Question

Consider the following triangle.

a = 6.0, b = 7.7, c = 13.6
Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle.
O Law of Sines
O Law of Cosines
Solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal pl
A =
B =
C =
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Answer 1

To determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle, we can compare the given information with the formulas for each law.

The Law of Sines states:

a / sin(A) = b / sin(B) = c / sin(C)

The Law of Cosines states:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, we are given the lengths of all three sides of the triangle (a = 6.0, b = 7.7, c = 13.6). Therefore, we have enough information to use the Law of Cosines to solve the triangle.

Using the Law of Cosines, we can find the measures of the angles:

c^2 = a^2 + b^2 - 2ab * cos(C)

(13.6)^2 = (6.0)^2 + (7.7)^2 - 2 * 6.0 * 7.7 * cos(C)

184.96 = 36 + 59.29 - 92.4 * cos(C)

184.96 = 95.29 - 92.4 * cos(C)

92.67 = -92.4 * cos(C)

cos(C) ≈ -1

Since the cosine of an angle cannot be greater than 1 or less than -1, it is not possible for the given triangle to have an angle with a cosine of -1. Therefore, the triangle is not solvable with the given side lengths.

In this case, the Law of Cosines is needed, but the triangle cannot be solved with the given information.