Determine the number of possible solutions if A = 48°, a = 5, and b = 6.
To determine the number of possible solutions, we need to analyze the given information and apply the Law of Sines or the Law of Cosines to solve the problem.
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. In other words:
a / sin(A) = b / sin(B) = c / sin(C)
Here, A, B, and C represent the angles of the triangle and a, b, and c represent the lengths of the corresponding sides.
Let's use the Law of Sines to find the unknown angle B. We have A = 48°, a = 5, and b = 6:
5 / sin(48°) = 6 / sin(B)
To solve this equation, we can cross multiply:
5 * sin(B) = 6 * sin(48°)
sin(B) = (6 * sin(48°)) / 5
Now, let's find the value of sin(B):
sin(B) = (6 * sin(48°)) / 5
sin(B) ≈ 0.723
To find the angle B, we can take the inverse sine (sin^(-1)) of 0.723:
B ≈ sin^(-1)(0.723)
Using a calculator, we can determine that B ≈ 46.76°.
Now that we have the values of angles A and B, let's find angle C:
Since the sum of the angles in a triangle is always 180°, we can calculate C:
C = 180° - A - B
C = 180° - 48° - 46.76°
C ≈ 85.24°
Now we have all three angle measures: A = 48°, B ≈ 46.76°, and C ≈ 85.24°.
Since the angles in a triangle determine the shape and size of the triangle, we have determined the unique triangle that satisfies the given information.
Therefore, there is only one possible solution (one unique triangle) if A = 48°, a = 5, and b = 6.