Remaining side (c): 45.17 miles
Remaining angle A: 42.0 degrees
Remaining angle B: 89.3 degrees
Solving for the Remaining Side and Angle in a Triangle
Given:
Triangle with sides:
a = 60.12 miles
b = 40.23 miles
Angle C (γ) = 48.7°
To find:
Remaining side (c)
Remaining angles (A and B)
Using the Law of Cosines:
The Law of Cosines states that for any triangle ΔABC with sides a, b, and c, and angle C opposite side c:
c² = a² + b² - 2ab * cos(C)
Finding side c:
Convert angle C to radians: C_rad = 48.7° * π/180° ≈ 0.8506 radians
Substitute known values: c² = 60.12² + 40.23² - 2 * 60.12 * 40.23 * cos(0.8506)
Solve for c: c ≈ 45.17 miles
Finding angle A:
Use the Law of Sines:
sin(A) / a = sin(C) / c
Substitute values: sin(A) / 60.12 = sin(0.8506) / 45.17
Solve for A: A ≈ 42.0 degrees
Finding angle B:
Use the Law of Sines:
sin(B) / b = sin(C) / c
Substitute values: sin(B) / 40.23 = sin(0.8506) / 45.17
Solve for B: B ≈ 89.3 degrees