Solve for the ambiguous triangle.
1. Given: A=45°; a=3; b=5
2. A=25°; a=3; b=6
To solve for the ambiguous triangle, we will use the Law of Sines and the Law of Cosines. The ambiguous case occurs when you are given an angle, a side opposite that angle, and another side adjacent to that angle. This situation can yield two possible triangles.
1. Given: A = 45°, a = 3, b = 5
To solve for the triangle in this case, proceed as follows:
Step 1: Apply the Law of Sines to find angle B:
sin(B) / b = sin(A) / a
sin(B) / 5 = sin(45°) / 3
To find the value of sin(B), cross-multiply:
3 * sin(B) = 5 * sin(45°)
Divide both sides by 3 to isolate sin(B):
sin(B) = (5 * sin(45°)) / 3
Now, take the inverse sine (sin^(-1)) of both sides to calculate the value of angle B:
B = sin^(-1)((5 * sin(45°)) / 3)
B ≈ 73.57°
Step 2: To find angle C, use the fact that the sum of angles in a triangle is 180°:
C = 180° - A - B
C = 180° - 45° - 73.57°
C ≈ 61.43°
So, for the given triangle, angle B ≈ 73.57° and angle C ≈ 61.43°.
2. Given: A = 25°, a = 3, b = 6
In this case, we can follow a similar process:
Step 1: Apply the Law of Sines to find angle B:
sin(B) / b = sin(A) / a
sin(B) / 6 = sin(25°) / 3
To find the value of sin(B), cross-multiply:
3 * sin(B) = 6 * sin(25°)
Divide both sides by 3 to isolate sin(B):
sin(B) = (6 * sin(25°)) / 3
Now, take the inverse sine (sin^(-1)) of both sides to calculate the value of angle B:
B = sin^(-1)((6 * sin(25°)) / 3)
B ≈ 64.93°
Step 2: To find angle C, use the fact that the sum of angles in a triangle is 180°:
C = 180° - A - B
C = 180° - 25° - 64.93°
C ≈ 90.07°
So, for this given triangle, angle B ≈ 64.93° and angle C ≈ 90.07°.
In both cases, we have solved for all angles in the ambiguous triangle.