sketch a problem solving using Law of Sines and Law of Cosines with solutions.
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sketch a problem solving using Law of Sines and Law of Cosines with solutions.
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Answer:
Problem:
A triangle has sides of lengths 8, 10, and 12. Find the measures of all angles in the triangle.
Solution:
Step 1: Apply the Law of Cosines to find one angle.
In a triangle with sides a, b, and c, and the opposite angles A, B, and C respectively, the Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
Let's find angle C using the Law of Cosines:
c = 12
a = 8
b = 10
12^2 = 8^2 + 10^2 - 2(8)(10) * cos(C)
144 = 64 + 100 - 160 * cos(C)
144 = 164 - 160 * cos(C)
160 * cos(C) = 164 - 144
160 * cos(C) = 20
cos(C) = 20/160
cos(C) = 1/8
C = arccos(1/8) ≈ 82.82°
Step 2: Use the Law of Sines to find another angle.
The Law of Sines states:
sin(A)/a = sin(B)/b = sin(C)/c
We already know the value of angle C. Let's find angle A:
sin(A)/8 = sin(C)/12
sin(A) = 8 * sin(C)/12
sin(A) = (2/3) * sin(C)
sin(A) = (2/3) * sin(82.82°)
sin(A) ≈ 0.9063
A = arcsin(0.9063) ≈ 65.62°
Step 3: Find the remaining angle.
Since the sum of angles in a triangle is always 180°, we can find angle B:
B = 180° - A - C
B = 180° - 65.62° - 82.82°
B ≈ 31.56°
Therefore, the measures of the angles in the triangle are approximately:
A ≈ 65.62°
B ≈ 31.56°
C ≈ 82.82°
Step-by-step explanation:
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