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Chapter 5: Problem 13

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$A=120^{\circ}, \quad b=6, \quad c=7$$

Short Answer

Expert verified
The measures of the triangle are: side 'a' equals approximately 11.92, angle 'B' is about 24.39 degrees and angle 'C' is about 35.61 degrees.

Step by step solution

01

Compute length of side 'a'

Using the law of cosines, the length of side 'a' can be calculated with the formula \(a^2 = b^2 + c^2 - 2bc \cos(A)\). Plugging in the given values, \(a^2 = 6^2 + 7^2 - 2(6)(7)\cos(120^{\circ})\). After calculating, the length of side 'a' is approximately 11.92.
02

Compute the measure of angle 'B'

The Law of Sines is applied to find the second angle 'B'. This law can be written in the formula \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\). We can rearrange this formula to solve for 'B': \[B = \sin^{-1}(\frac{b \sin(A)}{a})\]. Plugging in the values gives \[B =\sin^{-1}(\frac{6 \sin(120^{\circ})}{11.92})\]. After calculating, angle 'B' is approximately 24.39^{\circ}.
03

Compute the measure of angle 'C'

Finally, angle 'C' is calculated by subtracting angles 'A' and 'B' from 180 degrees because the total measures of angles in a triangle sums up to 180 degrees. So, \[C = 180^{\circ} - A - B = 180^{\circ} - 120^{\circ} - 24.39^{\circ}\]. After calculating, angle 'C' is approximately 35.61^{\circ}.

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