Answer:
- (a, c, A) = (13.74, 14.62, 70°)
- (b, a, A) = (3.08, 8.46, 70°)
- (c, A, B) = (8.25, 14.0°, 76.0°)
Step-by-step explanation:
These are all right triangles, so the Pythagorean theorem applies to side lengths. The acute angles are complementary. Here, we have used the sine relation for finding missing angles and sides.
1.
The Law of Sines can be useful.
c/sin(C) = b/sin(B)
c = sin(90°)·5/sin(20°) ≈ 14.62
Of course, angle A is the complement of angle B, so ...
A = 90° -20° = 70°
a/sin(A) = c/sin(C) ⇒ a = c·sin(70°) ≈ 13.74
The solution is (a, c, A) = (13.74, 14.62, 70°).
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2.
Again, the Law of Sines is useful.
b/sin(B) = c/sin(C)
b = sin(B)·c/sin(C) = sin(20°)·9/1 ≈ 3.08
A = 90° -B = 90° -20° = 70°
a = sin(A)·c/sin(C) = 9·sin(70°) ≈ 8.46
The solution is (b, a, A) = (3.08, 8.46, 70°).
Note your problem statement asks for the given angle B (20°). We have given the value of unknown angle A here (70°).
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3.
The Pythagorean theorem can be used to find c.
c² = a² +b²
c = √(a² +b²) = √(2² +8²) = √68 ≈ 8.25
Then the Law of Sines can be used to find the angles:
sin(A)/a = sin(C)/c
sin(A) = a/c·sin(90°) = a/c
A = arcsin(a/c) ≈ arcsin(2/8.24621) ≈ 14.0°
B = 90° -A = 76.0°
The solution is (c, A, B) = (8.25, 14.0°, 76.0°).
