The missing angle is 57 degrees, and the lengths of the other sides are approximately a = 6.03 and c = 6.51.
To solve the given triangle ABC, we need to find the missing angle and the lengths of the other sides.
Step 1: Given information
- BC = 7.8
- Angle B = 75 degrees
- Angle A = 48 degrees
Step 2: Find the missing angle
To find the missing angle, we can use the fact that the sum of the angles in a triangle is always 180 degrees.
Angle C = 180 - Angle A - Angle B
Angle C = 180 - 48 - 75
Angle C = 57 degrees
Step 3: Find the lengths of the other sides
To find the lengths of the other sides, we can use the Law of Sines or the Law of Cosines. Let's use the Law of Sines in this case.
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
sin(A)/a = sin(B)/b = sin(C)/c
We know:
- Angle A = 48 degrees
- Angle B = 75 degrees
- Angle C = 57 degrees
- BC = 7.8 (which is side b)
We can use the Law of Sines to find the lengths of the other sides:
sin(A)/a = sin(B)/b
sin(48)/a = sin(75)/7.8
Solving for a:
a = (7.8 * sin(48))/sin(75)
a ≈ 6.03
Similarly, we can find the length of side c using the Law of Sines:
sin(C)/c = sin(B)/b
sin(57)/c = sin(75)/7.8
Solving for c:
c = (7.8 * sin(57))/sin(75)
c ≈ 6.51