Problem

Solve the triangle.
\[
a=7, b=6, C=160^{\circ}
\]
$c \approx \square$ (Round to two decimal places as needed.)
$A \approx \square^{\circ}$ (Round to one decimal place as needed.)
$B \approx \square^{\circ}$ (Round to one decimal place as needed.)

Answer

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Answer

Final Answer: \(c \approx \boxed{12.80}\), \(A \approx \boxed{10.8^\circ}\), \(B \approx \boxed{9.2^\circ}\)

Steps

Step 1 :We are given two sides and an included angle of a triangle. We can use the Law of Cosines to find the third side, and then use the Law of Sines to find the other two angles.

Step 2 :The Law of Cosines is given by: \(c^2 = a^2 + b^2 - 2ab\cos(C)\)

Step 3 :The Law of Sines is given by: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\)

Step 4 :We can use these formulas to find the values of c, A, and B.

Step 5 :Given: a = 7, b = 6, C = 160 degrees

Step 6 :Convert C to radians: C = 2.792526803190927

Step 7 :Use the Law of Cosines to find c: c = 12.803678383418427

Step 8 :Use the Law of Sines to find A: A = 10.777090179609665 degrees

Step 9 :Use the Law of Sines to find B: B = 9.222909820390328 degrees

Step 10 :Final Answer: \(c \approx \boxed{12.80}\), \(A \approx \boxed{10.8^\circ}\), \(B \approx \boxed{9.2^\circ}\)

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