Given ABC with a = 7, b = 5, and mA= 19° , find the number of distinct solutions

I got 3 solutions

law of sines:

a/SinA=b/SinB=c/SinC
7/sin19deg=5/sinB
sinB=5*sin19deg/7=.233
B= arcsin.233=13.5 deg
Now you have A, B, a, b.
C=180-A-B
now solve for c using law of sines.

Your data illustrates the ambiguous case.

How can you have 3 solutions ???

To determine the number of distinct solutions in triangle ABC, we need to consider the given information:

1. Side lengths: a = 7 and b = 5.
2. Angle: mA = 19°.

Based on this information, let's analyze the possibilities:

Case 1: Side-Angle-Side (SAS) Congruence
Using the given information, we have an angle A (mA = 19°) and the corresponding side length a (a = 7). Additionally, we have side b (b = 5). By the SAS congruence, this information is sufficient to determine a triangle. Thus, we have one solution.

Case 2: Angle-Side-Angle (ASA) Congruence
In this case, we have angle A (mA = 19°), followed by the side length b (b = 5), and then angle B (since the sum of the interior angles of a triangle is 180°, angle B = 180° - 19° - 90° = 71°). Using ASA congruence, we can determine another triangle with this information. Therefore, we have a second solution.

Case 3: Two Triangles
In this case, we consider ambiguous case scenario, where there can be two different triangles formed with the given data. We already have solution 1 and solution 2. Now, let's consider a second angle A (mA = 19°) with a third side length c. We know that the sum of angles in a triangle is 180°; therefore, angle C = 180° - 19° - 90° = 71°. However, we need to determine the value of side c so that the given lengths can form a triangle.

To do this, we can use the Law of Sines, which states that the ratio of a side length to the sine of the opposite angle is the same for all three sides:

sin(A)/a = sin(B)/b = sin(C)/c

With A = 19°, a = 7, B = 71°, and b = 5, we can solve for c:

sin(19°)/7 = sin(71°)/5

By using trigonometric identities, we can find two solutions for c. Thus, we have two additional solutions for this ambiguous case.

Therefore, the total number of distinct solutions is 1 + 1 + 2 = 4.