Use the given parts of a right spherical triangle ACB to answer
C = 90°
B = 143° 44'
A = 60° 25'
To find the missing angles of the right spherical triangle ACB, we can use the fact that the sum of the angles in a triangle is always 180°. Since angle C is already given as 90°, we can find angle A by subtracting the sum of angles B and C from 180°:
A = 180° - (B + C)
A = 180° - (143° 44' + 90°)
Converting 143° 44' to decimal degrees:
143° 44' = 143 + (44/60) = 143.7333°
Substituting the values and performing the calculation:
A = 180° - (143.7333° + 90°)
A = 180° - 233.7333°
A = -53.7333°
Therefore, angle A in the right spherical triangle ACB is approximately -53.7333°.
Note: In a right spherical triangle, the sum of the three angles is 180°, but the individual angles can exceed 90°. In this case, angle A is negative, indicating that its position is on the opposite side of the sphere from angles B and C.
To find the missing sides and angles of the right spherical triangle ACB, we can use the following formulas:
1. The Law of Sines: sin(A)/sin(a) = sin(B)/sin(b) = sin(C)/sin(c)
2. The Law of Cosines: cos(a) = cos(b) * cos(c) + sin(b) * sin(c) * cos(A)
3. The Pythagorean Theorem: a^2 + b^2 = c^2
Using the given angles:
Angle A = 60° 25'
Angle B = 143° 44'
Angle C = 90°
Let's solve for the missing sides and angles step-by-step.
Step 1: Convert the given angles into decimal degrees.
Angle B = 143° 44' = 143 + (44/60) = 143.7333°
Angle A = 60° 25' = 60 + (25/60) = 60.4167°
Step 2: Solve for side a using the Law of Cosines.
cos(a) = cos(b) * cos(c) + sin(b) * sin(c) * cos(A)
cos(a) = cos(143.7333°) * cos(90°) + sin(143.7333°) * sin(90°) * cos(60.4167°)
cos(a) = 0 * 1 + 0 * 1 * cos(60.4167°)
cos(a) = 0
Since cos(a) = 0, we know that side a is 90 degrees. So a = 90°.
Step 3: Solve for side b using the Law of Sines.
sin(A)/sin(a) = sin(B)/sin(b)
sin(60.4167°)/sin(90°) = sin(143.7333°)/sin(b)
sin(60.4167°) * sin(b) = sin(143.7333°)
sin(b) = sin(143.7333°) / sin(60.4167°)
b = arcsin(sin(143.7333°) / sin(60.4167°))
Using a calculator to find the arc sine:
b ≈ arcsin(0.8768) ≈ 62.885°
Step 4: Solve for side c using the Pythagorean Theorem.
a^2 + b^2 = c^2
(90°)^2 + (62.885°)^2 = c^2
8100° + 3958.356° ≈ c^2
c^2 ≈ 12058.356°
c ≈ sqrt(12058.356°)
c ≈ 109.855
Therefore, the missing sides and angles of the right spherical triangle ACB are:
Side a ≈ 90°
Side b ≈ 62.885°
Side c ≈ 109.855°