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Consider a triangle ABC like the one below. Suppose that a=54,b=12, and c=51. (The figure is not drawn to scale.) Solve the triangle. Carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth. If there is more than one solution, use the button labeled "or".

User RobH
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1 Answer

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To solve the triangle ABC with side lengths a = 54, b = 12, and c = 51, we can use the Law of Cosines and the Law of Sines.

1. Find angle A using the Law of Cosines:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
cos(A) = (12^2 + 51^2 - 54^2) / (2 * 12 * 51)
cos(A) = (144 + 2601 - 2916) / 1224
cos(A) = -0.0123

Since the cosine value is negative, angle A is an obtuse angle.

2. Find angle B using the Law of Cosines:
cos(B) = (a^2 + c^2 - b^2) / (2 * a * c)
cos(B) = (54^2 + 51^2 - 12^2) / (2 * 54 * 51)
cos(B) = (2916 + 2601 - 144) / 5544
cos(B) = 0.9817

3. Find angle C using the Law of Cosines:
cos(C) = (a^2 + b^2 - c^2) / (2 * a * b)
cos(C) = (54^2 + 12^2 - 51^2) / (2 * 54 * 12)
cos(C) = (2916 + 144 - 2601) / 1296
cos(C) = 0.0289

4. Find angle C using the Law of Sines:
sin(C) / c = sin(A) / a
sin(C) = (sin(A) / a) * c
sin(C) = (sin(A) / 54) * 51
sin(C) = (sqrt(1 - cos(A)^2) / 54) * 51
sin(C) = (sqrt(1 - (-0.0123)^2) / 54) * 51
sin(C) = 0.8682

Now, we have the values of cos(C) and sin(C). We can use these values to find angle C.

C = arcsin(sin(C))
C = arcsin(0.8682)
C = 60.7 degrees

5. Find angle A using the sum of angles in a triangle:
A = 180 - B - C
A = 180 - 77.5 - 60.7
A = 41.8 degrees

Now, we have found the three angles of the triangle:
A = 41.8 degrees
B = 77.5 degrees
C = 60.7 degrees

Therefore, the triangle ABC has angles A = 41.8 degrees, B = 77.5 degrees, and C = 60.7 degrees.
User Popa
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