes to degrees you we just have to divide by 60 so that is where it's converted into degrees so that's what we have to find so time sign of 34 plus 36 over 60 so this is coming as 0 .7075 7075 and i just have to take an sign inverse of both the sides to get the value of a but there is definitely one thing which which which we need to take care over here sign is positive in quadrant one and quadrant two it means that angle a can be acute as well as optives so the acute angle will be 45 .03 degrees and another value of possible value of angle a is 180 minus 45 .03 degrees, which is going to be 134 .97 degrees.

02:38 134 .97 degrees.

02:41 All right.

02:44 So these are the two possibilities.

02:48 Now, if we talk about the first possibility, and i think if they have given in degrees, and they are saying that we have to round our answer to the simplify your or.

03:04 Answer and round to the nearest degree as needed around to the nearest minute as needed that's what we have to take care for the angles so let me do that uh i'm going to write two cases here case one is when angle a is 45 degrees 45 .03 degrees but i'm going to write it again uh in terms of degrees and minutes so that is definitely 45 degrees and the minutes will be converted uh sorry the rest degrees will be converted into minutes by multiplying with 60 that is coming as to the nearest minute that is coming as two minutes this is angle a one possible value and case 2 is when angle a will be let me just get this value of 134 .97 that's going to be sign inverse of 0 .7075 180 minus this value so this is coming as 134 degrees and 9 97 minutes, sorry, 97 .97, which will be converted into minutes.

04:11 I'm multiplying with 60.

04:13 So that's coming as 58 minutes.

04:15 Case 2 is this.

04:16 So angle c, let's call it a1 and let's call it a2.

04:20 So angle c1 is going to be, angle c1 is going to be, remember that the one which they are calling a1 is having the longer side c.

04:33 So let's see, whatever c comes out...

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